Abstract
In a M/M/N+M queue, when there are many customers waiting, it may be preferable to reject a new arrival rather than risk that arrival later abandoning without receiving service. On the other hand, rejecting new arrivals increases the percentage of time servers are idle, which also may not be desirable. We address these trade-offs by considering an admission control problem for a M/M/N+M queue when there are costs associated with customer abandonment, server idleness, and turning away customers. First, we formulate the relevant Markov decision process (MDP), show that the optimal policy is of threshold form, and provide a simple and efficient iterative algorithm that does not presuppose a bounded state space to compute the minimum infinite horizon expected average cost and associated threshold level. Under certain conditions we can guarantee that the algorithm provides an exact optimal solution when it stops; otherwise, the algorithm stops when a provided bound on the optimality gap is reached. Next, we solve the approximating diffusion control problem (DCP) that arises in the Halfin–Whitt many-server limit regime. This allows us to establish that the parameter space has a sharp division. Specifically, there is an optimal solution with a finite threshold level when the cost of an abandonment exceeds the cost of rejecting a customer; otherwise, there is an optimal solution that exercises no control. This analysis also yields a convenient analytic expression for the infinite horizon expected average cost as a function of the threshold level. Finally, we propose a policy for the original system that is based on the DCP solution, and show that this policy is asymptotically optimal. Our extensive numerical study shows that the control that arises from solving the DCP achieves a very similar cost to the control that arises from solving the MDP, even when the number of servers is small.
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References
Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York (1965)
Adusumilli, K.M., Hasenbein, J.J.: Dynamic admission and service rate control of a queue. Working Paper, University of Texas, Austin, Texas (2008)
Ata, B., Harrison, J.M., Shepp, L.A.: Drift rate control of a Brownian processing system. Ann. Appl. Probab. 15(2), 1145–1160 (2005)
Ata, B., Shneorson, S.: Dynamic control of an M/M/1 service system with adjustable arrival and service rates. Manag. Sci. 52(11), 1778–1791 (2006)
Armony, M., Ward, A.R.: Fair dynamic routing in large-scale heterogeneous server systems. Oper. Res. (2009). doi:10.1287/opre.1090.0777
Borovkov, A.A.: On limit laws for service processes in multi-channel systems. Sib. Math. J. 8, 746–763 (1967) (English Translation)
Boxma, O.J., de Waal, P.R.: Multiserver queues with impatient customers. ITC 14, 743–756 (1994)
Browne, S., Whitt, W.: Piecewise-Linear Diffusion Processes. Advances in Queueing: Theory, Methods, and Open Problems, pp. 463–480. CRC Press, Boca Raton (1995)
Chen, W., Huang, D., Kulkarni, A., Unnikrishnan, J., Zhu, Q., Mehta, P., Meyn, S., Wierman, A.: Approximate dynamic programming using fluid and diffusion approximations with applications to power management. Working Paper (2009)
Cil, E.B., Ormeci, E.L., Karaesmen, F.: Effects of system parameters on the optimal policy structure in a class of queueing control problems. Queueing Syst. 61(4), 273–304 (2008)
Garnett, O., Mandelbaum, A., Reiman, M.: Designing a call center with impatient customers. Manuf. Serv. Oper. Manag. 4(3), 208–227 (2002)
George, J.M., Harrison, J.M.: Dynamic control of a queue with adjustable service rate. Oper. Res. 49(5), 720–731 (2001)
Ghosh, A.P., Weerasinghe, A.P.: Optimal buffer size for a stochastic processing network in heavy traffic. Queueing Syst. 55(3), 147–159 (2007)
Ghosh, A.P., Weerasinghe, A.P.: Optimal buffer size and dynamic rate control for a queueing network with impatient customers in heavy traffic. Working Paper (2008)
Halfin, S., Whitt, W.: Heavy-traffic limits for queues with many exponential servers. Oper. Res. 29(3), 567–588 (1981)
Incoming Call Center Management Institute (ICMI). ICMI’s contact center outsourcing report—key findings. http://www.callcentermagazine.com/showArticle.jhtml?articleID=201805251 (2007)
Koole, G.: Monotonicity in Markov reward and decision chains: theory and applications. Found. Trends Stoch. Syst. 1(1), 1–76 (2006)
Koole, G., Pot, A.: A note on profit maximization and monotonicity for inbound call centers. Working Paper. Department of Mathematics, Vrije Universiteit Amsterdam, The Netherlands (2006)
Meyn, S.: Workload models for stochastic networks: value functions and performance evaluation. IEEE Trans. Autom. Control 50(8), 1106–1122 (2005)
Pang, G., Talreja, R., Whitt, W.: Martingale proofs of many-server heavy-traffic limits for Markovian queues. Probab. Surv. 4, 193–267 (2007)
Puterman, M.L.: Markov Decision Processes: Discrete Stochastic Dynamic Programming. Wiley, New York (1994)
Ward, A.R., Kumar, S.: Asymptotically optimal admission control of a queue with impatient customers. Math. Oper. Res. 33(1), 167–202 (2008)
Weerasinghe, A., Mandelbaum, A.: A many server controlled queuing system with impatient customers. Working Paper (2008)
Whitt, W.: Heavy traffic approximations for service systems with blocking. AT&T Bell Lab. Tech. J. 63, 689–708 (1984)
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Koçağa, Y.L., Ward, A.R. Admission control for a multi-server queue with abandonment. Queueing Syst 65, 275–323 (2010). https://doi.org/10.1007/s11134-010-9176-z
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DOI: https://doi.org/10.1007/s11134-010-9176-z
Keywords
- Admission control
- Customer abandonment
- Markov decision process
- Diffusion control problem
- Halfin–Whitt QED limit regime
- Average cost