Deriving the performance indices in product-form stochastic Petri nets: Open problems and simulation

Stochastic Petri nets are an important formalism used for the performance evaluation of computer and communication systems as well as other fields like bioin-formatics and logistics. Despite its high flexibility and modelling power, one of the problems of quantitative analyses based on stochastic Petri nets is the state space explosion, i.e., the high cardinality reached by the state space of even a structurally small SPN. As a consequence a direct analysis of the Markovian processes underlying the models is not feasible. Product-form Petri nets are a class of stochastic Petri nets whose invariant measure can be expressed as a product of functions, each of which depends only on a marking of a single place. Nevertheless, for the effective computation of the performance indices the computation of the stationary distribution is required. In this paper we propose a classification of product-form stochastic Petri nets based on the availability of algorithms for the computation of their stationary performance indices. Moreover, in case simulation is required, we introduce two stopping criteria that exploit the product-form property of the nets

Performance Bounds for Synchronized Queueing Networks

Las redes de Petri estocásticas constituyen un modelo unificado de las diferentes extensiones de redes de colas con sincronizaciones existentes en la literatura, válido para el diseño y análisis de prestaciones de sistemas informáticos distribuidos. En este trabajo se proponen técnicas de cálculo de cotas superiores e inferiores de las prestaciones de redes de Petri estocásticas en estado estacionario. Las cotas obtenidas son calculables en tiempo polinómico en el tamaño del modelo, por medio de la resolución de ciertos problemas de programación lineal definidos a partir de la matriz de incidencia de la red (en este sentido, las técnicas desarrolladas pueden considerarse estructurales). Las cotas calculadas dependen sólamente de los valores medios de las variables aleatorias que describen la temporización del sistema, y son independientes de los momentos de mayor orden. Esta independencia de la forma de las distribuciones de probabilidad asociadas puede considerarse como una útil generalización de otros resultados existentes para distribuciones particulares, puesto que los momentos de orden superior son, habitualmente, desconocidos en la realidad y difíciles de estimar. Finalmente, las técnicas desarrolladas se aplican al análisis de diferentes ejemplos tomados de la literatura sobre sistemas informáticos distribuidos y sistemas de fabricación. ******* Product form queueing networks have long been used for the performance evaluation of computer systems. Their success has been due to their capability of naturally expressing sharing of resources and queueing, that are typical situations of traditional computer systems, as well as to their efficient solution algorithms, of polynomial complexity on the size of the model. Unfortunately, the introduction of synchronization constraints usually destroys the product form solution, so that general concurrent and distributed systems are not easily studied with this class of models. Petri nets have been proved specially adequate to model parallel and distributed systems. Moreover, they have a well-founded theory of analysis that allows to investigate a great number of qualitative properties of the system. In the original definition, Petri nets did not include the notion of time, and tried to model only the logical behaviour of systems by describing the causal relations existing among events. This approach showed its power in the specification and analysis of concurrent systems in a way independent of the concept of time. Nevertheless the introduction of a timing specification is essential if we want to use this class of models for the performance evaluation of distributed systems. One of the main problems in the actual use of timed and stochastic Petri net models for the quantitative evaluation of large systems is the explosion of the computational complexity of the analysis algorithms. In general, exact performance results are obtained from the numerical solution of a continuous time Markov chain, whose dimension is given by the size of the state space of the model. Structural computation of exact performance measures has been possible for some subclasses of nets such as those with state machine topology. These nets, under certain assumptions on the stochastic interpretation are isomorphic to Gordon and Newell's networks, in queueing theory terminology. In the general case, efficient methods for the derivation of performance measures are still needed. Two complementary approaches to the derivation of exact measures for the analysis of distributed systems are the utilization of approximation techniques and the computation of bounds. Approximate values for the performance parameters are in general more efficiently derived than the exact ones. On the other hand, "exactness" only exists in theory! In other words, numerical algorithms must be applied in practice for the computation of exact values, therefore making errors is inevitable. Performance bounds are useful in the preliminary phases of the design of a system, in which many parameters are not known accurately. Several alternatives for those parameters should be quickly evaluated, and rejected those that are clearly bad. Exact (and even approximate) solutions would be computationally very expensive. Bounds become useful in these instances since they usually require much less computation effort. The computation of upper and lower bounds for the steady-state performance of timed and stochastic Petri nets is considered in this work. In particular, we study the throughput of transitions, defined as the average number of firings per time unit. For this measure we try to compute upper and lower bounds in polynomial time on the size of the net model, by means of proper linear programming problems defined from the incidence matrix of the net (in this sense, we develop structural techniques). These bounds depend only on the mean values and not on the higher moments of the probability distribution functions of the random variables that describe the timing of the system. The independence of the probability distributions can be viewed as a useful generalization of the performance results, since higher moments of the delays are usually unknown for real cases, and difficult to estimate and assess. From a different perspective, the obtained results can be applied to the analysis of queueing networks extended with some synchronization schemes. Monoclass queueing networks can be mapped on stochastic Petri nets. On the other hand, stochastic Petri nets can be interpreted as monoclass queueing networks augmented with synchronization primitives. Concerning the presentation of this manuscript, it should be mentioned that chapter 1 has been written with the object of giving the reader an outline of the stochastic Petri net model: its definition, terminology, basic properties, and related concepts, together with its deep relation with other classic stochastic network models. Chapter 2 is devoted to the presentation of the net subclasses considered in the rest of the work. The classification presented here is quite different from the one which is usual in the framework of Petri nets. The reason lies on the fact that our classification criterion, the computability of visit ratios for transitions, is introduced for the first time in the field of stochastic Petri nets in this work. The significance of that criterion is based on the important role that the visit ratios play in the computation of upper and lower bounds for the performance of the models. Nevertheless, classical important net subclasses are identified here in terms of the computability of their visit ratios from different parameters of the model. Chapter 3 is concerned with the computation of reachable upper and lower bounds for the most restrictive subclass of those presented in chapter 2: marked graphs. The explanation of this fact is easy to understand. The more simple is the model the more accessible will be the techniques an ideas for the development of good results. Chapter 4 provides a generalization for live and bounded free choice nets of the results presented in the previous chapter. Quality of obtained bounds is similar to that for strongly connected marked graphs: throughput lower bounds are reachable for bounded nets while upper bounds are reachable for 1-bounded nets. Chapter 5 considers the extension to other net subclasses, like mono-T-semiflow nets, FRT-nets, totally open deterministic systems of sequential processes, and persistent nets. The results are of diverse colours. For mono-T-semiflow nets and, therefore, for general FRT-nets, it is not possible (so far) to obtain reachable throughput bounds. On the other hand, for bounded ordinary persistent nets, tight throughput upper bounds are derived. Moreover, in the case of totally open deterministic systems of sequential processes the exact steady-state performance measures can be computed in polynomial time on the net size. In chapter 6 bounds for other interesting performance measures are derived from throughput bounds and from classical queueing theory laws. After that, we explore the introduction of more information from the probability distribution functions of service times in order to improve the bounds. In particular, for Coxian service delay of transitions it is possible to improve the throughput upper bounds of previous chapters which held for more general forms of distribution functions. This improvement shows to be specially fruitful for live and bounded free choice nets. Chapter 7 is devoted to case studies. Several examples taken from literature in the fields of distributed computing systems and manufacturing systems are modelled by means of stochastic Petri nets and evaluated using the techniques developed in previous chapters. Finally, some concluding remarks and considerations on possible extensions of the work are presented

