Algorithme Moyenne Déplacement Exponentielle C

Simulation des systèmes de simulation et de modélisation: le chemin le plus court vers les applications Ce site propose des informations sur la modélisation et la simulation de systèmes d'événements discrets. Il comprend des discussions sur la modélisation de la simulation descriptive, les commandes de programmation, les techniques d'estimation de la sensibilité, l'optimisation et la recherche de buts par simulation et l'analyse de simulation. Les avancées en matière de puissance de calcul, la disponibilité de la modélisation et de la simulation basées sur PC et la méthodologie de calcul efficace fournissent des modèles de simulation prescriptifs de pointe tels que l'optimisation pour poursuivre des recherches dans des processus d'analyse, de conception et de contrôle qui étaient auparavant hors de portée des modélisateurs Et les décideurs. Pour rechercher le site. Essayez E dit F ind dans la page Ctrl f. Entrez un mot ou une expression dans la boîte de dialogue, p. Quot optimizationquot ou quot sensitivityquot Si la première apparence de la phrase de mot n'est pas ce que vous recherchez, essayez F ind Next. Statistiques et probabilités pour la simulation Sujets en simulation descriptive Techniques de modélisation pour l'estimation de la sensibilité Techniques d'optimisation basées sur la simulation Métamodelage et les problèmes de recherche d'objectifs Techniques d'analyse de simulation Introduction Résumé Les utilisateurs, . La modélisation et la simulation de la conception de système de trade off est une bonne préparation pour les décisions de conception et d'ingénierie dans les emplois du monde réel. Dans ce site Web, nous étudions la modélisation et la simulation des systèmes informatiques. Nous avons besoin d'une bonne connaissance des techniques de modélisation de simulation et des systèmes simulés eux-mêmes. Le scénario décrit ci-dessus n'est qu'une situation où la simulation par ordinateur peut être utilisée efficacement. En plus de son utilisation comme outil pour mieux comprendre et optimiser la performance et / ou la fiabilité des systèmes, la simulation est également largement utilisée pour vérifier l'exactitude des conceptions. La plupart sinon tous les circuits intégrés numériques fabriqués aujourd'hui sont d'abord largement simulés avant d'être fabriqués pour identifier et corriger les erreurs de conception. La simulation au début du cycle de conception est importante parce que le coût de réparation des erreurs augmente considérablement plus tard dans le cycle de vie du produit que l'erreur est détectée. Une autre application importante de la simulation est le développement d'environnements virtuels. par exemple. pour s'entraîner. Analogue au holodeck dans le populaire programme de télévision de science-fiction Star Trek, les simulations génèrent des environnements dynamiques avec lesquels les utilisateurs peuvent interagir comme s'ils étaient vraiment là. De telles simulations sont largement utilisées aujourd'hui pour former du personnel militaire à des situations de champ de bataille, à une fraction du coût des exercices de course avec des réservoirs réels, des avions, etc. La modélisation dynamique dans les organisations est la capacité collective de comprendre les implications du changement dans le temps. Cette compétence est au cœur du succès du processus décisionnel stratégique. La disponibilité d'une modélisation et d'une simulation visuelles efficaces permet à l'analyste et au décideur de renforcer leur décision dynamique en répétant la stratégie pour éviter les pièges cachés. La simulation de système est l'imitation du fonctionnement d'un système réel, tel que le fonctionnement quotidien d'une banque, la valeur d'un portefeuille d'actions sur une période de temps ou le fonctionnement d'une chaîne de montage dans une usine, ou L'affectation du personnel d'un hôpital ou d'une entreprise de sécurité, dans un ordinateur. Au lieu de construire des modèles mathématiques étendus par des experts, les logiciels de simulation aisément disponibles ont permis de modéliser et d'analyser le fonctionnement d'un système réel par des non-experts, qui sont des gestionnaires mais pas des programmeurs. Une simulation est l'exécution d'un modèle, représenté par un programme informatique qui fournit des informations sur le système étudié. L'approche de simulation d'analyse d'un modèle s'oppose à l'approche analytique, où la méthode d'analyse du système est purement théorique. Comme cette approche est plus fiable, l'approche de simulation donne plus de souplesse et de commodité. Les activités du modèle sont des événements, qui sont activés à certains moments et affectent ainsi l'état global du système. Les points dans le temps où un événement est activé sont randomisés, donc aucune entrée de l'extérieur du système est nécessaire. Les événements existent de façon autonome et ils sont discrets donc entre l'exécution de deux événements rien ne se passe. Le SIMSCRIPT fournit une approche basée sur les processus d'écriture d'un programme de simulation. Avec cette approche, les composantes du programme se composent d'entités, qui combinent plusieurs événements connexes en un seul processus. Dans le domaine de la simulation, le concept de principe d'équivalence computationnelle a des implications bénéfiques pour le décideur. L'expérimentation simulée accélère et remplace efficacement les inquiétudes attendues en découvrant de nouvelles connaissances et des explications sur le comportement futur du système réel. Considérez le scénario suivant. Vous êtes le concepteur d'un nouveau commutateur pour les réseaux de mode de transfert asynchrone (ATM), une nouvelle technologie de commutation apparue sur le marché au cours des dernières années. Afin d'aider à assurer le succès de votre produit dans ce domaine est très concurrentiel, il est important que vous concevez le commutateur pour obtenir les meilleures performances possibles tout en maintenant un coût de fabrication raisonnable. Combien de mémoire doit être intégrée dans le commutateur Si la mémoire est associée à des liaisons de communication entrantes à des messages tampons à mesure qu'ils arrivent ou devrait-elle être associée à des liaisons sortantes pour contenir des messages concurrents pour utiliser le même lien De plus, quelle est la meilleure organisation de Composants matériels dans le commutateur Ce ne sont que quelques-unes des questions que vous devez répondre à venir avec un design. Avec l'intégration de l'intelligence artificielle, des agents et d'autres techniques de modélisation, la simulation est devenue un support de décision efficace et approprié pour les gestionnaires. En combinant l'émergence de la science de la complexité avec la technologie de simulation nouvellement popularisée, le groupe Emergent Solutions Group de PricewaterhouseCoopers construit un logiciel qui permet à la haute direction de jouer en toute sécurité ce que si des scénarios dans les mondes artificiels. Par exemple, dans un environnement de vente au détail de consommation, il peut être utilisé pour découvrir comment les rôles des consommateurs et des employés peuvent être simulés pour atteindre des performances de pointe. Statistiques pour les données corrélées Nous nous intéressons à n réalisations qui sont liées au temps, c'est-à-dire ayant n observations corrélées l'estimation de la moyenne est donnée par la moyenne S X i n, où la somme est au-dessus de i 1 à n. Où la somme est supérieure à j 1 à m, alors la variance estimée est: 1 43 2A S 2 n où S 2 l'estimation de variance habituelle rj, x le jième coefficient d'autocorrélation m le laps de temps maximum pour lequel les autocorrélations sont calculées, J 1, 2, 3. m En règle générale, le décalage maximal pour lequel les autocorrélations sont calculées devrait être d'environ 2 du nombre de n réalisations, bien que chaque rj, x puisse être testé pour déterminer s'il est significativement différent de zéro. Détermination de la taille de l'échantillon: On peut calculer la taille minimale de l'échantillon requise par n 1 43 2A S 2 t 2 (d 2 moyenne 2) Application: Un essai pilote a été effectué sur un modèle, les observations numérotées 150, la moyenne était de 205,74 minutes et la variance S 2 101, 921.54, l'estimation des coefficients de retard a été calculée comme suit: r 1, x 0,3301 r 2, x 0,2993, et r 3, x 0,1987. Calculer la taille minimale de l'échantillon pour s'assurer que l'estimation se situe à 43 d 10 de la moyenne vraie avec un 0,05. N (1.96) 2 (101.921.54) 1 43 2 (1-14) 0.3301 43 (1 - 24) 0.2993 43 (1-34) 0.1987 (0.1) 2 (205.74) 2 Qu'est-ce que le théor'eme de limite centrale? L'idée du théorème de la limite centrale (CLT) est que la moyenne d'un échantillon d'observations tirées d'une population ayant une distribution de forme quelconque est approximativement distribuée comme distribution normale si certaines conditions sont remplies. Dans les statistiques théoriques, il existe plusieurs versions du théorème de la limite centrale, selon la façon dont ces conditions sont spécifiées. Ils concernent les types d'hypothèses sur la répartition de la population mère (population à partir de laquelle l'échantillon est prélevé) et la procédure d'échantillonnage proprement dite. Une des versions les plus simples du théorème dit que si est un échantillon aléatoire de taille n (disons, n plus grand que 30) d'une population infinie, écart-type fini. La moyenne de l'échantillon normalisé converge vers une distribution normale standard ou, de manière équivalente, la moyenne de l'échantillon approche une distribution normale avec une moyenne égale à la moyenne de la population et l'écart-type égal à l'écart-type de la population divisé par la racine carrée de la taille de l'échantillon n. Cependant, dans les applications du théorème de la limite centrale aux problèmes pratiques d'inférence statistique, les statisticiens s'intéressent davantage à ce que la distribution approximative de la moyenne de l'échantillon suit une distribution normale pour les échantillons finis que la distribution limite elle-même. Un accord suffisamment proche avec une distribution normale permet aux statisticiens d'utiliser la théorie normale pour faire des inférences sur les paramètres de la population (comme la moyenne) en utilisant la moyenne de l'échantillon, quelle que soit la forme réelle de la population mère. Il est bien connu que quelle que soit la population mère, la variable normalisée aura une distribution moyenne 0 et un écart-type 1 sous échantillonnage aléatoire. De plus, si la population mère est normale, elle est distribuée exactement comme une variable normale normale pour tout entier positif n. Le théorème de la limite centrale indique le résultat remarquable que, même lorsque la population mère n'est pas normale, la variable normalisée est approximativement normale si la taille de l'échantillon est assez grande (par exemple gt 30). Il n'est généralement pas possible d'énoncer des conditions dans lesquelles l'approximation donnée par le théorème de limite centrale fonctionne et quelles tailles d'échantillon sont nécessaires avant que l'approximation devienne suffisamment bonne. En règle générale, les statisticiens ont utilisé la prescription selon laquelle si la distribution parente est symétrique et relativement courte, la moyenne de l'échantillon atteint une normalité approximative pour les échantillons plus petits que si la population parente est biaisée ou à queue longue. Dans cette leçon, nous étudierons le comportement de la moyenne d'échantillons de tailles différentes provenant d'une variété de populations parentales. L'examen des distributions d'échantillonnage des moyens d'échantillonnage calculés à partir d'échantillons de tailles différentes tirées d'une variété de distributions nous permet d'avoir un aperçu du comportement de la moyenne de l'échantillon dans ces conditions spécifiques et d'examiner la validité des lignes directrices mentionnées ci - Théorème de limite centrale en pratique. Sous certaines conditions, dans les grands échantillons, la distribution d'échantillonnage de la moyenne de l'échantillon peut être approximée par une distribution normale. La taille de l'échantillon nécessaire pour que l'approximation soit adéquate dépend fortement de la forme de la distribution parente. La symétrie (ou son absence) est particulièrement importante. Pour une distribution parentale symétrique, même très différente de la forme d'une distribution normale, une approximation adéquate peut être obtenue avec de petits échantillons (par exemple 10 ou 12 pour la distribution uniforme). Pour les distributions symétriques de parents à queue courte, la moyenne de l'échantillon atteint la normalité approximative pour les échantillons plus petits que si la population parente est oblique et à queue longue. Dans certains cas extrêmes (par exemple, binomial) des échantillons de dimensions dépassant de loin les recommandations typiques (par exemple 30) sont nécessaires pour une approximation adéquate. Pour certaines distributions sans premier et deuxième moments (par exemple, Cauchy), le théorème de limite centrale ne tient pas. Qu'est-ce qu'un modèle de moindres carrés? Beaucoup de problèmes dans l'analyse des données impliquent de décrire comment les variables sont liées. Le modèle le plus simple de tous les modèles décrivant la relation entre deux variables est un modèle linéaire ou linéaire. La méthode la plus simple d'ajustement d'un modèle linéaire consiste à baliser une ligne à travers les données d'un tracé. Une méthode plus élégante et conventionnelle est celle des moindres carrés, qui trouve la ligne minimisant la somme des distances entre les points observés et la ligne ajustée. Réaliser que le montage de la meilleure ligne par oeil est difficile, surtout quand il ya beaucoup de variabilité résiduelle dans les données. Sachez qu'il existe un lien simple entre les coefficients numériques de l'équation de régression et la pente et l'intersection de la droite de régression. Sachez qu'une seule statistique récapitulative comme un coefficient de corrélation ne raconte pas toute l'histoire. Un diagramme de dispersion est un complément essentiel à l'examen de la relation entre les deux variables. ANOVA: Analyse de Variance Les tests que nous avons appris jusqu'à ce point nous permettent de tester des hypothèses qui examinent la différence entre deux moyens seulement. L'analyse de Variance ou ANOVA nous permettra de tester la différence entre 2 ou plus de moyens. ANOVA fait cela en examinant le rapport de la variabilité entre deux conditions et la variabilité dans chaque condition. Par exemple, disons que nous offrons un médicament qui, selon nous, améliorera la mémoire d'un groupe de personnes et donnera un placebo à un autre groupe de personnes. Nous pouvons mesurer la performance de mémoire par le nombre de mots rappelés à partir d'une liste que nous demandons à tout le monde de mémoriser. Un test t permettrait de comparer la probabilité d'observer la différence dans le nombre moyen de mots rappelés pour chaque groupe. Un test ANOVA, en revanche, compare la variabilité que nous observons entre les deux conditions à la variabilité observée dans chaque condition. Rappelons que nous mesurons la variabilité comme la somme de la différence de chaque score par rapport à la moyenne. Lorsque nous calculons une ANOVA, nous utiliserons une formule de raccourci. Ainsi, lorsque la variabilité que nous prédisons (entre les deux groupes) est beaucoup plus grande que la variabilité que nous ne prédisons pas (au sein de chaque groupe), nous conclurons que nos traitements produisent Résultats différents. Fonction de densité exponentielle Une classe importante de problèmes de décision sous incertitude concerne le hasard entre les événements. Par exemple, la chance de la durée de la prochaine rupture d'une machine ne dépassant pas un certain temps, comme la machine à copier dans votre bureau de ne pas casser pendant cette semaine. La distribution exponentielle donne la répartition du temps entre des événements indépendants se produisant à un taux constant. Sa fonction de densité est: où l est le nombre moyen d'événements par unité de temps, ce qui est un nombre positif. La moyenne et la variance de la variable aléatoire t (temps entre les événements) sont de 1 l. Et 1 l 2. respectivement. Les applications comprennent l'évaluation probabiliste du temps entre l'arrivée des patients à la salle d'urgence d'un hôpital et l'arrivée des navires dans un port particulier. Commentaires: Cas particulier des distributions Weibull et gamma. Vous pouvez utiliser Exponential Applet pour effectuer vos calculs. Vous pouvez utiliser le test Lilliefors suivant pour Exponentially pour effectuer le test de qualité de l'ajustement. Processus de Poisson Une classe importante de problèmes de décision sous incertitude se caractérise par la faible probabilité d'occurrence d'un événement particulier tel qu'un accident. Donne la probabilité d'exactement x occurrences indépendantes pendant une période de temps donnée si les événements ont lieu indépendamment et à un taux constant. Peut également représenter le nombre d'occurrences sur des surfaces ou des volumes constants. Les énoncés suivants décrivent le processus de Poisson. Les occurrences des événements sont indépendantes. L'occurrence d'événements à partir d'un ensemble d'hypothèses dans un intervalle d'espace ou de temps n'a aucun effet sur la probabilité d'une seconde occurrence de l'événement dans le même intervalle ou dans tout autre intervalle. Théoriquement, un nombre infini d'occurrences de l'événement doit être possible dans l'intervalle. La probabilité d'occurrence unique de l'événement dans un intervalle donné est proportionnelle à la longueur de l'intervalle. Dans toute infime petite partie de l'intervalle, la probabilité de plus d'une occurrence de l'événement est négligeable. Poisson processus sont souvent utilisés, par exemple dans le contrôle de la qualité, la fiabilité, la demande d'assurance, le nombre entrant d'appels téléphoniques, et la théorie des files d'attente. Une application: Une des applications les plus utiles du processus de Poisson est dans le domaine de la théorie des files d'attente. Dans de nombreuses situations où des files d'attente se produisent, il a été montré que le nombre de personnes se joignant à la file d'attente dans