Delayed and Network Queues

Chapter 3Birth and Death Processes

3.1 Overviews of the Birth and Death Processes

In the previous chapter, we discussed both discrete- and continuous-time Markov processes with some examples. Before we discuss standard queues, we need to discuss another example of Markov chain and Markov process, which is the birth and death (B–D) processes. These processes may be discrete- or continuous-time. The discrete case is sometimes referred to as birth and death chain. Our discussion, however, is mainly on a continuous-time case, with a brief mention of discrete-time as well. Although the B–D process is important in itself, it has also been proven to be a useful tool for modeling queueing, circuit switches with a limited number of outgoing channels, reliability, inventory, performance evaluation of servers, demography, epidemiology, biology, and others. B–D processes are also good models for the flow of radioactive, cosmic, and other particles. In the economical sciences, B–D processes are used for describing the development of a number of enterprises in a particular area and manpower fluctuations.

B–D processes and chains with a finite or countable number of states also play a central role in stochastic modeling for applied probabilities. A number of books are available on various aspects of B–D models, for example, Gross and Harris (1998), Karlin and Taylor (1975), Kijima (1997), and Medhi (2003). Some classical probability texts such as Feller (1968) and Hoel et al. (1972) also ...