Investigation of a Neural Network Methodology to Predict Transient Performance in Fms

Most rapid analytical evaluative models for Flexible Manufacturing Systems (FMSs) are based on the steady-state performance. There is a practical need to develop robust, easy to construct, and transportable transient-state evaluative models for FMSs. This study proposes an ANN based metamodeling framework that can capture various post disruption system behaviors of FMS. The proposed ANN based meta-modeling scheme consists of a hierarchical taxonomy of mutilple ANNs. Each set of ANNs collectively represents a different part of the underlying system modeling domain. The taxonomical arrangement of multiple ANNs overcomes shortcomings often found in single ANN based meta-modeling schemes. These shortcomings are generally related to the limited knowledge acquisition capability of these schemes. The study uses an Extend based discrete simulation model that is built after an experimental FMS with a limited disruption trigger and handling capabilities. The simulation model is used to study various post-disruption behaviors by a given FMS and to study the feasibility of the proposed modeling scheme as a viable means to provide "lookahead" capability for a low level controller.Findings and Conclusions: The proposed ANN based metamodeling approach using multiple ANNs, in a taxonomically organized modeling structure, is an efficient way to capture multiple target performance index observation processes with a similar overall post-disruption behavior pattern. Despite its accuracy issues, this methodology was proven especially effective when it has to deal with noisy time series such as TIS at observation under a data rich environment. The study is to prove that the proposed methodology could be a viable means to model transient system behaviors. As long as individual observation processes of the selected performance index can keep their variances smaller among themselves, the accuracy of the overall model would be acceptable. This non-parametric performance modeling technique using hierarchically organized multiple ANNs, is worth further investigation.Industrial Engineering & Managemen

Methodologies synthesis

This deliverable deals with the modelling and analysis of interdependencies between critical infrastructures, focussing attention on two interdependent infrastructures studied in the context of CRUTIAL: the electric power infrastructure and the information infrastructures supporting management, control and maintenance functionality. The main objectives are: 1) investigate the main challenges to be addressed for the analysis and modelling of interdependencies, 2) review the modelling methodologies and tools that can be used to address these challenges and support the evaluation of the impact of interdependencies on the dependability and resilience of the service delivered to the users, and 3) present the preliminary directions investigated so far by the CRUTIAL consortium for describing and modelling interdependencies