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Convexity Results for the Erlang Delay and Loss Formulae When the Server Utilization Is Held Constant

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  • Arie Harel

    (Zicklin School of Business, Baruch College, City University of New York, New York, New York 10010)

Abstract

This paper proves a long-standing conjecture regarding the optimal design of the M / M / s queue. The classical Erlang delay formula is shown to be a convex function of the number of servers when the server utilization is held constant. This means that when the server utilization is held constant, the marginal decrease in the probability that all servers are busy in the M / M / s queue brought about by the addition of two extra servers is always less than twice the decrease brought about by the addition of one extra server. As a consequence, a method of marginal analysis yields the optimal number of servers that minimize the waiting and service costs when the server utilization is held constant. In addition, it is shown that the expected number of customers in the queue and in the system, as well as the expected waiting time and sojourn in the M / M / s queue, are convex in the number of servers when the server utilization is held constant. These results are useful in design studies involving capacity planning in service operations. The classical Erlang loss formula is also shown to be a convex function of the number of servers when the server utilization is held constant.

Suggested Citation

  • Arie Harel, 2011. "Convexity Results for the Erlang Delay and Loss Formulae When the Server Utilization Is Held Constant," Operations Research, INFORMS, vol. 59(6), pages 1420-1426, December.
    • Handle: RePEc:inm:oropre:v:59:y:2011:i:6:p:1420-1426
    DOI: 10.1287/opre.1110.0957
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    References listed on IDEAS

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    1. Arie Harel, 1990. "Convexity Properties of the Erlang Loss Formula," Operations Research, INFORMS, vol. 38(3), pages 499-505, June.
    2. Eylem Tekin & Wallace Hopp & Mark Van Oyen, 2009. "Pooling strategies for call center agent cross-training," IISE Transactions, Taylor & Francis Journals, vol. 41(6), pages 546-561.
    3. Arie Harel, 1988. "Sharp Bounds and Simple Approximations for the Erlang Delay and Loss Formulas," Management Science, INFORMS, vol. 34(8), pages 959-972, August.
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    7. Joel M. Calabrese, 1992. "Optimal Workload Allocation in Open Networks of Multiserver Queues," Management Science, INFORMS, vol. 38(12), pages 1792-1802, December.
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    Cited by:

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