Admission control for a multiserver queue with abandonment
 Yaşar Levent Koçağa,
 Amy R. Ward
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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 tradeoffs 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 manyserver 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.
 Abramowitz, M., Stegun, I.A. (1965) Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York
 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. (2005) Drift rate control of a Brownian processing system. Ann. Appl. Probab. 15: pp. 11451160 CrossRef
 Ata, B., Shneorson, S. (2006) Dynamic control of an M/M/1 service system with adjustable arrival and service rates. Manag. Sci. 52: pp. 17781791 CrossRef
 Armony, M., Ward, A.R.: Fair dynamic routing in largescale heterogeneous server systems. Oper. Res. (2009). doi:10.1287/opre.1090.0777
 Borovkov, A.A. (1967) On limit laws for service processes in multichannel systems. Sib. Math. J. 8: pp. 746763 CrossRef
 Boxma, O.J., Waal, P.R. (1994) Multiserver queues with impatient customers. ITC 14: pp. 743756
 Browne, S., Whitt, W. (1995) PiecewiseLinear Diffusion Processes. CRC Press, Boca Raton
 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. (2008) Effects of system parameters on the optimal policy structure in a class of queueing control problems. Queueing Syst. 61: pp. 273304 CrossRef
 Garnett, O., Mandelbaum, A., Reiman, M. (2002) Designing a call center with impatient customers. Manuf. Serv. Oper. Manag. 4: pp. 208227 CrossRef
 George, J.M., Harrison, J.M. (2001) Dynamic control of a queue with adjustable service rate. Oper. Res. 49: pp. 720731 CrossRef
 Ghosh, A.P., Weerasinghe, A.P. (2007) Optimal buffer size for a stochastic processing network in heavy traffic. Queueing Syst. 55: pp. 147159 CrossRef
 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. (1981) Heavytraffic limits for queues with many exponential servers. Oper. Res. 29: pp. 567588 CrossRef
 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. (2006) Monotonicity in Markov reward and decision chains: theory and applications. Found. Trends Stoch. Syst. 1: pp. 176 CrossRef
 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. (2005) Workload models for stochastic networks: value functions and performance evaluation. IEEE Trans. Autom. Control 50: pp. 11061122 CrossRef
 Pang, G., Talreja, R., Whitt, W. (2007) Martingale proofs of manyserver heavytraffic limits for Markovian queues. Probab. Surv. 4: pp. 193267 CrossRef
 Puterman, M.L. (1994) Markov Decision Processes: Discrete Stochastic Dynamic Programming. Wiley, New York
 Ward, A.R., Kumar, S. (2008) Asymptotically optimal admission control of a queue with impatient customers. Math. Oper. Res. 33: pp. 167202 CrossRef
 Weerasinghe, A., Mandelbaum, A.: A many server controlled queuing system with impatient customers. Working Paper (2008)
 Whitt, W. (1984) Heavy traffic approximations for service systems with blocking. AT&T Bell Lab. Tech. J. 63: pp. 689708
 Title
 Admission control for a multiserver queue with abandonment
 Journal

Queueing Systems
Volume 65, Issue 3 , pp 275323
 Cover Date
 20100701
 DOI
 10.1007/s111340109176z
 Print ISSN
 02570130
 Online ISSN
 15729443
 Publisher
 Springer US
 Additional Links
 Topics
 Keywords

 Admission control
 Customer abandonment
 Markov decision process
 Diffusion control problem
 Halfin–Whitt QED limit regime
 Average cost
 60J70
 60K25
 68M20
 90B22
 90C40
 Industry Sectors
 Authors

 Yaşar Levent Koçağa ^{(1)}
 Amy R. Ward ^{(1)}
 Author Affiliations

 1. Information and Operations Management Department, Marshall School of Business, University of Southern California, Los Angeles, USA