6Finite-Capacity Queues
Finite-capacity queues are very common in practice due to limited resources that all service providers face in real life. This results in the waiting space (whether it is real or virtual) being finite. In this chapter, the focus is on the finite-capacity queueing systems. Normally, the finite capacity systems are difficult to analyze mainly due to having two sets of boundary conditions (one near 0 and the other near the maximum size of the queue). Hence, one has to exploit the special structure of the matrices appearing in the generator. This is an integral part of matrix-analytic methods as envisioned and taught by Neuts.
This chapter is organized as follows. We split the study into two parts. The first one deals with the single-server case and is presented in section 6.1. The second part dealing with the multi-server case is considered in section 6.2. Within these sections different types of finite-capacity queueing models are studied.
We start with the most general setup for the queueing models and subsequently look at special cases. Some of these special cases might allow explicit solutions. The illustrative numerical examples will further assess the qualitative nature of the models.
6.1. Finite-capacity queues with single server
Here, we assume that the system has a finite capacity with only one server available for serving the customers and consider different scenarios for the arrivals and the services.
6.1.1. BMAP/PH/1/K queue
Earlier we studied ...
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