Software Performance and Scalability: A Quantitative Approach

Appendix B. Memoryless Property of the Exponential Distribution

	I have memories—but only a fool stores his past in the future.
	--David Gerrold

We mentioned in Chapter 4 that a Markov process is characterized by its unique property of memoryless-ness: the future states of the process are independent of its past history and depends solely on its present state. We further learnt that Poisson processes constitute a special class of Markov processes for which the event occurring patterns follow the Poisson distribution while the inter-arrival times and service times follow the exponential distribution. We have also learnt that if the event occurring patterns follow the Poisson distribution, then the inter-arrival times and service times follow the exponential distribution, or vice versa. It is important to understand that all these statements are supported by the fact that the exponential distribution is the only continuous distribution that possesses the unique property of memoryless-ness.

Now let's mathematically prove the memoryless property of the exponential distribution. Surprisingly, the proof is very simple.

First, let's state the following conditional probability law that

Equation B.1. 

which can be read as "given the event B, the probability of the event A is equal to the joint probability of A and B divided by the probability of the event B."

Let T be the variable representing the ...