Often it is said that the job is being processed at rate φ, meaning that the probability that the job is completed in the next infinitesimal time interval dt is φ dt, given it has not been completed already. The conditional probability that the job is completed within a further s seconds is given by the very same exponential distribution as would be used to determine the probability of the job being completed in s seconds starting from time 0. The memoryless property is this: that the distribution of remaining time until X completes in no way depends on t and is given by the same exponential distribution as X. Suppose t seconds have elapsed and X has not completed. Say that X, “completes” when the time corresponding to X is reached starting from 0). Suppose X is the random variable drawn from this distribution.
The density of the exponential is φ e −φ s and integration of this gives the corresponding distribution function 1 − e −φ s. However, the following property is more significant reason for its importance in queuing theory.
Indeed the distribution of virtually any positive random variable may lie approximated using the exponential ( Kelly, 1979). One reason is that the exponential can be used as a building block to construct other distributions as has been shown earlier. The exponential distribution is encountered frequently in queuing analysis. Phil Whiting, in Telecommunications Engineer's Reference Book, 1993 3.2.1 The memoryless property and the Poisson process
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