une période de temps donnée suit le modèle de Poisson. Par exemple, si le taux d'arrivées à une salle d'urgence est l par unité de temps (par exemple 1 heure), alors: P (n arrivées) l n e - l n La moyenne et la variance de la variable aléatoire n sont les deux l. Cependant, si la moyenne et la variance d'une variable aléatoire ayant des valeurs numériques égales, il n'est pas nécessaire que sa distribution soit un Poisson. P (0 arrivée) e - l P (1 arrivée) l e - l 1 P (2 arrivée) l 2 e - l 2 et ainsi de suite. En général: P (n1 arrivées) l Pr (n arrivées) n. Vous pouvez utiliser Poisson Applet pour effectuer vos calculs. Goodness-of-Fit for Poisson Remplacez les données numériques d'exemple par vos 14 paires de valeurs observées par leurs fréquences. Puis cliquez sur le bouton Calculer. Les boîtes vierges ne sont pas incluses dans les calculs. Lorsque vous entrez vos données pour passer d'une cellule à une cellule dans la matrice de données, utilisez la touche Tabulation et non la flèche ou entrez les touches. Fonction de densité uniforme Application: Donne la probabilité que l'observation se produise dans un intervalle particulier lorsque la probabilité d'occurrence dans cet intervalle est directement proportionnelle à la longueur d'intervalle. Exemple: Utilisé pour générer des nombres aléatoires dans l'échantillonnage et la simulation Monte Carlo. Commentaires: Cas particulier de la distribution bêta. La fonction de masse de la moyenne géométrique de n uniformes indépendants 0,1 est: P (X x) n x (n - 1) (Log1x n) (n -1) (n - 1). Z L U L - (1-U) On dit que L L a une répartition symétrique de Tukeys. Vous pouvez utiliser l'applet uniforme pour effectuer vos calculs. Quelques commandes SPSS utiles Pour plus de programmes SPSS utiles à l'analyse de la sortie d'entrée de simulation, visitez les rubriques Analyse des données. Générateur de nombres aléatoires Les générateurs de nombres aléatoires uniformes classiques présentent certains défauts majeurs, tels que la longueur de la courte période et le manque d'uniformité de la plus grande dimension. Cependant, il existe aujourd'hui une classe de générateurs relativement complexes qui est aussi efficace que les générateurs classiques tout en jouissant de la propriété d'une période beaucoup plus longue et d'une uniformité de plus grande dimension. Les programmes informatiques qui génèrent des nombres aléatoires utilisent un algorithme. Cela signifie que si vous connaissez l'algorithme et les valeurs des graines, vous pouvez prédire les nombres qui en résulteront. Parce que vous pouvez prédire les nombres qu'ils ne sont pas vraiment aléatoire - ils sont pseudo-aléatoires. A des fins statistiques, de bons générateurs de nombres pseudo-aléatoires sont assez bons. Le générateur de nombres aléatoires RANECU Un code FORTRAN pour un générateur de nombres aléatoires uniformes sur 0,1. RANECU est un générateur congruentiel linéaire multiplicatif adapté à une plate-forme 16 bits. Il combine trois générateurs simples et a une période dépassant 81012. Il est construit pour une utilisation plus efficace en prévoyant une séquence de tels numéros, LEN au total, à être retourné en un seul appel. Un ensemble de trois graines entières non nulles peut être fourni, à défaut de quoi un jeu par défaut est utilisé. Si elles sont fournies, ces trois graines, dans l'ordre, devraient se trouver dans les plages 1,32362, 1,31726 et 1,31656 respectivement. La routine de brassage en Visual Basic La méthode d'histogramme carré On nous donne un histogramme, avec des barres verticales ayant des hauteurs proportionnelles à la probabilité avec laquelle nous voulons produire une valeur indiquée par l'étiquette à la base. Un tel histogramme simple, posé plat, pourrait être: L'idée est de couper les barres en morceaux puis les rassembler dans un histogramme carré, toutes les hauteurs égales, avec chaque barre finale ayant une partie inférieure, ainsi qu'une partie supérieure indiquant où il Est venu de. Une seule variable aléatoire uniforme U peut alors être utilisée pour choisir l'une des barres finales et pour indiquer s'il faut utiliser la partie inférieure ou supérieure. Il existe de nombreuses façons de faire cette coupe et de réassembler le plus simple semble être l'algorithme de Robin des Bois: Prendre des plus riches pour amener les plus pauvres à la moyenne. ÉTAPE 1: L'histogramme d'origine (horizontal), hauteur moyenne 20: Prendre 17 de la bande a pour amener la bande e à la moyenne. Enregistrer le donneur et utiliser le vieux niveau pauvre pour marquer la partie inférieure du donataire: Ensuite, mettre d à la moyenne avec le donneur b. Enregistrer le donneur et utiliser le vieux niveau pauvre pour marquer la partie inférieure du donataire: Ensuite, amener un à la moyenne avec le donateur c. Enregistrer le donneur et utiliser le vieux niveau pauvre pour marquer la partie inférieure du donataire: Enfin, amener b jusqu'à la moyenne avec le donneur c. Noter le donneur et utiliser le vieux niveau pauvre pour marquer la partie inférieure du donataire: Nous avons maintenant un histogramme au carré, c'est-à-dire un rectangle avec 4 bandes de même surface, chaque bande avec deux régions. Une seule variable uniforme U peut être utilisée pour générer a, b, c, d, e avec les probabilités requises. 32. 27. 26. 12 .06. Configuration: Créer des tables, Soit j la partie entière de 15U, avec U uniforme dans (0,1). Si U lt Tj retourne Vj, sinon retourne VKj. Dans de nombreuses applications aucune table V n'est nécessaire: Vii et la procédure de génération devient Si U lt Tj retour j, sinon retour Kj. Références Lectures complémentaires: Aiello W. S. Rajagopalan, et R. Venkatesan, Conception de générateurs de nombres aléatoires pratiques et probablement bons, Journal of Algorithms. 29, 358-389, 1998. Dagpunar J. Principles of Random Variate Generation. Clarendon, 1988. Fishman G. Monte Carlo. Springer, 1996. James, Fortran version du générateur LEcuyer, Comput. Phys. Comm. . 60, 329-344, 1990. Knuth D. L'art de la programmation informatique, vol. 2. Addison-Wesley, 1998. LEcuyer P. Générateurs de nombres aléatoires combinés efficaces et portatifs, Comm. ACM, 31, 742-749, 774, 1988. LEcuyer P. Génération uniforme de nombres aléatoires, Ann. Op. Res. 53, 77-120, 1994. LEcuyer P. Génération de nombres aléatoires. Dans Handbook on Simulation. J. Banks (éd.), Wiley, 1998. Maurer U. Un test statistique universel pour les générateurs de bits aléatoires, J. Cryptology. 5, 89-105, 1992. Sobol I. et Y. Levitan, générateur de nombres pseudo-aléatoires pour ordinateurs personnels, ordinateurs mathématiques avec applications. 37 (4), 33-40, 1999. Tsang W-W. Algorithme d'arbre de décision pour l'équation de l'histogramme dans la génération de nombres aléatoires, Ars Combinatoria. 23A, 291-301, 1987 Test de l'aléatoire Nous avons besoin de tester à la fois le caractère aléatoire ainsi que l'uniformité. Les tests peuvent être classés en 2 catégories: tests empiriques ou statistiques et tests théoriques. Les tests théoriques traitent des propriétés du générateur utilisé pour créer la réalisation avec la distribution souhaitée, et ne regardez pas le nombre généré du tout. Par exemple, nous n'utiliserions pas un générateur de mauvaise qualité pour générer des nombres aléatoires. Les tests statistiques sont basés uniquement sur les observations aléatoires produites. Test de l'aléatoire: A. Test de l'indépendance: Tracer la réalisation x i vs x i1. S'il y a indépendance, le graphe ne montrera aucun motif distinctif du tout, mais sera parfaitement dispersé. B. Essais de course (run-ups, run-downs): Ceci est un test direct de l'hypothèse d'indépendance. Il ya deux statistiques de test à considérer: une basée sur une approximation normale et une autre utilisant des approximations numériques. Test basé sur l'approximation normale: Supposons que vous ayez N réalisations aléatoires. Soit a le nombre total d'exécutions dans une séquence. Si le nombre de trajets positifs et négatifs est supérieur à 20, la répartition de a est raisonnablement approximée par une distribution normale avec la moyenne (2N - 1) 3 et (16N - 29) 90. Rejette l'hypothèse d'indépendance ou d'existence de trajectoires Si Zo gt Z (1-alpha2) où Zo est le score Z. C. Tests de corrélation: Les nombres aléatoires montrent-ils une corrélation discernable? Calculez l'exemple de la fonction d'autocorrélation. Fréquence ou test de distribution uniforme: Utilisez le test de Kolmogorov-Smirimov pour déterminer si les réalisations suivent un U (0,1) Références Lectures complémentaires: Headrick T. Transformations rapides de polynômes de cinquième ordre pour générer des distributions non uniformes univariées et multivariées, Statistiques computationnelles et analyse de données . 40 (4), 685-711, 2002. Karian Z. et E. Dudewicz, Modern Statistical Systems et GPSS Simulation. CRC Press, 1998. Kleijnen J. et W. van Groenendaal, Simulation: A Statistical Perspective. Wiley, Chichester, 1992 Korn G. Les expériences statistiques réelles peuvent employer le logiciel de paquet de simulation, la pratique de modélisation de simulation et la théorie. 13 (1), 39-54, 2005. Lewis P. et E. Orav, Méthodologie de simulation pour statisticiens, analystes d'exploitation et ingénieurs. Wadsworth Inc. 1989 Madu Ch. Et Ch-H. Kuei, Conceptions statistiques expérimentales et analyse en modélisation de simulation. Greenwood Publishing Group, 1993. Pang K. Z. Yang, S. Hou et P. Leung, Production non uniforme de variables aléatoires par la méthode des bandes verticales, European Journal of Operational Research. 142 (3), 595-609, 2002. Robert C. et G. Casella, Méthodes statistiques de Monte Carlo. Springer, 1999. Modélisation Simulation La simulation en général est de prétendre que l'on traite avec une chose réelle tout en travaillant vraiment avec une imitation. Dans la recherche opérationnelle, l'imitation est un modèle informatique de la réalité simulée. Un simulateur de vol sur un PC est également un modèle informatique de certains aspects du vol: il montre sur l'écran les commandes et ce que le pilote (le jeune qui l'exploite) est censé voir du cockpit (son fauteuil). Pourquoi utiliser des modèles Pour voler un simulateur est plus sûr et moins cher que l'avion réel. Pour cette raison précisément, les modèles sont utilisés dans le commerce industriel et militaire: il est très coûteux, dangereux et souvent impossible de faire des expériences avec des systèmes réels. À condition que les modèles soient des descriptions adéquates de la réalité (ils sont valides), expérimenter avec eux peut économiser de l'argent, la souffrance et même le temps. Quand utiliser des simulations Des systèmes qui changent avec le temps, comme une station-service où les voitures vont et viennent (appelées systèmes dynamiques) et impliquent le hasard. Personne ne peut deviner à quel moment la prochaine voiture devrait arriver à la station, sont de bons candidats pour la simulation. La modélisation de systèmes dynamiques complexes nécessite théoriquement un trop grand nombre de simplifications et les modèles émergents peuvent ne pas être donc valables. La simulation ne nécessite pas beaucoup d'hypothèses simplificatrices, ce qui en fait le seul outil même en l'absence de hasard. Comment simuler Supposons que nous sommes intéressés par une station d'essence. Nous pouvons décrire le comportement de ce système graphiquement en traçant le nombre de voitures dans la station de l'état du système. Chaque fois qu'une voiture arrive le graphe augmente d'une unité tandis qu'une voiture de départ fait tomber le graphique d'une unité. Ce graphique (appelé chemin d'échantillon), pourrait être obtenu à partir de l'observation d'une station réelle, mais pourrait également être artificiellement construit. Une telle construction artificielle et l'analyse du chemin d'échantillon résultant (ou plus de chemins d'échantillons dans des cas plus complexes) consiste en la simulation. Types de simulations: Evénement discret. Le chemin d'échantillonnage ci-dessus se composait uniquement de lignes horizontales et verticales, car les arrivées et les départs de voitures se produisaient à des moments distincts, ce que nous appelons événements. Entre deux événements consécutifs, rien ne se produit - le graphique est horizontal. Lorsque le nombre d'événements est fini, nous appelons l'événement discret de simulation. Dans certains systèmes, l'état change tout le temps, pas seulement au moment de certains événements discrets. Par exemple, le niveau d'eau dans un réservoir avec des entrées et des sorties peut changer tout le temps. Dans de tels cas, la simulation continue est plus appropriée, bien que la simulation d'événements discrets puisse servir d'approximation. Examen plus poussé des simulations d'événements discrets. Comment est effectuée la simulation Les simulations peuvent être effectuées manuellement. Le plus souvent, cependant, le modèle de système est écrit soit comme un programme d'ordinateur (par exemple cliquez ici) ou comme une sorte d'entrée dans le logiciel de simulateur. Etat: Variable caractérisant un attribut dans le système tel que le niveau de stock en inventaire ou le nombre d'emplois en attente de traitement. Événement: Un événement à un moment donné qui peut changer l'état du système, comme l'arrivée d'un client ou le début d'un travail. Entité: Un objet qui traverse le système, comme des voitures dans une intersection ou des ordres dans une usine. Souvent, un événement (par exemple l'arrivée) est associé à une entité (par exemple, un client). File d'attente: une file d'attente n'est pas seulement une file d'attente physique de personnes, elle peut aussi être une liste de tâches, un tampon de produits finis en attente de transport ou n'importe quel endroit où les entités attendent quelque chose se produise pour une raison quelconque. Création: La création provoque l'arrivée d'une nouvelle entité dans le système à un moment donné. Planification: L'ordonnancement est l'acte d'assigner un nouvel événement futur à une entité existante. Variable aléatoire: Une variable aléatoire est une quantité qui est incertaine, comme le temps d'inter-croisement entre deux vols entrants ou le nombre de pièces défectueuses dans un envoi. Variable aléatoire: Une variable aléatoire est une variable aléatoire générée artificiellement. Distribution: Une distribution est la loi mathématique qui régit les caractéristiques probabilistes d'une variable aléatoire. Exemple simple: Construire une station-service de simulation avec une seule pompe desservie par un seul technicien. Supposons que l'arrivée des voitures ainsi que leurs temps de service sont aléatoires. D'abord identifiez les: états: nombre de voitures en attente de service et nombre de voitures servies à tout moment événements: arrivée des voitures, début du service, fin des entités de service: ce sont les voitures queue: la file d'attente des voitures devant le Pompe, en attente de service réalisations aléatoires: temps d'inter-croisement, distributions de temps de service: nous supposerons des distributions exponentielles pour le temps d'inter-croisement et le temps de service. Ensuite, spécifiez ce qu'il faut faire à chaque événement. L'exemple ci-dessus ressemblerait à ceci: En cas d'arrivée de l'entité: Créer l'arrivée suivante. Si le serveur est libre, envoyez l'entité pour le démarrage du service. Sinon, il rejoint la file d'attente. En cas de démarrage du service: Le serveur devient occupé. Planifiez la fin du service pour cette entité. En cas de fin de service: Serveur devient libre. Si les entités en attente dans la file d'attente: supprimer la première entité de la file d'attente l'envoyer pour le début du service. Une initiation est encore nécessaire, par exemple, la création de la première arrivée. Enfin, ce qui précède est traduit en code. Cela est facile avec une bibliothèque appropriée qui a des sous-programmes pour la création, l'ordonnancement, le calendrier approprié des événements, les manipulations de file d'attente, la génération de variables aléatoires et la collecte de statistiques. Comment simuler En plus de ce qui précède, le programme enregistre le nombre de voitures dans le système avant et après chaque changement, ainsi que la longueur de chaque événement. Développement de systèmes de simulation Les systèmes d'événements discrets (DES) sont des systèmes dynamiques qui évoluent dans le temps par l'apparition d'événements à des intervalles de temps éventuellement irréguliers. DES abondent dans les applications réelles. Les exemples incluent des systèmes de circulation, des systèmes de fabrication flexibles, des systèmes de communications informatiques, des lignes de production, des systèmes de durée de vie cohérente et des réseaux d'écoulement. La plupart de ces systèmes peuvent être modélisés en termes d'événements discrets dont l'occurrence fait passer le système d'un état à un autre. Lors de la conception, de l'analyse et de l'exploitation de tels systèmes complexes, on s'intéresse non seulement à l'évaluation des performances, mais aussi à l'analyse de sensibilité et à l'optimisation. Un système stochastique typique a un grand nombre de paramètres de contrôle qui peuvent avoir un impact significatif sur la performance du système. To establish a basic knowledge of the behavior of a system under variation of input parameter values and to estimate the relative importance of the input parameters, sensitivity analysis applies small changes to the nominal values of input parameters. For systems simulation, variations of the input parameter values cannot be made infinitely small. The sensitivity of the performance measure with respect to an input parameter is therefore defined as (partial) derivative. Sensitivity analysis is concerned with evaluating sensitivities (gradients, Hessian, etc.) of performance measures with respect to parameters of interest. It provides guidance for design and operational decisions and plays a pivotal role in identifying the most significant system parameters, as well as bottleneck subsystems. I have carried out research in the fields of sensitivity analysis and stochastic optimization of discrete event systems with an emphasis on computer simulation models. This part of lecture is dedicated to the estimation of an entire response surface of complex discrete event systems (DES) from a single sample path (simulation), such as the expected waiting