Dealing with Zero Density Using Piecewise Phase-Type Approximation

Abstract

Every probability distribution can be approximated up to a given precision by a phase-type distribution, i.e. a distribution encoded by a continuous time Markov chain (CTMC). However, an excessive number of states in the corresponding CTMC is needed for some standard distributions, in particular most distributions with regions of zero density such as uniform or shifted distributions. Addressing this class of distributions, we suggest an alternative representation by CTMC extended with discrete-time transitions. Using discrete-time transitions we split the density function into multiple intervals. Within each interval, we then approximate the density with standard phase-type fitting. We provide an experimental evidence that our method requires only a moderate number of states to approximate such distributions with regions of zero density. Furthermore, the usage of CTMC with discrete-time transitions is supported by a number of techniques for their analysis. Thus, our results promise an efficient approach to the transient analysis of a class of non-Markovian models.

Observation-Enhanced QoS Analysis of Component-Based Systems

“…To address the significant difficulties that delays within a process pose to PHD fitting, Korenčiak et al [69] have tackled probabilistic regions of zero density by using interval distributions to separate discrete and continuous features of distributions. Similar work [70] supports the synthesis of timeouts in fixed-delay CTMCs by using Markov decision processes.…”

“…Similar work [70] supports the synthesis of timeouts in fixed-delay CTMCs by using Markov decision processes. Unlike OMNI, [70] and [69] do not consider essential non-Markovian features of real data such as multimodal and long-tail distributions, and thus cannot handle empirical data that has these common characteristics.…”

Extension of PRISM by Synthesis of Optimal Timeouts in Fixed-Delay CTMC

“…This works for some models, however there are models for which such approximation can cause either a large error or a state space explosion (see, e.g. [7,2]). However, there is a formalism called fixed-delay CTMCs (fdCTMCs) [4,7,1] that is the requested extension of CTMCs by fixed-delay (fd) events, modeling the deterministic transitions or timeouts.…”

Efficient Timeout Synthesis in Fixed-Delay CTMC Using Policy Iteration

“…), a lot of research effort has been devoted to developing formalisms that generalize CTMC with fixed-delay transitions. Examples include deterministic and stochastic Petri nets [24], delayed CTMC [14], or fixed-delay CTMC (fdCTMC) [20], [5], [21].…”