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Operational Laws

Little’s Law Little’s law, Little’s result, or Little’s theorem is perhaps the most widely used formula in queueing theory was published by J. Little [25] in 1961. It is simple to state and intuitive, widely applicable, and depends only on weak assumptions about the properties of the queueing system. It says that the average number of customers in the system is equal to the average arrival rate of customer to the system multiplied by the average system time per customer. Historically, Little’s law has been written as L = λW Queueing Theory and its Applications, A Personal View 19 and in this usage it must be remembered that W is defined as mean response time, the mean time spent in the queue and at the server, and not just simply as the mean time spent waiting to be served, L refers to the average number of customers in the system and λ is the mean arrival rate. Little’s law can be applied when we relate L to the average number of customers waiting to receive service, Lq and W to the mean time spent waiting for service, Wq, that is another well-known form is Lq = λWq. The same applies also to the servicing aspect itself. In other words, Little’s law may be applied individually to the different parts of a queueing facility, namely the queue and the server. It may be applied even more generally than we have shown here. For example, it may be applied to separate parts of much larger queueing systems, such as subsystems in a queueing network. In such a case, L should be defined with respect to the number of customers in a subsystem and W with respect to the total time in that subsystem. Little’s law may also refer to a specific class of customer in a queueing system, or to subgroups of customers, and so on. Its range of applicability is very wide indeed. Finally, we comment on the amazing fact that the proof of Little’s law turns out to be independent of • specific assumptions regarding the arrival distribution A(t) • specific assumptions regarding the service time distribution B(t) • number of servers • particular queueing discipline Little’s law is important for three reasons: • because it is so widely applicable (it requires only very weak assumptions), it will be valuable to us in checking the consistency of measurement data • for example, in studying computer systems we frequently will find that we know two of the quantities related by Little’s law (say, the average number of requests in a system and the throughput of that system) and desire to know the third (the average system residence time, in this case)