Abstract
We devise a Monte Carlo based method for detecting whether a non-negative Markov chain is stable for a given set of parameter values. More precisely, for a given subset of the parameter space, we develop an algorithm that is capable of deciding whether the set has a subset of positive Lebesgue measure for which the Markov chain is unstable. The approach is based on a variant of simulated annealing, and consequently only mild assumptions are needed to obtain performance guarantees. The theoretical underpinnings of our algorithm are based on a result stating that the stability of a set of parameters can be phrased in terms of the stability of a single Markov chain that searches the set for unstable parameters. Our framework leads to a procedure that is capable of performing statistically rigorous tests for instability, which has been extensively tested using several examples of standard and non-standard queueing networks.
Highlights
The stability of a Markov chain is arguably among its most important properties
In essence the stability of a parameterized family of Markov processes can be summarized by the stability of a single Markov process, as generated by Algorithm 3.1 or Algorithm 3.2
The main contribution of this paper concerned the development of an automated procedure that determines if, for a specified set of parameter values, a given Markov chain is unstable
Summary
The stability of a Markov chain is arguably among its most important properties. In queueing applications it offers the guarantee that service has been sufficiently provisioned to cope with the load imposed on the network in the long run. For this reason the assessment of the stability of Markov chains has long been an area of intense research. The objective is often to determine the set of parameter values for which the system’s state does not diverge, referred to as the stability region, of a Markov chain. For many relatively standard Markov chains the stability region is expressed in terms of quantities related to the transition probabilities. Various (at first sight) counterintuitive results have been found; in particular, for specific queueing models “naïvely conjectured” conditions turn out to be insufficient to ensure stability
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