An application of the QBD process is modeling the number of jobs (queue length) in a MAP/PH/1/FCFS queue, where jobs arrive according to a Markovian arrival process (MAP), jobs are served in the order of their arrivals (FCFS), and their service demand has a PH distribution (as defined in Section 2.2). A MAP is used to model a job arrival process in Chapter 7. A PH distribution can also be used to model a job arrival process (PH renewal process), by specifying the interarrival time by the PH distribution. However, the interarrival times in a PH renewal process are i.i.d. random variables, while some applications require capturing the correlation in the interarrival times. The MAP generalizes the PH renewal process in the sense that the interarrival times in a MAP can be correlated.

For the purpose of understanding the later chapters, one only needs to know the definition of the MAP and a subclass of the MAP, the MMPP (Markov modulated Poisson process). In Appendix C, we will provide some basic properties of the MAP, so that a reader can further study the characteristics of the MAPs that we will use.