In this chapter we use the concepts of sample-path analysis and level-crossing, developed in Chapter 4, to model and analyze many single-station queueing systems in steady state. Section 5.1 considers several variants of the \(M(n)/M(n)/1\) queue. In Section 5.2 we develop Python code to handle the numerical aspects, and we apply it to a simple business case. Section 5.3 discusses the \(M^{X}/M/1\) queue and blocking policies. This analysis paves the way for deriving the formula for the expected waiting time of the \(M/G/1\) queue; Section 5.4 provides the details. Section 5.5 then derives the queue length distribution of the \(M/G/1\) queue. Section 5.6 discusses an example of the control of a queueing system. Section 5.7 and Section 5.8 provide models for inventory systems controlled by the base-stock policy or the \((Q,r)\) policy. Section 5.9 develops an algorithm to compute the long-run average cost for the \((s,S)\) and \((T,S)\) inventory policies. Finally, Section 5.10 formulates this cost computation as a hitting problem. The theory of hitting and stopping problems extends far beyond inventory control; it is, for instance, one of the cornerstones of mathematical finance, ruin theory in actuarial science, and connects to the theory of partial differential equations.