time of a customer in a queuing network, with respect to the controllable parameters of the system, such as service rates, buffer sizes and routing probabilities. With the response surfaces at hand, we are able to perform sensitivity analysis and optimization of a DES from a single simulation, that is, to find the optimal parameters of the system and their sensitivities (derivatives), with respect to uncontrollable system parameters, such as arrival rates in a queuing network. We identified three distinct processes. Descriptive Analysis includes: Problem Identification Formulation, Data Collection and Analysis, Computer Simulation Model Development, Validation, Verification and Calibration, and finally Performance Evaluation. Prescriptive Analysis: Optimization or Goal Seeking. These are necessary components for Post-prescriptive Analysis: Sensitivity, and What-If Analysis. The prescriptive simulation attempts to use simulation to prescribe decisions required to obtain specified results. It is subdivided into two topics - Goal Seeking and Optimization. Recent developments on single-run algorithms for the needed sensitivities (i. e. gradient, Hessian, etc.) make the prescriptive simulation feasible. Click on the image to enlarge it and THEN print it. Problem Formulation: Identify controllable and uncontrollable inputs. Identify constraints on the decision variables. Define measure of system performance and an objective function. Develop a preliminary model structure to interrelate the inputs and the measure of performance. Click on the image to enlarge it and THEN print it. Data Collection and Analysis: Regardless of the method used to collect the data, the decision of how much to collect is a trade-off between cost and accuracy. Simulation Model Development: Acquiring sufficient understanding of the system to develop an appropriate conceptual, logical and then simulation model is one of the most difficult tasks in simulation analysis. Model Validation, Verification and Calibration: In general, verification focuses on the internal consistency of a model, while validation is concerned with the correspondence between the model and the reality. The term validation is applied to those processes which seek to determine whether or not a simulation is correct with respect to the real system. More prosaically, validation is concerned with the question Are we building the right system. Verification, on the other hand, seeks to answer the question Are we building the system right Verification checks that the implementation of the simulation model (program) corresponds to the model. Validation checks that the model corresponds to reality. Calibration checks that the data generated by the simulation matches real (observed) data. Validation: The process of comparing the models output with the behavior of the phenomenon. In other words: comparing model execution to reality (physical or otherwise) Verification: The process of comparing the computer code with the model to ensure that the code is a correct implementation of the model. Calibration: The process of parameter estimation for a model. Calibration is a tweakingtuning of existing parameters and usually does not involve the introduction of new ones, changing the model structure. In the context of optimization, calibration is an optimization procedure involved in system identification or during experimental design. Input and Output Analysis: Discrete-event simulation models typically have stochastic components that mimic the probabilistic nature of the system under consideration. Successful input modeling requires a close match between the input model and the true underlying probabilistic mechanism associated with the system. The input data analysis is to model an element (e. g. arrival process, service times) in a discrete-event simulation given a data set collected on the element of interest. This stage performs intensive error checking on the input data, including external, policy, random and deterministic variables. System simulation experiment is to learn about its behavior. Careful planning, or designing, of simulation experiments is generally a great help, saving time and effort by providing efficient ways to estimate the effects of changes in the models inputs on its outputs. Statistical experimental-design methods are mostly used in the context of simulation experiments. Performance Evaluation and What-If Analysis: The what-if analysis is at the very heart of simulation models. Sensitivity Estimation: Users must be provided with affordable techniques for sensitivity analysis if they are to understand which relationships are meaningful in complicated models. Optimization: Traditional optimization techniques require gradient estimation. As with sensitivity analysis, the current approach for optimization requires intensive simulation to construct an approximate surface response function. Incorporating gradient estimation techniques into convergent algorithms such as Robbins-Monroe type algorithms for optimization purposes, will be considered. Gradient Estimation Applications: There are a number of applications which measure sensitivity information, (i. e. the gradient, Hessian, etc.), Local information, Structural properties, Response surface generation, Goal-seeking problem, Optimization, What-if Problem, and Meta-modelling Report Generating: Report generation is a critical link in the communication process between the model and the end user. A Classification of Stochastic Processes A stochastic process is a probabilistic model of a system that evolves randomly in time and space. Formally, a stochastic process is a collection of random variables all defined on a common sample (probability) space. The X(t) is the state while (time) t is the index that is a member of set T. Examples are the delay of the ith customer and number of customers in the queue at time t in an MM1 queue. In the first example, we have a discrete - time, continuous state, while in the second example the state is discrete and time in continuous. The following table is a classification of various stochastic processes. The man made systems have mostly discrete state. Monte Carlo simulation deals with discrete time while in discrete even system simulation the time dimension is continuous, which is at the heart of this site. Change in the States of the System A Classification of Stochastic Processes Simulation Output Data and Stochastic Processes To perform statistical analysis of the simulation output we need to establish some conditions, e. g. output data must be a covariance stationary process (e. g. the data collected over n simulation runs). Stationary Process (strictly stationary): A stationary stochastic process is a stochastic process with the property that the joint distribution all vectors of h dimension remain the same for any fixed h. First Order Stationary: A stochastic process is a first order stationary if expected of X(t) remains the same for all t. For example in economic time series, a process is first order stationary when we remove any kinds of trend by some mechanisms such as differencing. Second Order Stationary: A stochastic process is a second order stationary if it is first order stationary and covariance between X(t) and X(s) is function of t-s only. Again, in economic time series, a process is second order stationary when we stabilize also its variance by some kind of transformations such as taking square root. Clearly, a stationary process is a second order stationary, however the reverse may not hold. In simulation output statistical analysis we are satisfied if the output is covariance stationary . Covariance Stationary: A covariance stationary process is a stochastic process having finite second moments, i. e. expected of X(t) 2 be finite. Clearly, any stationary process with finite second moment is covariance stationary. A stationary process may have no finite moment whatsoever. Since a Gaussian process needs a mean and covariance matrix only, it is stationary (strictly) if it is covariance stationary. Two Contrasting Stationary Process: Consider the following two extreme stochastic processes: - A sequence Y 0 . Y 1 . of independent identically distributed, random-value sequence is a stationary process, if its common distribution has a finite variance then the process is covariance stationary. - Let Z be a single random variable with known distribution function, and set Z 0 Z 1 . Z. Note that in a realization of this process, the first element, Z 0, may be random but after that there is no randomness. The process i . i 0, 1, 2. is stationary if Z has a finite variance. Output data in simulation fall between these two type of process. Simulation outputs are identical, and mildly correlated (how mild It depends on e. g. in a queueing system how large is the traffic intensity r ). An example could be the delay process of the customers in a queueing system. Techniques for the Steady State Simulation Unlike in queuing theory where steady state results for some models are easily obtainable, the steady state simulation is not an easy task. The opposite is true for obtaining results for the transient period (i. e. the warm-up period). Gather steady state simulation output requires statistical assurance that the simulation model reached the steady state. The main difficulty is to obtain independent simulation runs with exclusion of the transient period. The two technique commonly used for steady state simulation are the Method of Batch means, and the Independent Replication. None of these two methods is superior to the other in all cases. Their performance depend on the magnitude of the traffic intensity. The other available technique is the Regenerative Method, which is mostly used for its theoretical nice properties, however it is rarely applied in actual simulation for obtaining the steady state output numerical results. Suppose you have a regenerative simulation consisting of m cycles of size n 1 . n 2,n m . respectivement. The cycle sums is: The overall estimate is: Estimate S y i S n i . the sums are over i1, 2. m The 100(1- a 2) confidence interval using the Z-table (or T-table, for m less than, say 30), is: Estimate 177 Z. S (n. m ) n S n i m, the sum is over i1, 2. m and the variance is: S 2 S (y i - n i . Estimate) 2 (m-1), the sum is over i1, 2. m Method of Batch Means: This method involves only one very long simulation run which is suitably subdivided into an initial transient period and n batches. Each of the batch is then treated as an independent run of the simulation experiment while no observation are made during the transient period which is treated as warm-up interval. Choosing a large batch interval size would effectively lead to independent batches and hence, independent runs of the simulation, however since number of batches are few on cannot invoke the central limit theorem to construct the needed confidence interval. On the other hand, choosing a small batch interval size would effectively lead to significant correlation between successive batches therefore cannot apply the results in constructing an accurate confidence interval. Suppose you have n equal batches of m observations each. The means of each batch is: mean i S x ij m, the sum is over j1, 2. m The overall estimate is: Estimate S mean i n, the sum is over i1, 2. n The 100(1- a 2) confidence interval using the Z-table (or T-table, for n less than, say 30), is: Estimate 177 Z. S where the variance is: S 2 S (mean i - Estimate) 2 (n-1), the sum is over i1, 2. n Method of Independent Replications: This method is the most popularly used for systems with short transient period. This method requires independent runs of the simulation experiment different initial random seeds for the simulators random number generator. For each independent replications of the simulation run it transient period is removed. For the observed intervals after the transient period data is collected and processed for the point estimates of the performance measure and for its subsequent confidence interval. Suppose you have n replications with of m observations each. The means of each replication is: mean i S x ij m, the sum is over j1, 2. m The overall estimate is: Estimate S mean i n, the sum is over i1, 2. n The 100(1- a 2) confidence interval using the Z-table (or T-table, for n less than, say 30), is: Estimate 177 Z. S where the variance is: S 2 S (mean i - Estimate) 2 (n-1), the sum is over i1, 2. n Further Reading: Sherman M. and D. Goldsman, Large-sample normality of the batch-means variance estimator, Operations Research Letters . 30, 319-326, 2002. Whitt W. The efficiency of one long run versus independent replications in steady-state simulation, Management Science . 37(6), 645-666, 1991. Determination of the Warm-up Period To estimate the long-term performance measure of the system, there are several methods such as Batch Means, Independent Replications and Regenerative Method. Batch Means is a method of estimating the steady-state characteristic from a single-run simulation. The single run is partitioned into equal size batches large enough for estimates obtained from different batches to be approximately independent. In the method of Batch Means, it is important to ensure that the bias due to initial conditions is removed to achieve at least a covariance stationary waiting time process. An obvious remedy is to run the simulation for a period large enough to remove the effect of the initial bias. During this warm-up period, no attempt is made to record the output of the simulation. The results are thrown away. At the end of this warm-up period, the waiting time of customers are collected for analysis. The practical question is How long should the warm-up period be. Abate and Whitt provided a relatively simple and nice expression for the time required (t p ) for an MM1 queue system (with traffic intensity r ) starting at the origin (empty) to reach and remain within 100p of the steady - state limit as follows: C( r )2 r ( r 2 4 r ) 4. Some notions of t p ( r ) as a function of r and p, are given in following table: Time ( t p ) required for an MM1 queue to reach and remain with 100p limits of the steady-state value. Although this result is developed for MM1 queues, it has already been established that it can serve as an approximation for more general i. e. GIG1 queues. Further Reading: Abate J. and W. Whitt, Transient behavior of regular Brownian motion, Advance Applied Probability . 19, 560-631, 1987. Chen E. and W. Kelton, Determining simulation run length with the runs test, Simulation Modelling Practice and Theory . 11, 237-250, 2003. Determination of the Desirable Number of Simulation Runs The two widely used methods for experimentation on simulation models are method of bath means, and independent replications. Intuitively one may say the method of independent replication is superior in producing statistically a good estimate for the systems performance measure. In fact, not one method is superior in all cases and it all depends on the traffic intensity r. After deciding what method is more suitable to apply, the main question is determination of number of runs. That is, at the planning stage of a simulation investigation of the question of number of simulation runs (n) is critical. The confidence level of simulation output drawn from a set of simulation runs depends on the size of data set. The larger the number of runs, the higher is the associated confidence. However, more simulation runs also require more effort and resources for large systems. Thus, the main goal must be in finding the smallest number of simulation runs that will provide the desirable confidence. Pilot Studies: When the needed statistics for number of simulation runs calculation is not available from existing database, a pilot simulation is needed. For large pilot simulation runs (n), say over 30, the simplest number of runs determinate is: where d is the desirable margin of error (i. e. the absolute error), which is the half-length of the confidence interval with 100(1- a ) confidence interval. S 2 is the variance obtained from the pilot run. One may use the following sample size determinate for a desirable relative error D in , which requires an estimate of the coefficient of variation (C. V. in ) from a pilot run with n over 30: These sample size determinates could also be used for simulation output estimation of unimodal output populations, with discrete or continuous random variables provided the pilot run size (n) is larger than (say) 30. The aim of applying any one of the above number of runs determinates is at improving your pilot estimates at feasible costs. You may like using the following Applet for determination of number of runs. Further Reading: Daz-Emparanza I, Is a small Monte Carlo analysis a good analysis Checking the size power and consistency of a simulation-based test, Statistical Papers . 43(4), 567-577, 2002. Whitt W. The efficiency of one long run versus independent replications in steady-state simulation, Management Science . 