This chapter combines and applies many analytical techniques we have studied thus far (Renewal-Reward, general transform analysis, and M/G/1 response time analysis) toward analyzing the problem of power management of a single server. The goal is to understand when a server should be turned off to save on power (sometimes called “power napping”) and when it should be left on.

In Section 27.1 we provide background on powering a server and state the specific power optimization problem that we address. To solve this problem, we first need to develop two more analysis topics.

The first topic is busy period analysis for the M/G/1. We have touched on busy periods in the exercise sections of prior chapters, but in Section 27.2, we go into much more depth in describing busy periods, including different types of busy periods and the Laplace transform of the busy period.

The second topic is the analysis of an M/G/1 with setup time, where the first job starting a busy period incurs an extra delay, known as the setup time. Setup times have also been discussed in earlier exercises; however, in Section 27.3 we consider their effect on the M/G/1.

Finally, in Section 27.4, we combine the analyses in Sections 27.2 and 27.3 to solve our power optimization problem.

The Power Optimization Problem

Consider the operation of a single-server system, specifically an M/G/1/FCFS queue. Thus far, we have only been concerned about the response time of the system.We now discuss the power usage.