With the tools developed in the previous chapters, we can simulate very general stochastic processes. While useful, we still need analytical models to make reasoning about such processes simpler; in this chapter we make a start with that.

As we have seen in Section 2.4, characterizing the transient behavior of stochastic systems is complicated, so we focus on the long-run time-average behavior. Arrival and service rates play a crucial role. A queueing system is stable only if the service rate exceeds the rate at which work arrives. Once stability is established, we can define performance measures such as the long-run average waiting time.

We derive some results fundamental to analyzing any stochastic process. These results are based on sample paths which thereby provide a unifying link between, on the one hand, the construction of a stochastic process in Chapter 2 and Chapter 3, and, on the other hand, the theoretical results that follow here. To illustrate the relations among these concepts, we provide two mind maps at the end of the chapter, see Fig. 1 and Fig. 2.

We keep the discussion intuitive and refer to M. El-Taha and S. Stidham Jr., Sample-Path Analysis of Queueing Systems, 1998 for proofs and further background.