37(6), 645-666, 1991. Determination of Simulation Runs Size At the planning stage of a simulation modeling the question of number of simulation runs (n) is critical. The following Java applets compute the needed Runs Size based on current avialable information ontained from a pilot simulation run, to achieve an acceptable accuracy andor risk. Enter the needed information, and then click the Calculate button. The aim of applying any one of the following number of simulation runs determinates is at improving your pilot estimates at a feasible cost. Notes: The normality condition might be relaxed for number of simulation runs over, say 30. Moreover, determination of number of simulation runs for mean could also be used for other unimodal simulation output distributions including those with discrete random variables, such as proportion, provided the pilot run is sufficiently large (say, over 30). Runs Size with Acceptable Absolute Precision Simulation Software Selection The vast amount of simulation software available can be overwhelming for the new users. The following are only a random sample of software in the market today: ACSL, APROS, ARTIFEX, Arena, AutoMod, CSIM, CSIM, Callim, FluidFlow, GPSS, Gepasi, JavSim, MJX, MedModel, Mesquite, Multiverse, NETWORK, OPNET Modeler, POSES, Simulat8, Powersim, QUEST, REAL, SHIFT, SIMPLE, SIMSCRIPT, SLAM, SMPL, SimBank, SimPlusPlus, TIERRA, Witness, SIMNON, VISSIM, and javasim. There are several things that make an ideal simulation package. Some are properties of the package, such as support, reactivity to bug notification, interface, etc. Some are properties of the user, such as their needs, their level of expertise, etc. For these reasons asking which package is best is a sudden failure of judgment. The first question to ask is for what purpose you need the software Is it for education, teaching, student-projects or research The main question is: What are the important aspects to look for in a package The answer depends on specific applications. However some general criteria are: Input facilities, Processing that allows some programming, Optimization capability, Output facilities, Environment including training and support services, Input-output statistical data analysis capability, and certainly the Cost factor. You must know which features are appropriate for your situation, although, this is not based on a Yes or No judgment. For description of available simulation software, visit Simulation Software Survey. Reference Further Reading: Nikoukaran J. Software selection for simulation in manufacturing: A review, Simulation Practice and Theory . 7(1), 1-14, 1999. Animation in Systems Simulation Animation in systems simulation is a useful tool. Most graphically based software packages have default animation. This is quite useful for model debugging, validation, and verification. This type of animation comes with little or no additional effort and gives the modeler additional insight into how the model. This type of animation comes with little or no additional effort and gives the modeler additional insight into how the model works. However, it augments the modeling tools available. The more realistic animation presents qualities which intend to be useful to the decision-maker in implementing the developed simulation model. There are also, good model management tools. Some tools have been developed which combined a database with simulation to store models, data, results, and animations. However, there is not one product that provides all of those capabilities. SIMSCRIPT II.5 Without computer one cannot perform any realistic dynamic systems simulation. SIMSCRIPT II.5 is a powerful, free-format, English-like simulation language designed to greatly simplify writing programs for simulation modelling. Programs written in SIMSCRIPT II.5 are easily read and maintained. They are accurate, efficient, and generate results which are acceptable to users. Unlike other simulation programming languages, SIMSCRIPT II.5 requires no coding in other languages. SIMSCRIPT II.5 has been fully supported for over 33 years. Contributing to the wide acceptance and success of SIMSCRIPT II.5 modelling are: A powerful worldview, consisting of Entities and Processes, provides a natural conceptual framework with which to relate real objects to the model. SIMSCRIPT II.5 is a modern, free-form language with structured programming constructs and all the built-in facilities needed for model development. Model components can be programmed so they clearly reflect the organization and logic of the modeled system. The amount of program needed to model a system is typically 75 less than its FORTRAN or C counterpart. A well designed package of program debug facilities is provided. The required tools are available to detect errors in a complex computer program without resorting an error. Simulation status information is provided, and control is optionally transferred to a user program for additional analysis and output. This structure allows the model to evolve easily and naturally from simple to detailed formulation as data becomes available. Many modifications, such as the choice of set disciplines and statistics are simply specified in the Preamble. You get a powerful, English-like language supporting a modular implementation. Because each model component is readable and self-contained, the model documentation is the model listing it is never obsolete or inaccurate. For more information contact SIMSCRIPT Guidelines for Running SIMSCRIPT on the VAX System System Dynamics and Discrete Event Simulation The modeling techniques used by system dynamics and discrete event simulations are often different at two levels: The modeler way of representing systems might be different, the underlying simulators algorithms are also different. Each technique is well tuned to the purpose it is intended. However, one may use a discrete event approach to do system dynamics and vice versa. Traditionally, the most important distinction is the purpose of the modeling. The discrete event approach is to find, e. g. how many resources the decision maker needs such as how many trucks, and how to arrange the resources to avoid bottlenecks, i. e. excessive of waiting lines, waiting times, or inventories. While the system dynamics approach is to prescribe for the decision making to, e. g. timely respond to any changes, and how to change the physical structure, e. g. physical shipping delay time, so that inventories, sales, production, etc. System dynamics is the rigorous study of problems in system behavior using the principles of feedback, dynamics and simulation. In more words system dynamics is characterized by: Searching for useful solutions to real problems, especially in social systems (businesses, schools, governments. ) and the environment. Using computer simulation models to understand and improve such systems. Basing the simulation models on mental models, qualitative knowledge and numerical information. Using methods and insights from feedback control engineering and other scientific disciplines to assess and improve the quality of models. Seeking improved ways to translate scientific results into achieved implemented improvement. Systems dynamics approach looks at systems at a very high level so is more suited to strategic analysis. Discrete event approach may look at subsystems for a detailed analysis and is more suited, e. g. to process re-engineering problems. Systems dynamics is indicative, i. e. helps us understand the direction and magnitude of effects (i. e. where in the system do we need to make the changes), whereas discrete event approach is predictive (i. e. how many resources do we need to achieve a certain goal of throughout). Systems dynamics analysis is continuous in time and it uses mostly deterministic analysis, whereas discrete event process deals with analysis in a specific time horizon and uses stochastic analysis. Some interesting and useful areas of system dynamics modeling approach are: Short-term and long term forecasting of agricultural produce with special reference to field crops and perennial fruits such as grapes, which have significant processing sectors of different proportions of total output where both demand and supply side perspectives are being considered. Long term relationship between the financial statements of balance sheet, income statement and cash flow statement balanced against scenarios of the stock markets need to seek a stablegrowing share price combined with a satisfactory dividend and related return on shareholder funds policy. Managerial applications include the development and evaluation of short-term and long-term strategic plans, budget analysis and assessment, business audits and benchmarking. A modeler must consider both as complementary tools to each other. Systems dynamic to look at the high level problem and identify areas which need more detailed analysis. Then, use discrete event modeling tools to analyze (and predict) the specific areas of interest. What Is Social Simulation Social scientists have always constructed models of social phenomena. Simulation is an important method for modeling social and economic processes. In particular, it provides a middle way between the richness of discursive theorizing and rigorous but restrictive mathematical models. There are different types of computer simulation and their application to social scientific problems. Faster hardware and improved software have made building complex simulations easier. Computer simulation methods can be effective for the development of theories as well as for prediction. For example, macro-economic models have been used to simulate future changes in the economy and simulations have been used in psychology to study cognitive mechanisms. The field of social simulation seems to be following an interesting line of inquiry. As a general approach in the field, a world is specified with much computational detail. Then the world is simulated (using computers) to reveal some of the non-trivial implications (or emergent properties) of the world. When these non trivial implications are made known (fed back) in world, apparently it constitutes some added values. Artificial Life is an interdisciplinary study enterprise aimed at understanding life-as-it-is and life-as-it-could-be, and at synthesizing life-like phenomena in chemical, electronic, software, and other artificial media. Artificial Life redefines the concepts of artificial and natural, blurring the borders between traditional disciplines and providing new media and new insights into the origin and principles of life. Simulation allows the social scientist to experiment with artificial societies and explore the implications of theories in ways not otherwise possible. Reference and Further Readings: Gilbert N. and K. Troitzsch, Simulation for the Social Scientist . Open University Press, Buckingham, UK, 1999. Sichman J. R. Conte, and N. Gilbert, (eds,), Multi-Agent Systems and Agent-Based Simulation . Berlin, Springer-Verlag, 1998. What Is Web-based Simulation Web-based simulation is quickly emerging as an area of significant interest for both simulation researchers and simulation practitioners. This interest in web-based simulation is a natural outgrowth of the proliferation of the World-Wide Web and its attendant technologies, e. g. HTML, HTTP, CGI, etc. Also the surging popularity of, and reliance upon, computer simulation as a problem solving and decision support systems tools. The appearance of the network-friendly programming language, Java, and of distributed object technologies like the Common Object Request Broker Architecture (CORBA) and the Object Linking and Embedding Component Object Model (OLECOM) have had particularly acute effects on the state of simulation practice. Currently, the researchers in the field of web-based simulation are interested in dealing with topics such as methodologies for web-based model development, collaborative model development over the Internet, Java-based modeling and simulation, distributed modeling and simulation using web technologies, and new applications. Parallel and Distributed Simulation The increasing size of the systems and designs requires more efficient simulation strategies to accelerate the simulation process. Parallel and distributed simulation approaches seem to be a promising approach in this direction. Current topics under extensive research are: Synchronization, scheduling, memory management, randomized and reactiveadaptive algorithms, partitioning and load balancing. Synchronization in multi-user distributed simulation, virtual reality environments, HLA, and interoperability. System modeling for parallel simulation, specification, re-use of modelscode, and parallelizing existing simulations. Language and implementation issues, models of parallel simulation, execution environments, and libraries. Theoretical and empirical studies, prediction and analysis, cost models, benchmarks, and comparative studies. Computer architectures, VLSI, telecommunication networks, manufacturing, dynamic systems, and biologicalsocial systems. Web based distributed simulation such as multimedia and real time applications, fault tolerance, implementation issues, use of Java, and CORBA. References Further Readings: Bossel H. Modeling Simulation . A. K. Peters Pub. 1994. Delaney W. and E. Vaccari, Dynamic Models and Discrete Event Simulation . Dekker, 1989. Fishman G. Discrete-Event Simulation: Modeling, Programming and Analysis . Springer-Verlag, Berlin, 2001. Fishwick P. Simulation Model Design and Execution: Building Digital Worlds . Prentice-Hall, Englewood Cliffs, 1995. Ghosh S. and T. Lee, Modeling Asynchronous Distributed Simulation: Analyzing Complex Systems . IEEE Publications, 2000. Gimblett R. Integrating Geographic Information Systems and Agent-Based Modeling: Techniques for Simulating Social and Ecological Processes . Oxford University Press, 2002. Harrington J. and K. Tumay, Simulation Modeling Methods: An Interactive Guide to Results-Based Decision . McGraw-Hill, 1998. Haas P. Stochastic Petri Net Models Modeling and Simulation . Springer Verlag, 2002. Hill D. Object-Oriented Analysis and Simulation Modeling . Addison-Wesley, 1996. Kouikoglou V. and Y. Phillis, Hybrid Simulation Models of Production Networks . Kluwer Pub. 2001. Law A. and W. Kelton, Simulation Modeling and Analysis . McGraw-Hill, 2000. Nelson B. Stochastic Modeling: Analysis Simulation . McGraw-Hill, 1995. Oakshott L., Business Modelling and Simulation . Pitman Publishing, London, 1997. Pidd M. Computer Simulation in Management Science . Wiley, 1998. Rubinstein R. and B. Melamed, Modern Simulation and Modeling . Wiley, 1998. Severance F. System Modeling and Simulation: An Introduction . Wiley, 2001. Van den Bosch, P. and A. Van der Klauw, Modeling, Identification Simulation of Dynamical Systems . CRC Press, 1994. Woods R. and K. Lawrence, Modeling and Simulation of Dynamic Systems . Prentice Hall, 1997. Techniques for Sensitivity Estimation Simulation continues to be the primary method by which engineers and managers obtain information about complex stochastic systems, such as telecommunication networks, health service, corporate planning, financial modeling, production assembly lines, and flexible manufacturing systems. These systems are driven by the occurrence of discrete events and complex interactions within these discrete events occur over time. For most discrete event systems (DES) no analytical methods are available, so DES must be studied via simulation. DES are studied to understand their performance, and to determine the best ways to improve their performance. In particular, one is often interested in how system performance depends on the systems parameter v, which could be a vector. DESs system performance is often measured as an expected value. Consider a system with continuous parameter v 206 V 205 R n . where V is an open set. Let be the steady state expected performance measure, where Y is a random vector with known probability density function (pdf), f(y v) depends on v, and Z is the performance measure. In discrete event systems, Monte Carlo simulation is usually needed to estimate J(v) for a given value v v 0 . By the law of large numbers converges to the true value, where y i . i 1, 2. n are independent, identically distributed, random vector realizations of Y from f (y v 0 ), and n is the number of independent replications. We are interested in sensitivities estimation of J(v) with respect to v. Applications of sensitivity information There are a number of areas where sensitivity information (the gradient, Hessian, etc.) of a performance measure J(v) or some estimate of it, is used for the purpose of analysis and control. In what follows, we single out a few such areas and briefly discuss them. Local information: An estimate for dJdv is a good local measure of the effect of on performance. For example, simply knowing the sign of the derivative dJdv at some point v immediately gives us the direction in which v should be changed. The magnitude of dJd also provides useful information in an initial design process: If dJdv is small, we conclude that J is not very sensitive to changes in. and hence focusing concentration on other parameters may improve performance. Structural properties: Often sensitivity analysis provides not only a numerical value for the sample derivative, but also an expression which captures the nature of the dependence of a performance measure on the parameter v. The simplest case arises when dJdv can be seen to be always positive (or always negative) for any sample path we may not be able to tell if the value of J(v) is monotonically increasing (or decreasing) in v. This information in itself is very useful in design and analysis. More generally, the form of dJdv can reveal interesting structural properties of the DES (e. g. monotonicity, convexity). Such properties must be exploited in order to determine optimal operating policies for some systems. Response surface generation: Often our ultimate goal is to obtain the function J(v), i. e. a curve describing how the system responds to different values of v. Since J(v) is unknown, one alternative is to obtain estimates of J(v) for as many values of v as possible. This is clearly a prohibitively difficult task. Derivative information, however may include not only first-order but also higher derivatives which can be used to approximate J(v). If such derivative information can be easily and accurately obtained, the task of response surface generation may be accomplished as well. Goal-seeking and What-if problems: Stochastic models typically depend upon various uncertain parameters that must be estimated from existing data sets. Statistical questions of how input parameter uncertainty propagates through the model into output parameter uncertainty is the so-called what-if analysis. A good answer to this question often requires sensitivity estimates. The ordinary simulation output results are the solution of a direct problem: Given the underlying pdf with a particular parameter value v. we may estimate the output function J(v). Now we pose the goal-seeking problem: given a target output value J 0 of the system and a parameterized pdf family, find an input value for the parameter, which generates such an output. There are strong motivations for both problems. When v is any controllable or uncontrollable parameter the decision maker is, for example, interested in estimating J(v) for a small change in v , the so called what-if problem, which is a direct problem and can be solved by incorporating sensitivity information