Steady-state analysis of infinite stochastic Petri nets: comparing the spectral expansion and the matrix-geometric method

Abstract

In this paper we investigate the eficiency of two solution approaches to infinite stochastic Petri nets: the matrix-geometric method and the spectral expansion method. W e first informally present infinite stochastic Petri nets, after which we describe, using uniform notation, the matrix-geometric and the spectral expansion method. W e put special emphasis on the numerical aspect of the solution procedures. Then, we investigate the suitability of these approaches t o account for batch-movements of tokens. W e then compare the two solution approaches when applied t o a larger modelling study of a fault-tolerant computer system. It turns out that the spectral expansion method is favorable in all cases, especially when more heavilg loaded systems are studied and when batch arrivals are incorporated i n the model.To the best of our knowledge, this paper is the first to compare the spectral expansion method, as advocated by Mitrani and Chakka, with the LatoucheRamaswami algorithm for the matrix-geometric case. Furthermore, our comparisons go well beyond the usual textbook cases, since we are able to generate, with our tool SPN2MGM, models that are much larger than those that have been assessed in the past.

Generalized QBD Processes, Spectral Expansion and Performance Modeling Applications

“…It is confirmed by a number of works that the spectral expansion method is better than the matrix geometric method in a number of aspects [5,32,33,39]. It is observed that the spectral expansion method is proved to be a mature technique for the performance analysis of various problems [5-17, 20-25, 27, 28, 31, 32, 38, 39, 49-51, 53, 54, 52, 58].…”

A Recurrent Solution of PH/M/c/N-Like and PH/M/c-Like Queues

“…Mitrani and Chakka [MIT95] compare the performance of the spectral expansion and matrix-geometric methods for an M/M/c-type queue. Haverkort and Ost [HAV97] also compare the spectral expansion method with the Latouche-Ramaswami algorithm for a model of a fault-tolerant system. Tran and Do [TRA00] present a comparison of the practical performance of matrix-geometric methods and of the spectral expansion for a specific quasi birth-and-death process.…”

Spectral Expansion Solutions for Markov-Modulated Queues

“…The latter arrive according to an independent Poisson process with rate ξ. When a fault occurs, the system instantaneously rolls back to the last established checkpoint; all transactions which arrived since that moment either remain in the queue, if they have not been processed, or return to it, in order to be processed again (it is assumed that repeated service times are resampled independently) (see [11,8]). This system can be modelled as an unbounded queue of (uncompleted) transactions, which is modulated by an environment consisting of completed transactions and checkpoints.…”

“…More recently, Grassmann [7] has discussed models where the eigenvalues can be isolated and determined very efficiently. Some comparisons between the spectral expansion and the matrix-geometric solutions can be found in [16] and in Haverkort and Ost [8]. The available evidence suggests that, where both methods are applicable, spectral expansion is faster even if the matrix R is computed by the most efficient algorithm.…”