Ergodic conditions for a class of state dependent queueing systems
Queueing system stability criteria are frequently formulated in the form of testing the sign of queue length (or waiting time) mean increments for heavy load conditions. In this paper we deal with queueing systems which behaviour can be represented by Markov chains on RN. As stability of such systems is equivalent to ergodicity of proper Markov chains, it must be strongly noticed that criteria of this form are valid only together with some additional set of conditions imposed on the considered chain. We shall suggest four comparatively easy to verify conditions and prove that this conditions are sufficient for considered criteria to be valid. We shall also show that suggested conditions are essential in the sense that skipping one of them turns the criteria to be false and apply the obtained results to check ergodicity of some important class of queueing systems, namely queues with waiting time dependent service times.
KeywordsMarkov Chain Queue Length Ergodic Condition General State Space Heavy Load Condition
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