Exact Methods for the Transient Analysis of Nonhomogeneous Continuous Time Markov Chains

Abstract

Most common performance and reliability models assume that rates associated with events such as arrivals, service completions, failures, repairs, etc. are all constant in time. Many practical systems, however, require time- (age-) dependent rates. For example, the use of Weibull failure rates is quite common in many reliability models. Likewise, most actual local area network (LAN) systems experience surges in the number of users that vary in magnitude over time. These surges may often be approximated by a periodic process. Therefore, nonhomogeneous continuous time Markov chains (CTMCs) may be well suited to model such systems. The transient analysis of time-varying linear systems is highly advanced in the field of systems and control theory. We present a review of some useful results, and then apply them to the analysis of nonhomogeneous CTMCs (especially periodic ones). One of the results of this analysis is, that for a certain class of useful nonhomogeneous CTMCs, a very simple method exists for transforming such a CTMC (and not just a periodic one) to an equivalent homogeneous CTMC that is then amenable to such homogeneous methods as Jensen’s method (also known as uniformization or randomization).