In Chapter 22 we studied the M/G/1/PS queue and derived simple closed-form solutions for πn, E[N], and E[T] (assuming G is any Coxian distribution).

In this chapter we move on to the M/G/1/FCFS queue. We have already had some exposure to thinking about the M/G/1/FCFS. Using the matrix-analytic techniques of Chapter 21, we saw that we could solve the M/PH/1/FCFS queue numerically, where PH represents an arbitrary phase-type distribution. However, we still do not have a simple closed-form solution for the M/G/1/FCFS that lets us understand the effect of load and the job size variability on mean response time.

This chapter introduces a simple technique, known as the “tagged job” technique, which allows us to obtain a simple expression for mean response time in the M/G/1/FCFS queue. The technique will not allow us to derive the variance of response time, nor will it help us understand the higher moments of the number of jobs in the M/G/1/FCFS – for those, we will need to wait until we get to transform analysis in Chapter 25. Nonetheless, the resulting simple formula for mean response time will lead to many insights about the M/G/1 queue and optimal system design for an M/G/1 system.

The Inspection Paradox

We motivate this chapter by asking several questions. We will come back to these questions repeatedly throughout the chapter. By the end of the chapter everything will be clear.