# American Institute of Mathematical Sciences

April  2012, 8(2): 299-323. doi: 10.3934/jimo.2012.8.299

## Minimizing equilibrium expected sojourn time via performance-based mixed threshold demand allocation in a multiple-server queueing environment

 1 Department of Industrial Engineering and Operations Research, University of California, Berkeley, United States 2 College of Management, Georgia Institute of Technology, 800 West Peachtree Street NW Atlanta, Georgia 30308-0520, United States 3 Advanced Modeling and Applied Computing Laboratory, Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong, China

Received  October 2010 Revised  August 2011 Published  April 2012

We study the optimal demand allocation policies to induce high service capacity and achieve minimum expected sojourn times in equilibrium in a queueing system with multiple strategic servers. We propose the mixed threshold allocation policy as an optimal state-dependent policy that induces optimal service capacity from strategic servers. Compensation to the server can be paid at customer allocation or upon job completion. Our study focuses on the use of a multiple-server mixed threshold allocation policy to replicate the demand of a given state-independent policy to achieve a symmetric equilibrium with lower expected sojourn time. The results indicate that, under both payment schemes, for any given multiple-server state-independent policy, there exists a multiple-server threshold policy that produces identical demand allocation and Nash equilibrium (if any). Moreover, the policy can be designed to minimize the expected sojourn time at a symmetric equilibrium. Furthermore, under the payment-at-allocation scheme, our results, combining with existing results on the optimality of the multiple-server linear allocation policy, show that the mixed threshold policy can achieve the maximum feasible service capacity and thus the minimum feasible equilibrium expected sojourn time. Hence, our results agree with previous two-server results and affirm that a trade-off between incentives and efficiency need not exist in the case of multiple servers.
Citation: Sin-Man Choi, Ximin Huang, Wai-Ki Ching. Minimizing equilibrium expected sojourn time via performance-based mixed threshold demand allocation in a multiple-server queueing environment. Journal of Industrial & Management Optimization, 2012, 8 (2) : 299-323. doi: 10.3934/jimo.2012.8.299
##### References:
 [1] G. Cachon and F. Zhang, Obtaining fast service in a queueing system via performance-based allocation of demand,, Management Science, 53 (2007), 408. doi: 10.1287/mnsc.1060.0636. [2] W. Ching, S. Choi and M. Huang, Optimal service capacity in a multiple-server queueing system: A game theory approach,, Journal of Industrial and Management Optimization, 6 (2010), 73. doi: 10.3934/jimo.2010.6.73. [3] S. Choi, X. Huang, W. Ching and M. Huang, Incentive effects of multiple-server queueing networks: the principal-agent perspective,, East Asian Journal on Applied Mathematics, 1 (2011), 379. [4] S. Choi, X. Huang and W. Ching, Inducing optimal service capacities via performance-based allocation of demand in a queueing system with multiple servers,, in, (2010). [5] T. Crabill, D. Gross and M. Magazine, A classified bibliography of research on optimal design and control of queues,, Operations Research, 25 (1977), 219. doi: 10.1287/opre.25.2.219. [6] S. Gilbert and Z. Weng, Incentive effects favor nonconsolidating queues in a service system: The principal-agent perspective,, Management Science, 44 (1998), 1662. doi: 10.1287/mnsc.44.12.1662. [7] E. Kalai, M. Kamien and M. Rubinovitch, Optimal service speeds in a competitive environment,, Management Science, 38 (1992), 1554. doi: 10.1287/mnsc.38.8.1154. [8] J. Laffont and D. Martimort, "The Theory of Incentives: The Principal-Agent Model,", Princeton University Press, (2002). [9] W. Lin and P. Kumar, Optimal control of a queueing system with two heterogeneous servers,, IEEE Trans. Automatic Control, 29 (1984), 696. doi: 10.1109/TAC.1984.1103637. [10] H. Luh and I. Viniotis, Threshold control policies for heterogeneous server systems,, Mathematical Methods of Operations Research, 55 (2002), 121. doi: 10.1007/s001860100168. [11] M. Osborne, "An Introduction to Game Theory,", Oxford University Press, (2004). [12] M. Rubinovitch, The slow server problem,, Journal of Applied Probability, 22 (1985), 205. doi: 10.2307/3213760. [13] F. Véricourt and Y. Zhou, On the incomplete results for the heterogeneous server problem,, Queueing Systems, 52 (2006), 189. doi: 10.1007/s11134-006-5067-8. [14] F. Zhang, "Coordination of Lead Times in Supply Chains,", Dissertation, (2004).

show all references

##### References:
 [1] G. Cachon and F. Zhang, Obtaining fast service in a queueing system via performance-based allocation of demand,, Management Science, 53 (2007), 408. doi: 10.1287/mnsc.1060.0636. [2] W. Ching, S. Choi and M. Huang, Optimal service capacity in a multiple-server queueing system: A game theory approach,, Journal of Industrial and Management Optimization, 6 (2010), 73. doi: 10.3934/jimo.2010.6.73. [3] S. Choi, X. Huang, W. Ching and M. Huang, Incentive effects of multiple-server queueing networks: the principal-agent perspective,, East Asian Journal on Applied Mathematics, 1 (2011), 379. [4] S. Choi, X. Huang and W. Ching, Inducing optimal service capacities via performance-based allocation of demand in a queueing system with multiple servers,, in, (2010). [5] T. Crabill, D. Gross and M. Magazine, A classified bibliography of research on optimal design and control of queues,, Operations Research, 25 (1977), 219. doi: 10.1287/opre.25.2.219. [6] S. Gilbert and Z. Weng, Incentive effects favor nonconsolidating queues in a service system: The principal-agent perspective,, Management Science, 44 (1998), 1662. doi: 10.1287/mnsc.44.12.1662. [7] E. Kalai, M. Kamien and M. Rubinovitch, Optimal service speeds in a competitive environment,, Management Science, 38 (1992), 1554. doi: 10.1287/mnsc.38.8.1154. [8] J. Laffont and D. Martimort, "The Theory of Incentives: The Principal-Agent Model,", Princeton University Press, (2002). [9] W. Lin and P. Kumar, Optimal control of a queueing system with two heterogeneous servers,, IEEE Trans. Automatic Control, 29 (1984), 696. doi: 10.1109/TAC.1984.1103637. [10] H. Luh and I. Viniotis, Threshold control policies for heterogeneous server systems,, Mathematical Methods of Operations Research, 55 (2002), 121. doi: 10.1007/s001860100168. [11] M. Osborne, "An Introduction to Game Theory,", Oxford University Press, (2004). [12] M. Rubinovitch, The slow server problem,, Journal of Applied Probability, 22 (1985), 205. doi: 10.2307/3213760. [13] F. Véricourt and Y. Zhou, On the incomplete results for the heterogeneous server problem,, Queueing Systems, 52 (2006), 189. doi: 10.1007/s11134-006-5067-8. [14] F. Zhang, "Coordination of Lead Times in Supply Chains,", Dissertation, (2004).
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