in the Taylors expansion of J(v) in the neighborhood of v. However, when v is a controllable input, the decision maker may be interested in the goal-seeking problem: what change in the input parameter will achieve a desired change in output value J(v). Another application of goal-seeking arises when we want to adapt a model to satisfy a new equality constraint (condition) for some stochastic function. The solution to the goal-seeking problem is to estimate the derivative of the output function with respect to the input parameter for the nominal system use this estimate in a Taylors expansion of the output function in the neighborhood of the parameter and finally, use Robbins-Monro (R-M) type of stochastic approximation algorithm to estimate the necessary controllable input parameter value within the desired accuracy. Optimization: Discrete-event simulation is the primary analysis tool for designing complex systems. However, simulation must be linked with a mathematical optimization technique to be effectively used for systems design. The sensitivity dJdv can be used in conjunction with various optimization algorithms whose function is to gradually adjust v until a point is reached where J(v) is maximized (or minimized). If no other constraints on v are imposed, we expect dJdv 0 at this point. Click on the image to enlarge it and THEN print it. Finite difference approximation Kiefer and Wolfowitz proposed a finite difference approximation to the derivative. One version of the Kiefer-Wolfwitz technique uses two-sided finite differences. The first fact to notice about the K-W estimate is that it requires 2N simulation runs, where N is the dimension of vector parameter q. If the decision maker is interested in gradient estimation with respect to each of the components of q. then 2N simulations must be run for each component of v. This is inefficient. The second fact is that it may have a very poor variance, and it may result in numerical calculation difficulties. Simultaneous perturbation methods The simultaneous perturbation (SP) algorithm introduced by Dr. J. Spall has attracted considerable attention. There has recently been much interest in recursive optimization algorithms that rely on measurements of only the objective function to be optimized, not requiring direct measurements of the gradient of the objective function. Such algorithms have the advantage of not requiring detailed modeling information describing the relationship between the parameters to be optimized and the objective function. For example, many systems involving complex simulations or human beings are difficult to model, and could potentially benefit from such an optimization approach. The simultaneous perturbation stochastic approximation (SPSA) algorithm operates in the same framework as the above K-W methods, but has the strong advantage of requiring a much lower number of simulation runs to obtain the same quality of result. The essential feature of SPSA, which accounts for its power and relative ease of use in difficult multivariate optimization problems--is the underlying gradient approximation that requires only TWO objective function measurements regardless of the dimension of the optimization problem (one variation of basic SPSA uses only ONE objective function measurement per iteration). The underlying theory for SPSA shows that the N-fold savings in simulation runs per iteration (per gradient approximation) translates directly into an N-fold savings in the number of simulations to achieve a given quality of solution to the optimization problem. In other words, the K-W method and SPSA method take the same number of iterations to converge to the answer despite the N-fold savings in objective function measurements (e. g. simulation runs) per iteration in SPSA. Perturbation analysis Perturbation analysis (PA) computes (roughly) what simulations would have produced, had v been changed by a small amount without actually making this change. The intuitive idea behind PA is that a sample path constructed using v is frequently structurally very similar to the sample path using the perturbed v. There is a large amount of information that is the same for both of them. It is wasteful to throw this information away and to start the simulation from scratch with the perturbed v. In PA, moreover, we can let the change approach zero to get a derivative estimator without numerical problems. We are interested in the affect of a parameter change on the performance measure. However, we would like to realize this change by keeping the order of events exactly the same. The perturbations will be so small that only the duration, not the order, of the states will be affected. This effect should be observed in three successive stages: Step 1: How does a change in the value of a parameter vary the sample duration related to that parameter Step 2: How does the change in an individual sample duration reflect itself as a change in a subsequent particular sample realization Step 3: Finally, what is the relationship between the variation of the sample realization and its expected value Score function methods Using the score function method, the gradient can be estimated simultaneously, at any number of different parameter values, in a single-run simulation. The basic idea is that, the gradient of the performance measure function, J( v ), is expressed as an expectation with respect to the same distribution as the performance measure function itself. Therefore, the sensitivity information can be obtained with little computational (not simulation) cost, while estimating the performance measure. It is well-known that the crude form of the SF estimator suffers from the problem of linear growth in its variance as the simulation run increases. However, in the steady-state simulation the variance can be controlled by run length. Furthermore, information about the variance may be incorporated into the simulation algorithm. A recent flurry of activity has attempted to improve the accuracy of the SF estimates. Under regenerative conditions, the estimator can easily be modified to alleviate this problem, yet the magnitude of the variance may be large for queueing systems with heavy traffic intensity. The heuristic idea is to treat each component of the system (e. g. each queue) separately, which synchronously assumes that individual components have local regenerative cycles. This approach is promising since the estimator remains unbiased and efficient while the global regenerative cycle is very long. Now we look at the general (non-regenerative) case. In this case any simulation will give a biased estimator of the gradient, as simulations are necessarily finite. If n (the length of the simulation) is large enough, this bias is negligible. However, as noted earlier, the variance of the SF sensitivity estimator increases with increase in n so, a crude SF estimator is not even approximately consistent. There are a number of ways to attack this problem. Most of the variations in an estimator comes from the score function. The variation is especially high, when all past inputs contribute to the performance and the scores from all are included. When one uses batch means, the variation is reduced by keeping the length of the batch small. A second way is to reduce the variance of the score to such an extent that we can use simulations long enough to effectively eliminate the bias. This is the most promising approach. The variance may be reduced further by using the standard variance reduction techniques (VRT), such as importance sampling. Finally, we can simply use a large number of iid replications of the simulation. Harmonic analysis Another strategy for estimating the gradient simulation is based on the frequency domain method, which differs from the time domain experiments in that the input parameters are deterministically varied in sinusoidal patterns during the simulation run, as opposed to being kept fixed as in the time domain runs. The range of possible values for each input factor should be identified. Then the values of each input factor within its defined range should be changed during a run. In time series analysis, t is the time index. In simulation, however, t is not necessarily the simulation clock time. Rather, t is a variable of the model, which keeps track of certain statistics during each run. For example, to generate the inter-arrival times in a queueing simulation, t might be the variable that counts customer arrivals. Frequency domain simulation experiments identify the significant terms of the polynomial that approximates the relationship between the simulation output and the inputs. Clearly, the number of simulation runs required to identify the important terms by this approach is much smaller than those of the competing alternatives, and the difference becomes even more conspicuous as the number of parameters increases. Conclusions Further Readings PA and SF (or LR) can be unified. Further comparison of the PA and SF approaches reveals several interesting differences. Both approaches require an interchange of expectation and differentiation. However, the conditions for this interchange in PA depend heavily on the nature of the problem, and must be verified for each application, which is not the case in SF. Therefore, in general, it is easier to satisfy SF unbiased conditions. PA assumes that the order of events in the perturbed path is the same as the order in the nominal path, for a small enough change in v. allowing the computation of the sensitivity of the sample performance for a particular simulation. For example, if the performance measure is the mean number of customer in a busy period, the PA estimate of the gradient with respect to any parameter is zero The number of customers per busy period will not change if the order of events does not change. In terms of ease of implementation, PA estimators may require considerable analytical work on the part of algorithm developer, with some customization for each application, whereas SF has the advantage of remaining a general definable algorithm whenever it can be applied. Perhaps the most important criterion for comparison lies in the question of accuracy of an estimator, typically measured through its variance. If an estimator is strongly consistent, its variance is gradually reduced over time and ultimately approaches to zero. The speed with which this happens may be extremely important. Since in practice, decisions normally have to be made in a limited time, an estimator whose variance decreases fast is highly desirable. In general, when PA does provide unbiased estimators, the variance of these estimators is small. PA fully exploits the structure of DES and their state dynamics by extracting the needed information from the observed sample path, whereas SF requires no knowledge of the system other than the inputs and the outputs. Therefore when using SF methods, variance reduction is necessary. The question is whether or not the variance can be reduced enough to make the SF estimator useful in all situations to which it can be applied. The answer is certainly yes. Using the standard variance reduction techniques can help, but the most dramatic variance reduction occurs using new methods of VR such as conditioning, which is shown numerically to have a mean squared error that is essentially the same as that of PA. References Further Readings: Arsham H. Algorithms for Sensitivity Information in Discrete-Event Systems Simulation, Simulation Practice and Theory . 6(1), 1-22, 1998. Fu M. and J-Q. Hu, Conditional Monte Carlo: Gradient Estimation and Optimization Applications . Kluwer Academic Publishers, 1997. Rubinstein R. and A. Shapiro, Discrete Event Systems: Sensitivity Analysis and Stochastic Optimization by the Score Function Method . John Wiley Sons, 1993. Whitt W. Minimizing delays in the GIG1 queue, Operations Research . 32(1), 41-51, 1984. Simulation-based Optimization Techniques Discrete event simulation is the primary analysis tool for designing complex systems. Simulation, however, must be linked with a optimization techniques to be effectively used for systems design. We present several optimization techniques involving both continuous and discrete controllable input parameters subject to a variety of constraints. The aim is to determine the techniques most promising for a given simulation model. Many man-made systems can be modeled as Discrete Event Systems (DES) examples are computer systems, communication networks, flexible manufacturing systems, production assembly lines, and traffic transportation systems. DES evolve with the occurrence of discrete events, such as the arrival of a job or the completion of a task, in contrast with continuously variable dynamic processes such as aerospace vehicles, which are primarily governed by differential equations. Owing to the complex dynamics resulting from stochastic interactions of such discrete events over time, the performance analysis and optimization of DES can be difficult tasks. At the same time, since such systems are becoming more widespread as a result of modern technological advances, it is important to have tools for analyzing and optimizing the parameters of these systems. Analyzing complex DES often requires computer simulation. In these systems, the objective function may not be expressible as an explicit function of the input parameters rather, it involves some performance measures of the system whose values can be found only by running the simulation model or by observing the actual system. On the other hand, due to the increasingly large size and inherent complexity of most man-made systems, purely analytical means are often insufficient for optimization. In these cases, one must resort to simulation, with its chief advantage being its generality, and its primary disadvantage being its cost in terms of time and money. Even though, in principle, some systems are analytically tractable, the analytical effort required to evaluate the solution may be so formidable that computer simulation becomes attractive. While the price for computing resources continue to dramatically decrease, one nevertheless can still obtain only a statistical estimate as opposed to an exact solution. For practical purposes, this is quite sufficient. These man-made DES are costly, and therefore it is important to operate them as efficiently as possible. The high cost makes it necessary to find more efficient means of conducting simulation and optimizing its output. We consider optimizing an objective function with respect to a set of continuous andor discrete controllable parameters subject to some constraints. Click on the image to enlarge it and THEN print it. The above figure illustrates the feedback loop application. Although the feedback concept is not a simulation but a systemic concept, however, whatever paradigm we use one can always incorporate feedback. For example, consider a discrete event system (DES) model that employs resources to achieve certain tasksprocesses, by only incorporating decision rules regarding how to manage the stocks and thence how the resource will be deployed depending on the stock level, clearly, in the system structure there are feedback loops. Usually when modelers choose a DES approach they often model the system as open loop or nearly open loop system, making the system behave as if there where no superior agent controlling the whole productionservice process. Closing the loops should be an elemental task that simulation modeler should take care of, even if the scope does not involve doing it, there must be awareness of system behavior, particularly if there is known to be that the system if under human decision making processesactivities. In almost all simulation models, an expected value can express the systems performance. Consider a system with continuous parameter v 206 V, where V is the feasible region. Let be the steady state expected performance measure, where Y is a random vector with known probability density function (pdf), f(y v) depends on v, and Z is the performance measure. In discrete event systems, Monte Carlo simulation is usually needed to estimate J(v) for a given value v v 0 . By the law of large numbers converges to the true value, where y i . i 1, 2. n are independent, identically distributed, random vector realizations of Y from f (y v 0 ), and n is the number of independent replications. The aim is to optimize J(v) with respect to v. We shall group the optimization techniques for simulation into seven broad categories namely, Deterministic Search, Pattern Search, Probabilistic Search, Evolutionary Techniques, Stochastic Approximation, Gradient Surface, and some Mixtures of the these techniques Click on the image to enlarge it and THEN print it. Deterministic search techniques A common characteristic of deterministic search techniques is that they are basically borrowed from deterministic optimization techniques. The deterministic objective function value required in the technique is now replaced with an estimate obtained from simulation. By having a reasonably accurate estimate, one hopes that the technique will perform well. Deterministic search techniques include heuristic search, complete enumeration, and random search techniques. Heuristic search technique The heuristic search technique is probably most commonly used in optimizing response surfaces. It is also the least sophisticated scheme mathematically, and it can be thought of as an intuitive and experimental approach. The analyst determines the starting point and stopping rule based on previous experience with the system. After setting the input parameters (factors) to levels that appear reasonable, the analyst makes a simulation run with the factors set at those levels and computes the value of the response function. If it appears to be a maximum (minimum) to the analyst, the experiment is stopped. Otherwise the analyst changes parameter settings and makes another run. This process continues until the analyst believes that the output has been optimized. Suffice it to say that, if the analyst is not intimately familiar with the process being simulated, this procedure can turn into a blind search and can expend an inordinate amount of time and computer resources without producing results commensurate with input. The heuristic search