Software Performance and Scalability: A Quantitative Approach

Appendix C. M/M/1 Queues at Steady State

	The theory of probabilities is at bottom nothing but common sense reduced to calculus; it enables us to appreciate with exactness that which accurate minds feel with a sort of instinct for which often times they are unable to account.
	--Pierre-Simon Laplace

In this appendix, we demonstrate how to analyze the behavior of a queuing model using the theoretical framework about random processes we formulated in Appendices A and B. The simplest queuing model, the M/M/1 queues, is chosen for this purpose. Although extremely simple, the M/M/1 model is sufficient for demonstrating the standard procedure of deriving various performance metrics of a queuing system based on the theoretical framework established for describing stochastic processes. In addition, the M/M/1 model reveals many basic facets of a wide range of more complex queuing models.

The analysis of M/M/1 queuing model can be carried out by applying the limiting state probability results of birth–death chains obtained previously. Let's start with reviewing the major results of birth–death chains presented in Appendix A. We assume that the reader is already familiar with some of the basic concepts and metrics about queuing systems introduced in Chapter 4 of this book.

REVIEW OF BIRTH–DEATH CHAINS

The key results of birth–death chains are the limiting state probabilities as repeated below for the initial state probability π₀ and limiting state probability π_n, together with the normalization ...