journ time for each customer is E[D], in notation equilibrium. Suppose further that the time average number of cus- tomers within the system is E[N]. Then Little's law gives Equation Symbol Deterministic ~E[D] = E[N] (3.1) Erlangian with k degrees of freedom Little's law is an accounting identity and can be understood as General such. Suppose customers are charged at a rate of £1 per second they Hyperexponential with j degrees of freedom Markovian Arriving Departing Customer Customer Markovian is equivalent to a Poisson process, if the customer population is infinite, and to exponential for the service distribution. The Erlangian distribution is just the sum of k independent, identi- System cally distributed exponential variates. Sometimes this is referred to as a server having k exponential service 'phases'. See Figure 3.4(a) illustrates (This distribution is also referred to as the gamma.) The hyperexponential is constructed from the exponential distribu- tion as well but the customer undergoes a single service phase Figure 3.3 Illustration of Little's law allocated at random from a number of possible phases, see Figure 3.4(h). Models based on the exponential, Markov process and Markov chains 3/5 To see this must be the case, consider the following example. A random job from a class with exponential processing time require- ments, with rate parameter Q, has been the sole task within a computer for t seconds. The total processing requirement is X. Let us compute the probability that the job will take at most a further S seconds to be completed. The proportion of random jobs whose processing times lie in the interval (t,t+s] is given by the area A under the graph in Figure 3.5, as marked. However this particular Figure 3.4 An Erlangian E3 and a Hyperexponential H2 server: (a) E3; (b) H2 Some examples of queues represented in Kendall queuing nota- tion are: 1. D/G/1; single server queue with deterministic arrivals and an arbitrary service time. 2. M/M/3/10; a queue with Poisson arrivals, three exponential servers 