Framework for Analysis of Queueing Systems with Correlated Arrival Processes and Simultaneous Service of a Restricted Number of Customers in Scenarios with an Infinite Buffer and Retrials

In this paper, we create a framework for the uniform algorithmic analysis of queueing systems with the Markov arrival process and the simultaneous service of a restricted number of customers, described by a multidimensional Markov chain. This chain behaves as the finite-state quasi-death process between successive service-beginning epochs, with jumps occurring at these epochs. Such a description of the service process generalizes many known mechanisms of restricted resource sharing and is well suited for describing various future mechanisms. Scenarios involving customers who cannot enter service upon arrival, access via waiting in an infinite buffer, and access via retrials are considered. We compare the generators of the multidimensional Markov chains describing the operation of queueing systems with a buffer and with retrials and show that the sufficient conditions for the ergodicity of these systems coincide. The computation of the stationary distributions of these chains is briefly discussed. The results can be used for performance evaluation and capacity planning of various queueing models with the Markov arrival process and a variety of different service mechanisms that provide simultaneous service to many customers.