Timed Comparisons of Semi-Markov Processes

TL;DR

This paper explores the comparison of semi-Markov processes based on their time-dependent behavior, introducing a 'faster than' relation, analyzing its computational complexity, and proposing approximation algorithms for specific cases.

Contribution

It defines the 'faster than' relation for semi-Markov processes, studies its complexity, and provides algorithms for time-bounded and unambiguous cases.

Findings

The 'faster than' relation is undecidable in general.

An additive approximation algorithm is proposed for slow residence-time functions.

A coNP algorithm is developed for unambiguous semi-Markov processes.

Abstract

Semi-Markov processes are Markovian processes in which the firing time of the transitions is modelled by probabilistic distributions over positive reals interpreted as the probability of firing a transition at a certain moment in time. In this paper we consider the trace-based semantics of semi-Markov processes, and investigate the question of how to compare two semi-Markov processes with respect to their time-dependent behaviour. To this end, we introduce the relation of being "faster than" between processes and study its algorithmic complexity. Through a connection to probabilistic automata we obtain hardness results showing in particular that this relation is undecidable. However, we present an additive approximation algorithm for a time-bounded variant of the faster-than problem over semi-Markov processes with slow residence-time functions, and a coNP algorithm for the exact faster-than problem over unambiguous semi-Markov processes.