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Optimal Operation of an M / M /2 Queue with Removable Servers

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  • Colin E. Bell

    (University of Tennessee, Knoxville, Tennessee)

Abstract

The form of the optimal policy is investigated in an average cost, infinite horizon M / M /2 problem where the number of servers working can be adjusted at arrival or service completion epochs. The costs considered are linear holding costs, linear servers' wages and set-up (shut-down) charges per server turned on (off). It is shown that an optimal policy has a hysteresis form characterized by four parameters, R 1 , R 2 , S 0 , and S 1 , denoting numbers of customers in the system when the number of working servers should be adjusted upward to 1, 2, and downward to 0, 1, respectively. Allowing for the possibilities that S 0 = −1 (or S 1 = −1) denoting the fact that the number of working servers are never adjusted downward to 0 (or 1), the following relationships between R 1 , R 2 , S 0 , and S 1 are shown: (i) R i ≥ i , i = 1, 2, (ii) R 2 ≥ R 1 , (iii) −1 ≤ S 0 ≤ S 1 , (iv) R 2 > S 1 , and (v) S 0 ≤ 0. Such a policy has the property that it never reacts to the arrival of a customer by immediately reducing the number of working servers and never reacts to a service completion by immediately increasing the number of working servers.

Suggested Citation

  • Colin E. Bell, 1980. "Optimal Operation of an M / M /2 Queue with Removable Servers," Operations Research, INFORMS, vol. 28(5), pages 1189-1204, October.
    • Handle: RePEc:inm:oropre:v:28:y:1980:i:5:p:1189-1204
    DOI: 10.1287/opre.28.5.1189
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    Cited by:

    1. Chakravarthy, Srinivas R. & Agnihothri, Saligrama R., 2008. "A server backup model with Markovian arrivals and phase type services," European Journal of Operational Research, Elsevier, vol. 184(2), pages 584-609, January.
    2. R. E. Lillo, 2000. "Optimal Operating Policy for an M/G/1 Exhaustive Server-Vacation Model," Methodology and Computing in Applied Probability, Springer, vol. 2(2), pages 153-167, August.
    3. Zhang, Zhe G. & Tian, Naishuo, 2004. "An analysis of queueing systems with multi-task servers," European Journal of Operational Research, Elsevier, vol. 156(2), pages 375-389, July.
    4. Bradley, James R., 2005. "Optimal control of a dual service rate M/M/1 production-inventory model," European Journal of Operational Research, Elsevier, vol. 161(3), pages 812-837, March.
    5. Tian, Naishuo & Zhang, Zhe George, 2006. "A two threshold vacation policy in multiserver queueing systems," European Journal of Operational Research, Elsevier, vol. 168(1), pages 153-163, January.
    6. Tianhua Zhang & Juliang Zhang & Fu Zhao & Yihong Ru & John W. Sutherland, 2020. "Allocating resources for a restaurant that serves regular and group-buying customers," Electronic Commerce Research, Springer, vol. 20(4), pages 883-913, December.
    7. Wei Li & Attahiru Sule Alfa, 2000. "Optimal policies for M/M/m queue with two different kinds of (N, T)‐policies," Naval Research Logistics (NRL), John Wiley & Sons, vol. 47(3), pages 240-258, April.
    8. Zhe George Zhang, 2009. "Performance Analysis of a Queue with Congestion-Based Staffing Policy," Management Science, INFORMS, vol. 55(2), pages 240-251, February.
    9. Hernan Caceres & Dongchen Yu & Alexander Nikolaev, 2018. "Evaluating shortfall distributions in periodic inventory systems with stochastic endogenous demands and lead-times," Annals of Operations Research, Springer, vol. 271(2), pages 405-427, December.
    10. Boronico, Jess S. & Siegel, Philip H., 1998. "Capacity planning for toll roadways incorporating consumer wait time costs," Transportation Research Part A: Policy and Practice, Elsevier, vol. 32(4), pages 297-310, May.

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