Abstract
In this paper, we deal with the lumpability approach to cope with the state space explosion problem inherent to the computation of the stationary performance indices of large stochastic models. The lumpability method is based on a state aggregation technique and applies to Markov chains exhibiting some structural regularity. Moreover, it allows one to efficiently compute the exact values of the stationary performance indices when the model is actually lumpable. The notion of quasi-lumpability is based on the idea that a Markov chain can be altered by relatively small perturbations of the transition rates in such a way that the new resulting Markov chain is lumpable. In this case, only upper and lower bounds on the performance indices can be derived. Here, we introduce a novel notion of quasi-lumpability, named proportional lumpability, which extends the original definition of lumpability but, differently from the general definition of quasi-lumpability, it allows one to derive exact stationary performance indices for the original process. We then introduce the notion of proportional bisimilarity for the terms of the performance process algebra PEPA. Proportional bisimilarity induces a proportional lumpability on the underlying continuous-time Markov chains. Finally, we prove some compositionality results and show the applicability of our theory through examples.
Highlights
In the context of performance evaluation of computer systems, continuous-time Markov chains (CTMCs) constitute the underlying semantics model of a plethora of modelling formalism such as Stochastic Petri nets [30], Stochastic Automata Networks (SAN) [31], queuing networks [7] and a class of Markovian process algebras (MPAs), e.g. [20,21]
We introduce a novel notion of quasi-lumpability, named proportional lumpability, which extends the original definition of lumpability but, differently from the general definition of quasi-lumpability, it allows one to derive exact stationary performance indices for the original process when the stationary probabilities of the perturbed one are known
In [26], we introduced a novel notion of lumpability, named proportional lumpability that as the notion of quasi-lumpability, extends the original definition of ordinary lumpability but, differently from the general definition of quasi-lumpability, it allows us to derive an exact solution of the original process
Summary
In the context of performance evaluation of computer systems, continuous-time Markov chains (CTMCs) constitute the underlying semantics model of a plethora of modelling formalism such as Stochastic Petri nets [30], Stochastic Automata Networks (SAN) [31], queuing networks [7] and a class of Markovian process algebras (MPAs), e.g. [20,21]. A structural-based approach to lumping for SPNs is studied in [3,5], where structural symmetries of the net are exploited to derive a lumped underlying CTMC in an efficient way These works share with our results the consequence of reducing the computational complexity of the stationary performance indices of the model. The notion of strong equivalence introduced in [21] for processes expressed as terms of the Performance Evaluation Process Algebra (PEPA) always induces a lumping of the CTMC underlying a PEPA process, in general the opposite is not true With respect to these works, our results on PEPA models generalize the notion of strong equivalence and allow us to deal with underlying Markov chains that are not directly lumpable.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
