25 - Transform Analysis

Summary

This chapter is a very brief introduction to the wonderful world of transforms. One can think of the transform of a random variable as an onion. This onion is an expression that contains inside it all the moments of the random variable. Getting the moments out of the onion is not an easy task, however, and may involve some tears as the onion is peeled, where the “peeling process” involves differentiating the transform. The first moment is stored in the outermost layer of the onion and thus does not require too much peeling to reach. The second moment is stored a little deeper, the third moment even deeper (more tears), etc. Although getting the moments is painful, it is entirely straightforward how to do it – just keep peeling the layers.

Transforms are a hugely powerful analysis technique. For example, until now we have only learned how to derive the mean response time, E[T], for the M/G/1. However, by the end of the next chapter, we will be able to derive the transform of T, which will allow us to obtain any desired moment of T.

The subject of transforms is very broad. In this chapter we only cover a small subset, namely those theorems that aremost applicable in analyzing the performance of queues. We use transforms heavily in analyzing scheduling algorithms in Part VII.