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 remark ably simple product-form stationary distributions both at arrival epochs an d in continuous time. Although this method works well also in several varia nts of this system, it breaks down when customers arrive in batches. The re sulting GI(x)/M/1 system has no tractable stationary distribution. In this paper we use a recent result of Miyazawa and Taylor (1997) to obtain a stoc hastic upper bound for the GI(x)/M/1 system. We also introduce a class of c ontinuous-time Markov chains which are related to the original GI(x)/M/1 em bedded Markov chain that are shown to have modified geometric stationary di stributions. We use them to obtain easily computed stochastic lower bounds for the GI(x)/M/1 system. Numerical studies demonstrate the quality of thes e bounds.