Introduction to Stochastic Networks

Volume 44 of the series Applications of Mathematics pp 135-154

Little Laws

  • Richard SerfozoAffiliated withSchool of Industrial and Systems Engineering, Georgia Institute of Technology

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In a Markovian, regenerative, or stationary network, the average sojourn times of customers in a sector of the network can often be obtained from a Little law. Specifically, a Little law for a service system says that the average sojourn time W of a customer in the system and the average queue length L of the system are related by L = λW, where λ is the average arrival rate of units to the system. This fundamental relation is a law of averages or law of large numbers when the quantities L, λ, W are “limits” of averages. It is also a law of expectations when the quantities are expected values. This chapter focuses on Little laws of averages, which are based on sample path analysis. The next chapter covers Little laws of expectations for stationary systems, which are based on Palm probability analysis.