and room for 10 customers, including those in service. Figure 3.5 Conditional probability that the job is M/M/1//20; a queue with exponential interarrival times with completed within S seconds rate proportional to the number of customers not in queue, a single exponential server, and infinite waiting room. The cus- tomer population is 20. job is one which has taken t seconds already and it is the proportion of all such jobs which are completed within a further seconds that In the case of queuing networks one may wish to specify the queue is required. The probability a job takes at least t seconds is given by types whilst ignoring the nature of the arrival process to it al- the tail region B. Thus the conditional distribution of the remaining together. In such cases it may be written, for example, that there is time until the job is completed, given that t seconds have elapsed, is a network of ./M/1 queues meaning only that there is a single server as in Equation 3.2. at each of the nodes with exponentially distributed service times. Pr 1 < X Area A Area Models based on the exponential, Markov process and Markov chains 3.2.1 The memoryless property and the Poisson process e-4' e-4(1+5) (3.2) = e-41 1 - e - 4 The exponential distribution is encountered frequently in queuing analysis. One reason is that the exponential can be used as a building This is seen to be independent of t. The conditional probability block to construct other distributions as has been shown earlier. that the job is completed within a further s seconds is given by the Indeed the distribution of virtually any positive random variable very same exponential distribution as would be used to determine may be approximated using the exponential (Kelly, 1979). How- the probability of the job being completed in S seconds starting from ever, the following property is more significant reason for its im- time (). Often it is said that the job is being processed at rate Q, portance in queuing theory. meaning that the probability that the job is completed i