can be ineffective and inefficient in the hand of a novice. Complete enumeration and random techniques The complete enumeration technique is not applicable to continuous cases, but in discrete space v it does yield the optimal value of the response variable. All factors ( v ) must assume a finite number of values for this technique to be applicable. Then, a complete factorial experiment is run. The analyst can attribute some degree of confidence to the determined optimal point when using this procedure. Although the complete enumeration technique yields the optimal point, it has a serious drawback. If the number of factors or levels per factor is large, the number of simulation runs required to find the optimal point can be exceedingly large. For example, suppose that an experiment is conducted with three factors having three, four, and five levels, respectively. Also suppose that five replications are desired to provide the proper degree of confidence. Then 300 runs of the simulator are required to find the optimal point. Hence, this technique should be used only when the number of unique treatment combinations is relatively small or a run takes little time. The random search technique resembles the complete enumeration technique except that one selects a set of inputs at random. The simulated results based on the set that yields the maximum (minimum) value of the response function is taken to be the optimal point. This procedure reduces the number of simulation runs required to yield an optimal result however, there is no guarantee that the point found is actually the optimal point. Of course, the more points selected, the more likely the analyst is to achieve the true optimum. Note that the requirement that each factor assumes only a finite number of values is not a requirement in this scheme. Replications can be made on the treatment combinations selected, to increase the confidence in the optimal point. Which strategy is better, replicating a few points or looking at a single observation on more points, depends on the problem. Response surface search Response surface search attempts to fit a polynomial to J(v). If the design space v is suitably small, the performance function J(v) may be approximated by a response surface, typically a first order, or perhaps quadratic order in v. possibly after transformation, e. g. log ( v ). The response surface method (RSM) requires running the simulation in a first order experimental design to determine the path of steepest descent. Simulation runs made along this path continue, until one notes no improvement in J(v). The analyst then runs a new first order experimental design around the new optimal point reached, and finds a new path of steepest descent. The process continues, until there is a lack of fit in the fitted first order surface. Then, one runs a second order design, and takes the optimum of the fittest second order surface as the estimated optimum. Although it is desirable for search procedures to be efficient over a wide range of response surfaces, no current procedure can effectively overcome non-unimodality (surfaces having more than one local maximum or minimum). An obvious way to find the global optimal would be to evaluate all the local optima. One technique that is used when non-unimodality is known to exist, is called the Las Vegas technique. This search procedure estimates the distribution of the local optima by plotting the estimated J( v ) for each local search against its corresponding search number. Those local searches that produce a response greater than any previous response are then identified and a curve is fitted to the data. This curve is then used to project the estimated incremental response that will be achieved by one more search. The search continues until the value of the estimated improvement in the search is less than the cost of completing one additional search. It should be noted that a well-designed experiment requires a sufficient number of replications so that the average response can be treated as a deterministic number for search comparisons. Otherwise, since replications are expensive, it becomes necessary to effectively utilize the number of simulation runs. Although each simulation is at a different setting of the controllable variables, one can use smoothing techniques such as exponential smoothing to reduce the required number of replications. Pattern search techniques Pattern search techniques assume that any successful set of moves used in searching for an approximated optimum is worth repeating. These techniques start with small steps then, if these are successful, the step size increases. Alternatively, when a sequence of steps fails to improve the objective function, this indicates that shorter steps are appropriate so we may not overlook any promising direction. These techniques start by initially selecting a set of incremental values for each factor. Starting at an initial base point, they check if any incremental changes in the first variable yield an improvement. The resulting improved setting becomes the new intermediate base point. One repeats the process for each of the inputs until one obtains a new setting where the intermediate base points act as the initial base point for the first variable. The technique then moves to the new setting. This procedure is repeated, until further changes cannot be made with the given incremental values. Then, the incremental values are decreased, and the procedure is repeated from the beginning. When the incremental values reach a pre-specified tolerance, the procedure terminates the most recent factor settings are reported as the solution. Conjugate direction search The conjugate direction search requires no derivative estimation, yet it finds the optimum of an N-dimensional quadratic surface after, at most, N-iterations, where the number of iterations is equal to the dimension of the quadratic surface. The procedure redefines the n dimensions so that a single variable search can be used successively. Single variable procedures can be used whenever dimensions can be treated independently. The optimization along each dimension leads to the optimization of the entire surface. Two directions are defined to be conjugate whenever the cross-product terms are all zero. The conjugate direction technique tries to find a set of n dimensions that describes the surface such that each direction is conjugate to all others. Using the above result, the technique attempts to find two search optima and replace the n th dimension of the quadratic surface by the direction specified by the two optimal points. Successively replacing the original dimension yields a new set of n dimensions in which, if the original surface is quadratic, all directions are conjugate to each other and appropriate for n single variable searches. While this search procedure appears to be very simple, we should point out that the selection of appropriate step sizes is most critical. The step size selection is more critical for this search technique because - during axis rotation - the step size does not remain invariant in all dimensions. As the rotation takes place, the best step size changes, and becomes difficult to estimate. Steepest ascent (descent) The steepest ascent (descent) technique uses a fundamental result from calculus ( that the gradient points in the direction of the maximum increase of a function), to determine how the initial settings of the parameters should be changed to yield an optimal value of the response variable. The direction of movement is made proportional to the estimated sensitivity of the performance of each variable. Although quadratic functions are sometimes used, one assumes that performance is linearly related to the change in the controllable variables for small changes. Assume that a good approximation is a linear form. The basis of the linear steepest ascent is that each controllable variable is changed in proportion to the magnitude of its slope. When each controllable variable is changed by a small amount, it is analogous to determining the gradient at a point. For a surface containing N controllable variables, this requires N points around the point of interest. When the problem is not an n-dimensional elliptical surface, the parallel-tangent points are extracted from bitangents and inflection points of occluding contours. Parallel tangent points are points on the occluding contour where the tangent is parallel to a given bitangent or the tangent at an inflection point. Tabu search technique An effective technique to overcome local optimality for discrete optimization is the Tabu Search technique. It explores the search space by moving from a solution to its best neighbor, even if this results in a deterioration of the performance measure value. This approach increases the likelihood of moving out of local optima. To avoid cycling, solutions that were recently examined are declared tabu (Taboo) for a certain number of iterations. Applying intensification procedures can accentuate the search in a promising region of the solution space. In contrast, diversification can be used to broaden the search to a less explored region. Much remains to be discovered about the range of problems for which the tabu search is best suited. Hooke and Jeeves type techniques The Hooke and Jeeves pattern search uses two kinds of moves namely, an exploratory and a pattern move. The exploratory move is accomplished by doing a coordinate search in one pass through all the variables. This gives a new base point from which a pattern move is made. A pattern move is a jump in the pattern direction determined by subtracting the current base point from the previous base point. After the pattern move, another exploratory move is carried out at the point reached. If the estimate of J(v) is improved at the final point after the second exploratory move, it becomes the new base point. If it fails to show improvement, an exploratory move is carried out at the last base point with a smaller step in the coordinate search. The process stops when the step gets small enough. Simplex-based techniques The simplex-based technique performs simulation runs first at the vertices of the initial simplex i. e. a polyhedron in the v - space having N1 vertices. A subsequent simplex (moving towards the optimum) are formed by three operations performed on the current simplex: reflection, contraction, and expansion. At each stage of the search process, the point with the highest J(v) is replaced with a new point foundvia reflection through the centroid of the simplex. Depending on the value of J(v) at this new point, the simplex is either expanded, contracted, or unchanged. The simplex technique starts with a set of N1 factor settings. These N1 points are all the same distance from the current point. Moreover, the distance between any two points of these N1 points is the same. Then, by comparing their response values, the technique eliminates the factor setting with the worst functional value and replaces it with a new factor setting, determined by the centroid of the N remaining factor settings and the eliminated factor setting. The resulting simplex either grows or shrinks, depending on the response value at the new factor settings. One repeats the procedure until no more improvement can be made by eliminating a point, and the resulting final simplex is small. While this technique will generally performance well for unconstrained problems, it may collapse to a point on a boundary of a feasible region, thereby causing the search to come to a premature halt. This technique is effective if the response surface is generally bowl - shaped even with some local optimal points. Probabilistic search techniques All probabilistic search techniques select trial points governed by a scan distribution, which is the main source of randomness. These search techniques include random search, pure adaptive techniques, simulated annealing, and genetic methods. Random search A simple, but very popular approach is the random search, which centers a symmetric probability density function (pdf) e. g. the normal distribution, about the current best location. The standard normal N(0, 1) is a popular choice, although the uniform distribution U-1, 1 is also common. A variation of the random search technique determines the maximum of the objective function by analyzing the distribution of J(v) in the bounded sub-region. In this variation, the random data are fitted to an asymptotic extreme-value distribution, and J is estimated with a confidence statement. Unfortunately, these techniques cannot determine the location of J. which can be as important as the J value itself. Some techniques calculate the mean value and the standard deviation of J(v) from the random data as they are collected. Assuming that J is distributed normally in the feasible region. the first trial, that yields a J-value two standard deviations within the mean value, is taken as a near-optimum solution. Pure adaptive search Various pure adaptive search techniques have been suggested for optimization in simulation. Essentially, these techniques move from the current solution to the next solution that is sampled uniformly from the set of all better feasible solutions. Evolutionary Techniques Nature is a robust optimizer. By analyzing natures optimization mechanism we may find acceptable solution techniques to intractable problems. Two concepts that have most promise are simulated annealing and the genetic techniques. Simulated annealing Simulated annealing (SA) borrows its basic ideas from statistical mechanics. A metal cools, and the electrons align themselves in an optimal pattern for the transfer of energy. In general, a slowly cooling system, left to itself, eventually finds the arrangement of atoms, which has the lowest energy. The is the behavior, which motivates the method of optimization by SA. In SA we construct a model of a system and slowly decrease the temperature of this theoretical system, until the system assumes a minimal energy structure. The problem is how to map our particular problem to such an optimizing scheme. SA as an optimization technique was first introduced to solve problems in discrete optimization, mainly combinatorial optimization. Subsequently, this technique has been successfully applied to solve optimization problems over the space of continuous decision variables. SA is a simulation optimization technique that allows random ascent moves in order to escape the local minima, but a price is paid in terms of a large increase in the computational time required. It can be proven that the technique will find an approximated optimum. The annealing schedule might require a long time to reach a true optimum. Genetic techniques Genetic techniques (GT) are optimizers that use the ideas of evolution to optimize a system that is too difficult for traditional optimization techniques. Organisms are known to optimize themselves to adapt to their environment. GT differ from traditional optimization procedures in that GT work with a coding of the decision parameter set, not the parameters themselves GT search a population of points, not a single point GT use objective function information, not derivatives or other auxiliary knowledge and finally, GT use probabilistic transition rules, not deterministic rules. GT are probabilistic search optimizing techniques that do not require mathematical knowledge of the response surface of the system, which they are optimizing. They borrow the paradigms of genetic evolution, specifically selection, crossover, and mutation. Selection: The current points in the space are ranked in terms of their fitness by their respective response values. A probability is assigned to each point that is proportional to its fitness, and parents (a mating pair) are randomly selected. Crossover: The new point, or offspring, is chosen, based on some combination of the genetics of the two parents. Mutation: The location of offspring is also susceptible to mutation, a process, which occurs with probability p, by which a offspring is replaced randomly by a new offspring location. A generalized GT generates p new offspring at once and kills off all of the parents. This modification is important in the simulation environment. GT are well suited for qualitative or policy decision optimization such as selecting the best queuing disciplines or network topologies. They can be used to help determine the design of the system and its operation. For applications of GT to inventory systems, job-shop, and computer time-sharing problems. GT do not have certain shortcomings of other optimization techniques, and they will usually result in better calculated optima than those found with the traditionally techniques. They can search a response surface with many local optima and find (with a high probability) the approximate global optimum. One may use GT to find an area of potential interest, and then resort to other techniques to find the optimum. Recently, several classical GT principles have been challenged. Differential Evolution. Differential Evolution (DE) is a genetic type of algorithm for solving continuous stochastic function optimization. The basic idea is to use vector differences for perturbing the vector population. DE adds the weighted difference between two population vectors to a third vector. This way, no separate probability distribution has to be used, which makes the scheme completely self-organizing. A short comparison When performing search techniques in general, and simulated annealing or genetic techniques specifically, the question of how to generate the initial solution arises. Should it be based on a heuristic rule or on a randomly generated one Theoretically, it should not matter, but in practice this may depend on the problem. In some cases, a pure random solution systematically produces better final results. On the other hand, a good initial solution may lead to lower overall run times. This can be important, for example, in cases where each iteration takes a relatively long time therefore, one has to use some clever termination rule. Simulation time is a crucial bottleneck in an optimization process. In many cases, a simulation is run several times with different initial solutions. Such a technique is most robust, but it requires the maximum number of replications compared with all other techniques. The pattern search technique applied to small problems with no constraints or qualitative input parameters requires fewer replications than the GT. GT, however, can easily handle constraints, and have lower computational complexity. Finally, simulated annealing can be embedded within the Tabu search to construct a probabilistic technique for global optimization. References Further Readings: Choi D.