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
- 1 March 1976
- journal article
- Published by Cambridge University Press (CUP) in Journal of Applied Probability
- Vol. 13 (1) , 195-199
- https://doi.org/10.2307/3212684
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) = Qk–i/Qk, where the Q's are easily calculated by recurrence in terms of an arbitrary Q0 ≠ 0. We augment Takács's theorem by showing that P(k | i) = bk–i/bk, where bn is the mean busy period in the M/G/1 queue with finite waiting room of size n; that is, if we take Q0 equal to the mean service time, then Qn =bn.Keywords
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