Theoretical proof demonstrates the validity of L = λW in queuing processes, highlighting a universal relationship between queue size, arrival rate, and waiting time.
In a queuing process, let 1/λ be the mean time between the arrivals of two consecutive units, L be the mean number of units in the system, and W be the mean time spent by a unit in the system. It is shown that, if the three means are finite and the corresponding stochastic processes strictly stationary, and, if the arrival process is metrically transitive with nonzero mean, then L = λW.
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John D. C. Little (1961) studied this question.
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