Many practical problems can be represented by a small finite-state CTMC. When this happens, one is always happy. A finite-state CTMC, whose transition rates are numbers (not variables), can always be solved, given enough computational power, because it simply translates to a finite set of linear simultaneous equations. When transition rates are arbitrary parameters (λ's and μ's and such), the chain might still be solvable via symbolic manipulation, provided that the number of equations is not too great. Section 16.1 provides additional practice with setting up and solving finite-state CTMCs.

Unfortunately, many systems problems involve unbounded queues that translate into infinite-state CTMCs. We have already seen the M/M/1 and the M/M/k, which involve just a single queue and are solvable, even though the number of states is infinite. However, as we move to queueing networks (systems with multiple queues), we see that we need to track the number of jobs in each queue, resulting in a chain that is infinite in more than one dimension. At first such chains seem entirely intractable. Fortunately, it turns out that a very large class of such chains is easily solvable in closed form. This chapter, starting with Section 16.2 on time-reversibility and leading into Section 16.3 on Burke's theorem, provides us with the foundations needed to develop the theory of queueing networks, which will be the topic of the next few chapters.