Geometric-form bounds for the GI(x)/M/1 queueing system

The GI/M/1 queueing system was long ago studied by considering the embedded discrete-time Markov chain at arrival epochs and was proved to have remarkably simple product-form stationary distributions both at arrival epochs and in continuous time. Although this method works well also in several variants of this system, it breaks down when customers arrive in batches. The resulting GI(x)/M/1 system has no tractable stationary distribution. In this paper we use a recent result of Miyazawa and Taylor to obtain a stochastic upper bound for the GI(x)/M/1 system. We also introduce a class of continuous-time Markov chains which are related to the original GI(x)/M/1 embedded Markov chain that are shown to have modified geometric stationary distributions. We use them to obtain easily computed stochastic lower bounds for the GI(x)/M/1 system. Numerical studies demonstrate the quality of these bounds.