Transient Analysis of Rewarded Continuous Time Markov Models by Regenerative Randomization with Laplace Transform Inversion

Abstract

In this paper we develop a variant, regenerative randomization with Laplace transform inversion, of a previously proposed method (the regenerative randomization method) for the transient analysis of rewarded continuous time Markov models. Those models find applications in dependability and performability analysis of computer and telecommunication systems. The variant differs from regenerative randomization in that the truncated transformed model obtained in that method is solved using a Laplace transform inversion algorithm instead of standard randomization. As regenerative randomization, the variant requires the selection of a regenerative state on which the performance of the method depends. For a class of models, class C', including typical failure/repair models, a natural selection for the regenerative state exists and, with that selection, theoretical results are available assessing the performance of the method in terms of "visible" characteristics. Using dependability class C' models of moderate size of a RAID 5 architecture we compare the performance of the variant with those of regenerative randomization and randomization with steady-state detection for irreducible models, and with those of regenerative randomization and standard randomization for models with absorbing states. For irreducible models, the new variant seems to be about as fast as randomization with steady-state detection for models which are not too small when the initial probability distribution is concentrated in the regenerative state, and significantly faster than regenerative randomization when the model is stiff and not very large. For stiff models with absorbing states, the new variant is much faster than standard randomization and significantly faster than regenerative randomization when the model is not very large. In addition, the variant seems to be able to achieve stringent accuracy levels safely.

Computing the Expected Markov Reward Rates with Stationarity Detection and Relative Error Control

“…The first one is taken from (Carrasco, 2003b) and corresponds to a RAID 5 storage system (Chen et al, 1994) with the architecture shown in Figure 1. The system comprises 40 × 5 disks, 5 controllers, 3 hot spare disks, and 1 hot spare controller.…”

“…The initial probability is one for the fully operational state and is zero for the remaining states. As in (Carrasco, 2003b), we assume that if unavailable disks do not belong to the same string, when one of them becomes available the remaining unavailable disks still belong to different strings whenever their number is ≥ 2. The CTMC has 14 081 states and 94 405 transition rates.…”