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Bulò. A general algorithm to compute the steadystate solution of productform cooperating Markov chains
 In Proc. of MASCOTS 2009
, 2009
"... Abstract—In the last few years several new results about productform solutions of stochastic models have been formulated. In particular, the Reversed Compound Agent Theorem (RCAT) and its extensions play a pivotal role in the characterization of cooperating stochastic models in productform. Alth ..."
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Abstract—In the last few years several new results about productform solutions of stochastic models have been formulated. In particular, the Reversed Compound Agent Theorem (RCAT) and its extensions play a pivotal role in the characterization of cooperating stochastic models in productform. Although these results have been used to prove several wellknown theorems (e.g., Jackson queueing network and Gnetwork solutions) as well as novel ones, to the best of our knowledge, an automatic tool to derive the productform solution (if present) of a generic cooperation among a set of stochastic processes, is not yet developed. In this paper we address the problem of solving the nonlinear system of equations that arises from the application of RCAT. We present an iterative algorithm that is the base of a software tool currently under development. We illustrate the algorithm, discuss the convergence and the complexity, compare it with previous algorithms defined for the analysis of the Jackson networks and the Gnetworks. Several tests have been conducted involving the solutions of a (arbitrary) large number of cooperating processes in productform by RCAT. I.
Methodological Construction of ProductForm Stochastic PetriNets for Performance Evaluation
"... obtained by a compositional technique for the first time, by combining small SPNs with productforms in a hierarchical manner. In this way, performance engineering methodology is enhanced by the greatly improved efficiency endowed to the steady state solution of a much wider range of Markov models. ..."
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obtained by a compositional technique for the first time, by combining small SPNs with productforms in a hierarchical manner. In this way, performance engineering methodology is enhanced by the greatly improved efficiency endowed to the steady state solution of a much wider range of Markov models. Previous methods have relied on analysis of the whole net and so are not incremental – hence they are intractable in all but small models. We show that the productform condition for open nets depends, in general, on the transition rates, whereas closed nets have only structural conditions for a productform, except in rather pathological cases. Both the “building blocks ” formed by the said small SPNs and their compositions are solved for their productforms using the Reversed Compound Agent Theorem (RCAT), which, to date, has been used exclusively in the context of processalgebraic models. The resulting methodology provides a powerful, general and rigorous route to productforms in large stochastic models and is illustrated by several detailed examples. I.
doi:10.1093/comjnl/bxp021 Turning Back Time—What Impact on Performance?
, 2009
"... Consistent with the divideandconquer approach to problem solving, a recursive result is presented in the domain of stochastic modelling that derives productform solutions for the steady state probabilities of certain networks composed from interacting Markov chains. Practical applications include ..."
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Consistent with the divideandconquer approach to problem solving, a recursive result is presented in the domain of stochastic modelling that derives productform solutions for the steady state probabilities of certain networks composed from interacting Markov chains. Practical applications include multitasking operating systems, communication channels and multitiered storage systems. The approach is also applied to the computation of response time quantiles, which are vital in transaction processing, computer communication service level agreements and other operational systems. The joint probability distribution of the sojourn times of a tagged task at each node in a network is determined by noting that this is the same in both the forward and reversed processes. In this way, existing results for response time probability densities in tandem, treelike and overtakefree Markovian queueing networks are quickly and systematically obtained. We further show how to apply the method in more general networks.
AutoCAT: Automated ProductForm Solution of Stochastic Models
"... Abstract We propose algorithms to automatically generate exact and approximate productform solutions for large Markov processes that cannot be solved by direct numerical methods. Focusing on models that can be described as cooperating Markov processes, which include queueing networks and stochastic ..."
