PH-graphs for analyzing shortest path problems with correlated traveling times

Abstract

This paper presents a new approach to model weighted graphs with correlated weights at the edges. Such models are important to describe many real world problems like routing in computer networks or finding shortest paths in traffic models under realistic assumptions. Edge weights are modeled by phase type distributions (PHDs), a versatile class of distributions based on continuous time Markov chains (CTMCs). Correlations between edge weights are introduced by adding dependencies between the PHDs of adjacent edges using transfer matrices. The new model class, denoted as PH graphs (PHGs), allows one to formulate many shortest path problems as the computation of an optimal policy in a continuous time Markov decision process (CTMDP). The basic model class is defined, methods to parameterize the required PHDs and transfer matrices based on measured data are introduced and the formulation of basic shortest path problems as solutions of CTMDPs with the corresponding solution algorithms are also provided. Numerical examples for some typical stochastic shortest path problems demonstrate the usability of the new approach.