ABSTRACT
This chapter contains a partial study of the M/M/1 queue from the point of view of large deviations. This is a partial study because nearly any question one can ask of the M/M/1 queue can be answered. Its steady-state and transient distributions are known explicitly. The chapter begins with formal derivations of most of the results of the chapter. Once the large deviations principle are established, it proceeds to justify the Freidlin-Wentzell theory for the M/M/1 process. Then, after justifying using these calculations to calculate the probability of hitting isolated points, the chapter finishes with a new calculation: how long are the busy periods of an M/M/1 queue, and how large can the process get during long busy periods?.