-H. Cooperative mutation based evolutionary programming for continuous function optimization, Operations Research Letters . 30, 195-201, 2002. Reeves C. and J. Rowe, Genetic Algorithms: Principles and Perspectives . Kluwer, 2002. Saviotti P. (Ed.), Applied Evolutionary Economics: New Empirical Methods and Simulation Techniques . Edward Elgar Pub. 2002. Wilson W. Simulating Ecological and Evolutionary Systems in C . Cambridge University Press, 2000. Stochastic approximation techniques Two related stochastic approximation techniques have been proposed, one by Robbins and Monro and one by Kiefer and Wolfowitz. The first technique was not useful for optimization until an unbiased estimator for the gradient was found. Kiefer and Wolfowitz developed a procedure for optimization using finite differences. Both techniques are useful in the optimization of noisy functions, but they did not receive much attention in the simulation field until recently. Generalization and refinement of stochastic approximation procedures give rise to a weighted average, and stochastic quasi-gradient methods. These deal with constraints, non-differentiable functions, and some classes of non-convex functions, among other things. Kiefer-Wolfowitz type techniques Kiefer and Wolfowitz proposed a finite difference approximation to the derivative. One version of the Kiefer-Wolfwitz technique uses two-sided finite differences. The first fact to notice about the K-W estimate is that it requires 2N simulation runs, where N is the dimension of vector parameter v. If the decision maker is interested in gradient estimation with respect to each of the components of v. then 2N simulations must be run for each component of v. This is inefficient. The second fact is that it may have a very poor variance, and it may result in numerical calculation difficulties. Robbins-Monro type techniques The original Robbins-Monro (R-M) technique is not an optimization scheme, but rather a root finding procedure for functions whose exact values are not known but are observed with noise. Its application to optimization is immediate: use the procedure to find the root of the gradient of the objective function. Interest was renewed in the R-M technique as a means of optimization, with the development of the perturbation analysis, score function (known also as likelihood ratio method), and frequency domain estimates of derivatives. Optimization for simulated systems based on the R-M technique is known as a single-run technique. These procedures optimize a simulation model in a single run simulation with a run length comparable to that required for a single iteration step in the other methods. This is achieved essentially be observing the sample values of the objective function and, based on these observations, updating the values of the controllable parameters while the simulation is running, that is, without restarting the simulation. This observing-updating sequence is done repeatedly, leading to an estimate of the optimum at the end of a single-run simulation. Besides having the potential of large computational savings, this technique can be a powerful tool in real-time optimization and control, where observations are taken as the system is evolving in time. Gradient surface method One may combine the gradient-based techniques with the response surface methods (RSM) for optimization purposes. One constructs a response surface with the aid of n response points and the components of their gradients. The gradient surface method (GSM) combines the virtue of RSM with that of the single - run, gradient estimation techniques such as Perturbation Analysis, and Score Function techniques. A single simulation experiment with little extra work yields N 1 pieces of information i. e. one response point and N components of the gradient. This is in contrast to crude simulation, where only one piece of information, the response value, is obtained per experiment. Thus by taking advantage of the computational efficiency of single-run gradient estimators. In general, N-fold fewer experiments will be needed to fit a global surface compared to the RSM. At each step, instead of using Robbins-Monro techniques to locate the next point locally, we determine a candidate for the next point globally, based on the current global fit to the performance surface. The GSM approach has the following advantages The technique can quickly get to the vicinity of the optimal solution because its orientation is global 23, 39. Thus, it produces satisfying solutions quickly Like RSM, it uses all accumulated information And, in addition, it uses gradient surface fitting, rather than direct performance response-surface fitting via single-run gradient estimators. This significantly reduces the computational efforts compared with RSM. Similar to RSM, GSM is less sensitive to estimation error and local optimality And, finally, it is an on-line technique, the technique may be implemented while the system is running. A typical optimization scheme involves two phases: a Search Phase and an Iteration Phase. Most results in analytic computational complexity assume that good initial approximations are available, and deal with the iteration phase only. If enough time is spent in the initial search phase, we can reduce the time needed in the iteration phase. The literature contains papers giving conditions for the convergence of a process a process has to be more than convergent in order to be computationally interesting. It is essential that we be able to limit the cost of computation. In this sense, GSM can be thought of as helping the search phase and as an aid to limit the cost of computation. One can adopt standard or simple devices for issues such as stopping rules. For on-line optimization, one may use a new design in GSM called single direction design. Since for on-line optimization it may not be advisable or feasible to disturb the system, random design usually is not suitable. Post-solution analysis Stochastic models typically depend upon various uncertain and uncontrollable input parameters that must be estimated from existing data sets. We focus on the statistical question of how input-parameter uncertainty propagates through the model into output - parameter uncertainty. The sequential stages are descriptive, prescriptive and post-prescriptive analysis. Rare Event Simulation Large deviations can be used to estimate the probability of rare events, such as buffer overflow, in queueing networks. It is simple enough to be applied to very general traffic models, and sophisticated enough to give insight into complex behavior. Simulation has numerous advantages over other approaches to performance and dependability evaluation most notably, its modelling power and flexibility. For some models, however, a potential problem is the excessive simulation effort (time) required to achieve the desired accuracy. In particular, simulation of models involving rare events, such as those used for the evaluation of communications and highly-dependable systems, is often not feasible using standard techniques. In recent years, there have been significant theoretical and practical advances towards the development of efficient simulation techniques for the evaluation of these systems. Methodologies include: Techniques based on importance sampling, The restart method, and Hybrid analyticsimulation techniques among newly devised approaches. Conclusions Further Readings With the growing incidence of computer modeling and simulation, the scope of simulation domain must be extended to include much more than traditional optimization techniques. Optimization techniques for simulation must also account specifically for the randomness inherent in estimating the performance measure and satisfying the constraints of stochastic systems. We described the most widely used optimization techniques that can be effectively integrated with a simulation model. We also described techniques for post-solution analysis with the aim of theoretical unification of the existing techniques. All techniques were presented in step-by-step format to facilitate implementation in a variety of operating systems and computers, thus improving portability. General comparisons among different techniques in terms of bias, variance, and computational complexity are not possible. However, a few studies rely on real computer simulations to compare different techniques in terms of accuracy and number of iterations. Total computational effort for reduction in both the bias andvariance of the estimate depends on the computational budget allocated for a simulation optimization. No single technique works effectively andor efficiently in all cases. The simplest technique is the random selection of some points in the search region for estimating the performance measure. In this technique, one usually fixes the number of simulation runs and takes the smallest (or largest) estimated performance measure as the optimum. This technique is useful in combination with other techniques to create a multi-start technique for global optimization. The most effective technique to overcome local optimality for discrete optimization is the Tabu Search technique. In general, the probabilistic search techniques, as a class, offer several advantages over other optimization techniques based on gradients. In the random search technique, the objective function can be non-smooth or even have discontinuities. The search program is simple to implement on a computer, and it often shows good convergence characteristics in noisy environments. More importantly, it can offer the global solution in a multi-modal problem, if the technique is employed in the global sense. Convergence proofs under various conditions are given in. The Hooke-Jeeves search technique works well for unconstrained problems with less than 20 variables pattern search techniques are more effective for constrained problems. Genetic techniques are most robust and can produce near-best solutions for larger problems. The pattern search technique is most suitable for small size problems with no constraint, and it requires fewer iterations than the genetic techniques. The most promising techniques are the stochastic approximation, simultaneous perturbation, and the gradient surface methods. Stochastic approximation techniques using perturbation analysis, score function, or simultaneous perturbation gradient estimators, optimize a simulation model in a single simulation run. They do so by observing the sample values of the objective function, and based on these observations, the stochastic approximation techniques update the values of the controllable parameters while the simulation is running and without restarting the simulation. This observing-updating sequence, done repeatedly, leads to an estimate of the optimum at the end of a single-run simulation. Besides having the potential of large savings in computational effort in the simulation environment, this technique can be a powerful tool in real-time optimization and control, where observations are taken as the system is evolving over time. Response surface methods have a slow convergence rate, which makes them expensive. The gradient surface method combines the advantages of the response surface methods (RSM) and efficiency of the gradient estimation techniques, such as infinitesimal perturbation analysis, score function, simultaneous perturbation analysis, and frequency domain technique. In the gradient surface method (GSM) the gradient is estimated, and the performance gradient surface is estimated from observations at various points, similar to the RSM. Zero points of the successively approximating gradient surface are then taken as the estimates of the optimal solution. GSM is characterized by several attractive features: it is a single run technique and more efficient than RSM at each iteration step, it uses the information from all of the data points rather than just the local gradient it tries to capture the global features of the gradient surface and thereby quickly arrive in the vicinity of the optimal solution, but close to the optimum, they take many iterations to converge to stationary points. Search techniques are therefore more suitable as a second phase. The main interest is to figure out how to allocate the total available computational budget across the successive iterations. For when the decision variable is qualitative, such as finding the best system configuration, a random or permutation test is proposed. This technique starts with the selection of an appropriate test statistic, such as the absolute difference between the mean responses under two scenarios. The test value is computed for the original data set. The data are shuffled (using a different seed) the test statistic is computed for the shuffled data and the value is compared to the value of the test statistic for the original, un-shuffled data. If the statistics for the shuffled data are greater than or equal to the actual statistic for the original data, then a counter c, is incremented by 1. The process is repeated for any desired m number of times. The final step is to compute (c1)(m1), which is the significant level of the test. The null hypothesis is rejected if this significance level is less than or equal to the specified rejection level for the test. There are several important aspects to this nonparametric test. First, it enables the user to select the statistic. Second, assumptions such as normality or equality of variances made for the t-test, ranking-and-selection, and multiple-comparison procedures, are no longer needed. A generalization is the well-known bootstrap technique. What Must Be Done computational studies of techniques for systems with a large number of controllable parameters and constraints. effective combinations of several efficient techniques to achieve the best results under constraints on computational resources. development of parallel and distributed schemesdevelopment of an expert system that incorporates all available techniques. References Further Readings: Arsham H. Techniques for Monte Carlo Optimizing, Monte Carlo Methods and Applications . 4(3), 181-230, 1998. Arsham H. Stochastic Optimization of Discrete Event Systems Simulation, Microelectronics and Reliability . 36(10), 1357-1368, 1996. Fu M. and J-Q. Hu, Conditional Monte Carlo: Gradient Estimation and Optimization Applications . Kluwer Academic Publishers, 1997. Rollans S. and D. McLeish, Estimating the optimum of a stochastic system using simulation, Journal of Statistical Computation and Simulation . 72, 357 - 377, 2002. Rubinstein R. and A. Shapiro, Discrete Event Systems: Sensitivity Analysis and Stochastic Optimization by the Score Function Method . John Wiley Sons, 1993. Metamodeling and the Goal seeking Problems The simulation models although simpler than the real-world system, are still a very complex way of relating input (v) to output J(v). Sometimes a simpler analytic model may be used as an auxiliary to the simulation model. This auxiliary model is often referred to as a metamodel. In many simulation applications such as systems analysis and design applications, the decision maker may not be interested in optimization but wishes to achieve a certain value for J(v), say J 0 . This is the goal-seeking problem. given a target output value J 0 of the performance and a parameterized pdf family, one must find an input value for the parameter, which generates such an output. Metamodeling The simulation models although simpler than the real-world system, are still a very complex way of relating input (v) to output J(v). Sometimes a simpler analytic model may be used as an auxiliary to the simulation model. This auxiliary model is often referred to as a metamodel. There are several techniques available for metamodeling including: design of experiments, response surface methodology, Taguchi methods, neural networks, inductive learning, and kriging. Metamodeling may have different purposes: model simplification and interpretation, optimization, what-if analysis, and generalization to models of the same type. The following polynomial model can be used as an auxiliary model. where d v v-v 0 and the primes denote derivatives. This metamodel approximates J(v) for small d v. To estimate J(v) in the neighborhood of v 0 by a linear function, we need to estimate the nominal J(v) and its first derivative. Traditionally, this derivative is estimated by crude Monte Carlo i. e. finite difference which requires rerunning the simulation model. Methods which yield enhanced efficiency and accuracy in estimating, at little additional computational (Not simulation) cost, are presented in this site. The Score Function method of estimating the first derivative is: where Sf(y v) f(y v)d Lnf(y v) dv is the Score function and differentiations is with respect to v, provided that, f(y v) exist, and f(y v) is positive for all v in V. The Score function approach can be extended in estimating the second and higher order of derivatives. For example, an estimate for the second derivative based on the Score Function method is: Where S and H S S 2 are the score and information functions, respectively, widely used in statistics literature, such as in the construction of Cramer-Rao bounds. By having gradient and Hessian in our disposal, we are able to construct a second order local metamodel using the Taylors series. An Illustrative Numerical Example: For most complex reliability systems, the performance measures such as mean time to failure (MTTF) are not available in analytical form. We resort to Monte Carlo Simulation (MCS) to estimate MTTF function from a family of single-parameter density functions of the components life with specific value for the parameter. The purpose of this section is to solve the inverse problem, which deals with the calculation of the components life