Queueing Systems with Structured Markov Chains

In the previous chapters we studied queueing systems with different interarrival and service time distributions. Chapter 7 is devoted to the analysis of queueing systems with exponential interarrival and service time distributions. The number of customers in these queueing systems is characterized by CTMCs with a generally nonhomogeneous birth-and-death structure. In contrast, Chap. 8 is devoted to the analysis of queueing systems with nonexponential interarrival and service time distributions. It turns out that far more complex analysis approaches are required for the analysis of queues with nonexponential interarrival and service time distributions. In this chapter we introduce queueing systems whose interarrival and service time distributions are nonexponential, but they can be analyzed with CTMCs. Indeed in this chapter we demonstrate the use of the results of Chap. 5 for the analysis of queueing systems with phase-type (PH) distributed interarrival and service times or with arrival and service processes that are MAPs. The main message of this chapter is that in queueing models the presence of PH or MAP processes instead of exponential distributions results in a generalization of the underlying CTMCs from birth-and-death processes to quasi-birth-and-death (QBDs) processes.