On the relationship between the distribution of maximal queue length in the M/G/ 1 queue and the mean busy period in the M/G /1/ n queue

Abstract

Takács has shown that, in the M/G /1 queue, the probability P ( k | i ) that the maximum number of customers present simultaneously during a busy period that begins with i customers present is P ( k | i ) = Q k – i / Q k , where the Q 's are easily calculated by recurrence in terms of an arbitrary Q 0 ≠ 0. We augment Takács's theorem by showing that P ( k | i ) = b k – i / b k , where b n is the mean busy period in the M/G /1 queue with finite waiting room of size n ; that is, if we take Q 0 equal to the mean service time, then Q n = b n .

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