parameters (such as MTTF) of a homogeneous subsystem, given a desired target MTTF for the system. A stochastic approximation algorithm is used to estimate the necessary controllable input parameter within a desired range of accuracy. The potential effectiveness is demonstrated by simulating a reliability system with a known analytical solution. Consider the coherent reliability sub-system with four components component 1, and 2 are in series, and component 3 and 4 also in series, however these two series of components are in parallel, as illustrated in the following Figure. All components are working independently and are homogeneous i. e. manufactured by an identical process, components having independent random lifetimes Y1, Y2, Y3, and Y4, which are distributed exponentially with rates v v 0 0.5. The system lifetime is Z (Y1,Y2,Y3,Y4 v 0 ) max min (Y3,Y4), min (Y1,Y2). It is readily can be shown that the theoretical expected lifetime of this sub-system is The underlying pdf for this system is: f(y v) v 4 exp(-v S y i ), the sum is over i 1, 2, 3, 4. Applying the Score function method, we have: S(y) f (y v) f(y v) 4v - S y i . the sum is over i 1, 2, 3, 4. H(y) f (y v) f(y v) v 2 ( S y i ) 2 - 8v ( S y i ) 12 v 2 , the sums are over i 1, 2, 3, 4. The estimated average lifetime and its derivative for the nominal system with v v 0 0.5, are: respectively, where Y i, j is the j th observation for the i th component (i 1, 2, 3, 4). We have performed a Monte Carlo experiment for this system by generating n 10000 independent replications using SIMSCRIPT II.5 random number streams 1 through 4 to generate exponential random variables Y1, Y2, Y3, Y4. respectively, on a VAX system. The estimated performance is J(0.5) 1.5024, with a standard error of 0.0348. The first and second derivatives estimates are -3.0933 and 12.1177 with standard errors of 0.1126 and 1.3321, respectively. The response surface approximation in the neighborhood of v 0.5 is: J(v) 1.5024 (v - 0.5) (-3.0933) (v - 0.5) 2 (12.1177)2 6.0589v 2 - 9.1522v 4.5638 A numerical comparison based on exact and the approximation by this metamodel reveals that the largest absolute error is only 0.33 for any v in the range of 0.40, 0.60. This error could be reduced by either more accurate estimates of the derivatives andor using a higher order Taylor expansion. A comparison of the errors indicates that the errors are smaller and more stable in the direction of increasing v. This behavior is partly due to the fact that lifetimes are exponentially distributed with variance 1v. Therefore, increasing v causes less variance than the nominal system (with v 0.50). Goal seeking problem In many systems modeling and simulation applications, the decision maker may not be interested in optimization but wishes to achieve a certain value for J(v), say J 0 . This is the goal-seeking problem. given a target output value J 0 of the performance and a parameterized pdf family, one must find an input value for the parameter, which generates such an output. When is a controllable input, the decision maker may be interested in the goal-seeking problem: namely, what change of the input parameter will achieve a desired change in the output value. Another application of the goal-seeking problem arises when we want to adapt a model to satisfy a new equality constraint with some stochastic functions. We may apply the search techniques, but the goal-seeking problem can be considered as an interpolation based on a meta-model. In this approach, one generates a response surface function for J(v). Finally, one uses the fitted function to interpolate for the unknown parameter. This approach is tedious, time-consuming, and costly moreover, in a random environment, the fitted model might have unstable coefficients. For a given J(v) the estimated d v, using the first order approximation is: provided that the denominator does not vanish for all v 0 in set V. The Goal-seeker Module: The goal-seeking problem can be solved as a simulation problem. By this approach, we are able to apply variance reduction techniques (VRT) used in the simulation literature. Specifically, the solution to the goal-seeking problem is the unique solution of the stochastic equation J(v) - J 0 0. The problem is to solve this stochastic equation by a suitable experimental design, to ensure convergence. The following is a Robbins - Monro (R-M) type technique. where d j is any divergent sequence of positive numbers. Under this conditions, d v J 0 - J(v j ) converges to approach zero while dampening the effect of the simulation random errors. These conditions are satisfied, for example, by the harmonic sequence d j 1j. With this choice, the rate of reduction of di is very high initially but may reduce to very small steps as we approach the root. Therefore, a better choice is, for example d j 9 (9 j). This technique involves placing experiment i1 according to the outcome of experiment i immediately preceding it, as is depicted in the following Figure: Under these not unreasonable conditions, this algorithm will converge in mean square moreover, it is an almost sure convergence. Finally, as in Newtons root-finding method, it is impossible to assert that the method converges for just any initial v v 0 . even though J(v) may satisfy the Lipschits condition over set V. Indeed, if the initial value v 0 is sufficiently close to the solution, which is usually the case, then this algorithm requires only a few iterations to obtain a solution with very high accuracy. An application of the goal-seeker module arises when we want to adapt a model to satisfy a new equality constraint (condition) for some stochastic function. The proposed technique can also be used to solve integral equations by embedding the Importance Sampling techniques within a Monte Carlo sampling. One may extend the proposed methodology to the inverse problems with two or more unknown parameters design by considering two or more relevant outputs to ensure uniqueness. By this generalization we could construct a linear (or even nonlinear) system of stochastic equations to be solved simultaneously by a multidimensional version of the proposed algorithm. The simulation design is more involved for problems with more than a few parameters. References and Further Readings: Arsham H. The Use of Simulation in Discrete Event Dynamic Systems Design, Journal of Systems Science . 31(5), 563-573, 2000. Arsham H. Input Parameters to Achieve Target Performance in Stochastic Systems: A Simulation-based Approach, Inverse Problems in Engineering . 7(4), 363-384, 1999. Arsham H. Goal Seeking Problem in Discrete Event Systems Simulation, Microelectronics and Reliability . 37(3), 391-395, 1997. Batmaz I. and S. Tunali, Small response surface designs for metamodel estimation, European Journal of Operational Research . 145(3), 455-470, 2003. Ibidapo-Obe O. O. Asaolu, and A. Badiru, A New Method for the Numerical Solution of Simultaneous Nonlinear Equations, Applied Mathematics and Computation . 125(1), 133-140, 2002. Lamb J. and R. Cheng, Optimal allocation of runs in a simulation metamodel with several independent variables, Operations Research Letters . 30(3), 189-194, 2002. Simpson T. J. Poplinski, P. Koch, and J. Allen, Metamodels for Computer-based Engineering Design: Survey and Recommendations, Engineering with Computers . 17(2), 129-150, 2001. Tsai C-Sh. Evaluation and optimisation of integrated manufacturing system operations using Taguchs experiment design in computer simulation, Computers And Industrial Engineering . 43(3), 591-604, 2002. What-if Analysis Techniques Introduction The simulation models are often subject to errors caused by the estimated parameter(s) of underlying input distribution function. What-if analysis is needed to establish confidence with respect to small changes in the parameters of the input distributions. However the direct approach to what-if analysis requires a separate simulation run for each input value. Since this is often inhibited by cost, as an alternative, what people are basically doing in practice is to plot results and use a simple linear interpolationextrapolation. This section presents some simulation-based techniques that utilize the current information for estimating performance function for several scenarios without any additional simulation runs. Simulation continues to be the primary method by which system analysts obtain information about analysis of complex stochastic systems. In almost all simulation models, an expectedvalue can express the systems performance. Consider a system with continuous parameter v 206 V, where V is the feasible region. Let be the steady state expected performance measure, where Y is a random vector with known probability density function (pdf), f(y v) depends on v, and Z is the performance measure. In discrete event systems, Monte Carlo simulation is usually needed to estimate J(v) for a given value v. By the law of large numbers where y i . i 1, 2. n are independent, identically distributed, random vector realizations of Y from f (y v ), and n is the number of independent replications. This is an unbiased estimator for J(v) and converges to J(v) by law of large numbers. There are strong motivations for estimating the expected performance measure J(v) for a small change in v to v d v, that is to solve the so-called what if problem. The simulationist must meet managerial demands to consider model validation and cope with uncertainty in the estimation of v. Adaptation of a model to new environments also requires an adjustment in v. An obvious solution to the what if problem is the Crude Monte Carlo (CMC) method, which estimates J(v d v) for each v separately by rerunning the system for each v d v. Therefore costs in CPU time can be prohibitive The use of simulation as a tool to design complex computer stochastic systems is often inhibited by cost. Extensive simulation is needed to estimate performance measures for changes in the input parameters. As as an alternative, what people are basically doing in practice is to plot results of a few simulation runs and use a simple linear interpolationextrapolation. In this section we consider the What-if analysis problem by extending the information obtained from a single run at the nominal value of parameter v to the closed neighborhood. We also present the use of results from runs at two or more points over the intervening interval. We refer to the former as extrapolation and the latter as interpolation by simulation. The results are obtained by some computational cost as opposed to simulation cost . Therefore, the proposed techniques are for estimating a performance measure at multiple settings from a simulation at a nominal value. Likelihood Ratio (LR) Method A model based on Radon-Nikodym theorem to estimate J(v d v) for stochastic systems in a single run is as follows: where the likelihood ratio W is: W f(y v d v) f(y v) adjusts the sample path, provided f(y v) does not vanish. Notice that by this change of probability space, we are using the common realization as J(v). The generated random vector y is roughly representative of Y, with f(v). Each of these random observations, could also hypothetically came from f(v d v). W weights the observations according to this phenomenon. Therefore, the What-if estimate is: which is based on only one sample path of the system with parameter v and the simulation for the system with v d v is not required. Unfortunately LR produces a larger variance compared with CMC. However, since E(W)1, the following variance reduction techniques (VRT) may improve the estimate. Exponential Tangential in Expectation Method In the statistical literature the efficient score function is defined to be the gradient S(y) d Ln f(y v) dv We consider the exponential (approximation) model for J(v d v) in a first derivative neighborhood of v by: J(v d v) E Z(y). exp d vS(y) Eexp( d S(y)) Now we are able to estimate J(v d v) based on n independent replications as follows: Taylor Expansion of Response Function The following linear Taylor model can be used as an auxiliary model. J(v d v) J(v) d v. J (v) . where the prime denotes derivative. This metamodel approximates J(v d v)) for small d v. For this estimate, we need to estimate the nominal J(v) and its first derivative. Traditionally, this derivative is estimated by crude Monte Carlo i. e. finite difference, which requires rerunning the simulation model. Methods which yield enhanced efficiency and accuracy in estimating, at little additional cost, are of great value. There are few ways to obtain efficiently the derivatives of the output with respect to an input parameter as presented earlier on this site. The most straightforward method is the Score Function (SF). The SF approach is the major method for estimating the performance measure and its derivative, while observing only a single sample path from the underlying system. The basic idea of SF is that the derivative of the performance function, J(v), is expressed as expectation with respect to the same distribution as the performance measure itself. Therefore, for example, using the estimated values of J(v) and its derivative J(v), the estimated J(v d v) is: VarJ(v d v) VarJ(v) ( d v) 2 VarJ(v) 2 d v CovJ(v), J(v). This variation is needed for constructing a confidence interval for the perturbed estimate. Interpolation Techniques Given two points, v1 and v2 (scalars only) sufficiently close, one may simulate at these two points then interpolates for any desired points in between. Assuming the given v1 and v2 are sufficiently close and looks for the best linear interpolation in the sense of minimum error on the interval. Clearly, Similar to the Likelihood Ratio approach, this can be written as: where the likelihood ratios W1 and W2 are W1 f(y v) f(y v1) and W2 f(y v) f(y v2), respectively. One obvious choice is f f(y v1) f(y v1)f(y v2). This method can easily extended to k-point interpolation. For 2-point interpolation, if we let f to be constant within the interval 0, 1, then the linear interpolated what-if estimated value is: where the two estimates on the RHS of are two independent Likelihood Ratio extrapolations using the two end-points. We define f as the f in this convex combination with the minimum error in the estimate. That is, it minimizes By the first order necessary and sufficient conditions, the optimal f is: Thus, the best linear interpolation for any point in interval v1, v2 is: which is the optimal interpolation in the sense of having minimum variance. Conclusions Further Readings Estimating system performance for several scenarios via simulation generally requires a separate simulation run for each scenario. In some very special cases, such as the exponential density f(y v)ve - vy. one could have obtained the perturbed estimate using Perturbation Analysis directly as follow. Clearly, one can generate random variate Y by using the following inverse transformation: where Ln is the natural logarithm and U i is a random number distributed Uniformly 0,1. In the case of perturbed v, the counterpart realization using the same U i is Clearly, this single run approach is limited, since the inverse transformation is not always available in closed form. The following Figure illustrates the Perturbation Analysis Method: Since the Perturbation Analysis Approach has this serious limitation, for this reason, we presented some techniques for estimating performance for several scenarios using a single-sample path, such as the Likelihood Ratio method, which is illustrated in the following Figure. Research Topics: Items for further research include: i) to introduce efficient variance reduction and bias reduction techniques with a view to improving the accuracy of the existing and the proposed methods ii) to incorporate the result of this study in a random search optimization technique. In this approach one can generate a number of points in the feasible region uniformly distributed on the surface of a hyper-sphere each stage the value of the performance measure is with a specified radius centered at a starting point. At estimated at the center (as a nominal value). Perturbation analysis is used to estimate the performance measure at the sequence of points on the hyper-sphere. The best point (depending whether the problem is max or min) is used as the center of a smaller hyper - sphere. Iterating in this fashion one can capture the optimal solution within a hyper-sphere with a specified small enough radius. Clearly, this approach could be considered as a sequential self-adaptive optimization technique. iii) to estimate the sensitivities i. e. the gradient, Hessian, etc. of J(v) can be approximated using finite difference. For example the first derivative can be obtained in a single run using the Likelihood Ratio method as follows: the sums are over all i, i 1, 2, 3. n, where The last two estimators may induce some variance reductions. iv) Other interpolation techniques are also possible. The most promising one is based on Kriging. This technique gives more weight to neighboring realizations, and is widely used in geo-statistics. Other items for further research include some experimentation on large and complex systems such as a large Jacksonian network with routing that includes feedback loops in order to study the efficiency of the presented technique. References Further Readings: Arsham H. Performance Extrapolation in Discrete-event Systems Simulation, Journal of Systems Science . 27(9), 863-869, 1996. Arsham H. A Simulation Technique for Estimation in Perturbed Stochastic Activity Networks, Simulation . 58(8), 258-267, 1992. Arsham H. Perturbation Analysis in Discrete-Event Simulation, Modelling and Simulation . 11(1), 21-28, 1991. Arsham H. What-if Analysis in Computer Simulation Models: A Comparative Survey with Some Extensions, Mathematical and Computer Modelling . 13(1), 101-106, 1990. Arsham H. Feuerverger, A. McLeish, D. Kreimer J. and Rubinstein R. Sensitivity analysis and the what-if problem in simulation analysis, Mathematical and Computer Modelling . 12(1), 193-219, 1989. PDF Version The Copyright Statement: The fair use, according to the 1996 Fair Use Guidelines for Educational Multimedia. of materials presented on this Web site is permitted for non-commercial and classroom purposes only. 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