Chapter 6 - Non-Markovian Queueing Systems

Abstract

This chapter discusses non-Markovian queueing systems and the systems where the distributions of the inter-arrival and service times are not necessarily exponential and the distributions are assumed independent. A model of the type M/G/1 is examined, where G denotes the fact that the distribution of service time is general. The process [N(t), t ≥ 0], where N(t) gives the state of the system or the system size at time t, is a non-Markovian process. However, the analysis of such a process could be based on a Markovian process that can be extracted out of it. There are a number of techniques or approaches that are used for this purpose, such as the embedded-Markov-chain technique, the supplementary-variable technique, and Lindley's integral-equation method. The chapter focuses on the embedded-Markov-chain technique for a system with a Poisson input, the Pollaczek–Khinchin formula for the M/G/1 model, queues with a finite input source (M/G/1//N system), and systems with limited waiting spaces (M/G/1/K systems). Concepts related to busy period, the Mx/G/1 model with bulk arrival, multiserver model, and queues with Markovian arrival process are discussed in the chapter.