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Abstract We propose algorithms to automatically generate exact and approximate productform solutions for large Markov processes that cannot be solved by direct numerical methods. Focusing on models that can be described as cooperating Markov processes, which include queueing networks and stochastic Petri nets as special cases, it is shown that finding a global optimum for a nonconvex quadratic program is sufficient for determining a productform solution. Such problems are notoriously hard to solve due to the inherent difficulty of searching over nonconvex sets. Using a potential theory for Markov processes, convexification techniques and a class of linear constraints that follow from stochastic characterization of productform solutions, we obtain a family of linear programming relaxations that can be solved efficiently. A sequence of these linear programs is solved under increasingly tighter constraints to determine the exact productform solution of a model when one exists. This approach is then extended to obtain approximate solutions for nonproductform models. Finally, our new techniques are validated with examples and increasingly complex case studies, which show the effectiveness of the method on both conventional and novel performance models. 1
Automated productforms with MEERCAT Abstract — The Reversed Compound Agent Theorem (RCAT) is
"... a compositional result that uses Markovian process algebra to derive the reversed process of two cooperating continuous time Markov chains at equilibrium, under certain conditions. From this reversed process, together with the given, forward process, the joint state probabilities can be expressed as ..."
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a compositional result that uses Markovian process algebra to derive the reversed process of two cooperating continuous time Markov chains at equilibrium, under certain conditions. From this reversed process, together with the given, forward process, the joint state probabilities can be expressed as a productform. We introduce MEERCAT, the first implementation of the RCAT, which classifies models where the theorem can be applied, and generates their productform solutions. I.
unknown title
, 2010
"... pp. xx–xx A tool for the numerical solution of cooperating Markov chains in productform ..."
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pp. xx–xx A tool for the numerical solution of cooperating Markov chains in productform
I. SUMMARY OF THE TUTORIAL
"... Productform models are a class of Markovian models whose steadystate solution can be computed efficiently thanks to a separable equilibrium probability distribution. They first appeared in queuing theory with the wellknow result of Jackson networks. Many research efforts have been devoted to the ..."
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Productform models are a class of Markovian models whose steadystate solution can be computed efficiently thanks to a separable equilibrium probability distribution. They first appeared in queuing theory with the wellknow result of Jackson networks. Many research efforts have been devoted to the identification of new productform models both for queueing networks and other formalisms (e.g. for Markovian process algebra or for stochastic Petri nets etc.). The results presented in [1], [2], [3] allow the computation of the productform solutions of Markovian models via structural analysis of their underlying processes and constraints on the rates of the corresponding reversed processes. These results greatly simplify the derivation of productform solutions, as they offer a constructive way to establish whether a Continuous Time Markov Chain
Turning Back Time What Impact on Performance?
"... Consistent with the divideandconquer approach to problem solving, a recursive result is presented in the domain of stochastic modelling that derives productform solutions for the steady state probabilities of certain networks composed from interacting Markov chains. Practical applications includ ..."
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Consistent with the divideandconquer approach to problem solving, a recursive result is presented in the domain of stochastic modelling that derives productform solutions for the steady state probabilities of certain networks composed from interacting Markov chains. Practical applications include multitasking operating systems, communication channels and multitiered storage systems. The approach is also applied to the computation of response time quantiles, which are vital in transaction processing, computer communication service level agreements and other operational systems. The joint probability distribution of the sojourn times of a tagged task at each node in a network is determined by noting that this is the same in both the forward and reversed processes. In this way, existing results for response time probability densities in tandem, treelike, and overtakefree Markovian queueing networks are quickly and systematically obtained. We further show how to apply the method in more general networks. 1
Discussant Contributions for the Computer Journal Lecture
"... A resurgence in productforms Interest in the stochastic behaviour of queueing networks began in the 1960s with the work by Jackson, Gordon, Newell and others [1–3]. These authors derived the socalled productform solutions for the equilibrium probabilities of the joint state in such a continuous t ..."
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A resurgence in productforms Interest in the stochastic behaviour of queueing networks began in the 1960s with the work by Jackson, Gordon, Newell and others [1–3]. These authors derived the socalled productform solutions for the equilibrium probabilities of the joint state in such a continuous time Markov chain (CTMC). From the preceding lecture, of course, these networks are special cases of Gnetworks, where there are no negative customers, triggers, signals, etc. After a number of generalizations of Jackson networks in the 1970s and 1980s, it was thought by many that a productform for a stochastic network would only exist if that network satisfied a condition called partial or local balance [4, 5]. Essentially, this specifies a specific way by which the global balance equations of probability fluxes into, and out of, a given state (the steadystate