Ergodicity Bounds for the Markovian Queue With Time-Varying Transition Intensities, Batch Arrivals and One Queue Skipping Policy

Abstract

In this paper we revisit the Markovian queueing system with a single server, infinite capacity queue and the special queue skipping policy. Customers arrive in batches, but are served one by one according to any conservative discipline. The size of the arriving batch becomes known upon its arrival and at any time instant the total number of customers in the system is also known. According to the adopted queue skipping policy if a batch, which size is greater than the current system size, arrives to the system, all current customers in the system are removed from it and the new batch is placed in the queue. Otherwise the new batch is lost. The distribution of the total number of customers in the system is under consideration under assumption that the arrival intensity $\lambda(t)$ and/or the service intensity $\mu(t)$ are non-random functions of time. We provide the method for the computation of the upper bounds for the rate of convergence of system size to the limiting regime, whenever it exists, for any bounded $\lambda(t)$ and $\mu(t)$ (not necessarily periodic) and any distribution of the batch size. For periodic intensities $\lambda(t)$ and/or $\mu(t)$ and light-tailed distribution of the batch size it is shown how the obtained bounds can be used to numerically compute the limiting distribution of the queue size with the given error. Illustrating numerical examples are provided.