Dealing with Zero Density Using Piecewise Phase-type Approximation

TL;DR

This paper introduces a novel method for approximating distributions with zero density regions using a combination of continuous-time Markov chains and discrete-time transitions, improving efficiency and accuracy.

Contribution

It proposes an extended CTMC model with discrete-time transitions to better approximate distributions with zero density regions, reducing the number of states needed.

Findings

Requires only a moderate number of states for approximation.

Supports efficient transient analysis of non-Markovian models.

Effective for distributions with zero density regions.

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.