r in the queue in Eq. (23.11) rises sharply to infinity as p gets close to one. 23.3.2 Waiting probabilities The most important law in queuing theory is Little's law. It says that the average wait- ing time in a queuing system is the average service time multiplied by the average number in the queue, as shown in Eq. (23.12): (23.12) Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. Traffic Engineering for Voice and Data Capacity and Quality Principles where W = average time in the system L = average number in system l = average interarrival time The queuing system can be very general; it does not have to be M/M/1 for Little's law to hold. While this sounds obvious, the average of a multiplication product is typically not the same as the product of their individual averages. Queues are a special case. Equation (23.13) shows that the average number waiting in the queue Lq is the av- erage number in the entire system L minus the average number being served p6: (23.13) Little's law gives us the average time in an M/M/1 queue in Eq. (23.14): (23.14) Notice that this average waiting time is proportional to the average service time 1/. In a packet system, this is the average packet size divided by the speed of the packet pipe. And p is the utilization, the packet traffic divided by the capacity of the pipe leav- ing the store-and-forward environment of a packet switch. However, we want more than the average waiting time. We want to know what frac- tion of the packets have to wait more than a given amount of time. More specifically, we want to engineer the packet pipe to have enough capacity that the probability of a packet waiting too long is small enough to satisfy the QoS requirements. Given that an arrival has to wait in an M/M/1 queue, the waiting time follows an ex- ponential distribution, as shown in Eq. (23.15): Prob(wait>t) =Prob(wait)e-tu(1-