I was able to numerically "show" that an M/M/n queue is concave increasing in the number for $n$ servers, which makes intuitive sense. However is there a formal analytical proof for it? Where can I find it?


2 Answers 2


Dyer and Proll (1977)1 showed that for an M/M/c queue, the mean waiting time is a strictly decreasing and convex function of c.


[1] Dyer, M. E., Proll, L. G. (1977). On the Validity of Marginal Analysis for Allocating Servers in M/M/c Queues. Management Science. 23(9):1019-1022.

  • $\begingroup$ For a broader overview see Yu et al. (2006). $\endgroup$ May 17, 2020 at 19:54
  • $\begingroup$ Thank you so much. $\endgroup$
    – Paul
    May 18, 2020 at 9:52
  • $\begingroup$ @TheSimpliFire The Yu et al link has deprecated. $\endgroup$
    – Galen
    Dec 20, 2023 at 3:18
  • $\begingroup$ @Galen I've edited the link. $\endgroup$ Dec 20, 2023 at 8:55

You can derive them from the balance equations. If you check Taha's or Lieberman's Introduction to OR books, you can find the proofs.


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