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Unformatted text preview: Computer Networks A simple queueing system Saad Mneimneh Computer Science Hunter College of CUNY New York Thanks to Andrey Markov. 1 Introduction We now consider a simple queueing system called M/M/1. This terminology arises from a standard notational system in queueing theory, first suggested by David Kendel in 1953. The three parts of the notation describe: the inter-arrival time distribution, the service time distribution, and the number of servers The letter M stands for memoryless, i.e. the Poisson process with expo- nential distribution. Other possibilities include D for deterministic inter-arrival times, and G denoting a general distribution of inter-arrival times. Therefore, the M/M/1 queueing system consists of customers arriving according to a Pois- son process with rate , and one server with a service time per customer that is exponentially distributed with rate . This of course does not completely describe the system; for instance, what is the size of the queue where customers wait for service? We will assume an infinite queue for now as depicted below: 1 server packets/sec packets/sec infinite queue Figure 1: The M/M/1 queueing system for packets (server is transmitter) We already know a lot about this system from simple application of Littles theorem. For instance, N = T , where N is the average number of customers in the system and T is the average delay per customer. Similarly, N Q = W , where N Q is the average number of customers in the queue (i.e. not being currently served), and W is the average waiting time in the queue per customer. Moreover, = / , where is the average number of customers being served, which can be also interpreted as the server utilization, throughput, or efficiency. 1 Note also that N = N Q + and T = W + 1 / . Therefore, it is enough to determining one of N , T , N Q , and W , to determines all. One approach would be to first determine N ....
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- Spring '08
- Computer Networks