Part V involves the analysis of multi-server and multi-queue systems.

We start in Chapter 14 with the M/M/k server farm model, where k servers all work “cooperatively” to handle incoming requests from a single queue. We derive simple closed-form formulas for the distribution of the number of jobs in the M/M/k. We then exploit these formulas in Chapter 15 to do capacity provisioning for the M/M/k. Specifically, we answer questions such as, “What is the minimum number of servers needed to guarantee that only a small fraction of jobs are delayed?” We derive simple answers to these questions in the form of square-root staffing rules. In these two chapters and the exercises therein, we also consider questions pertaining to resource allocation, such as whether a single fast server is superior to many slow servers, and whether a single central queue is superior to having a queue at each server.

We then move on to analyzing networks of queues, consisting of multiple servers, each with its own queue, with probabilistic routing of packets (or jobs) between the queues. In Chapter 16 we build up the fundamental theory needed to analyze networks of queues. This includes time-reversibility and Burke's theorem. In Chapter 17, we apply our theory to Jackson networks of queues. We prove that these have product form, and we derive the limiting distribution of the number of packets at each queue. Our proofs introduce the concept of Local Balance, which we use repeatedly in derivations throughout the book.