The first step to analyze a stochastic process¹¹ In our case, queueing or inventory processes. is to model and simulate it. In this chapter we discuss how to do this by means of discrete-time models of queueing systems. Here, discrete time means that we chop up time in intervals of equal duration, and study how the system behaves from one period to the next.²² Such a period can be an hour, or a day; in fact, any amount of time that makes sense in the context of the model. For hospitals, it may be days, for supermarkets it may be 5 minutes.

Section 2.1 and Section 2.2 show how to make a simulation model of a case of mental health department. Section 2.3 uses simulation to obtain general insight into how queueing systems behave. Section 2.4 provides motivation as to why we concentrate on long-run behavior for queueing systems if we want to use mathematics to analyze such systems. Section 2.5 presents suitable recursions for single-item inventories in discrete time, and demonstrates in particular how to design good rules to control stochastic inventory systems. In Section 2.6 we show how to set up a simulation for such systems.