We prove a strong (and seemingly odd) result about the M/M/c queue: the reciprocal of the average sojourn time is a concave function of the traffic intensity. We use this result to show that the average itself is jointly convex in arrival and service rates. The standard deviation has the same properties. Also, we determine conditions under which these properties are exhibited by a standard approximation for the M/G/c queue. These results are useful in design studies for telecommunications and production systems.
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Harel et al. (1987) studied this question.
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