he fifth symbol is the customer population, the size of the community that can request service. If the last symbol or the last two symbols are left out (e.g., M/M/1), then there is an unlimited population, and M = 80. The significance of a noninfinite value M is that a bigger queue has fewer customers out there and has a lower arrival rate. Let us concentrate on the M/M/1 queue whose full five-letter form is M/M/1/00/00. This is a queue with random exponential interarrival time and random exponential service time. We also assume this is a first-come, first-served (FCFS) queue, where the order of service is the order of arrival. There are prioritized queues, but M/M/1 is not one of them. Do we really believe that packet data have perfectly memoryless interarrivals and an exactly exponential distribution of packet size? Of course not. However, it is a good starting point, and research has shown that M/M/1 is a reasonably good model for net- work traffic. There are only two parameters for an M/M/1 queue. The mean interarrival time is 1, and the mean service rate is , so the mean service time is 1/. We use the service rate rather than time to follow the same pattern used in Fig. 23.1 for circuit-switched traf- fic engineering. The utilization, or traffic intensity, of an M/M/1 queue is the fraction of time the server is busy. The utilization p is the mean interarrival time l divided by the mean service rate , as shown in Eq. (23.9): (23.9) That p is the utilization makes sense because the server, on average, does 1/ work every l time interval. Each arrival coming into the queue will receive immediate ser- vice when the server is not busy, 1 - p of the time, and will have to wait in the queue when the server is busy, the other p of the time. The M/M/1 queue does not make sense if p 1 because the queue backs up endlessly. The distribution of the length of the M/M/1 queue over time is an exponential distri- bution as shown in Eq. (23.10): Prob(n in queue) = (1 p)p (23.10) Average in (23.11) Notice that the average numbe