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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

J. Laffont and D. Martimort, "The Theory of Incentives: The Principal-Agent Model,", Princeton University Press, (2002).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

M. Osborne, "An Introduction to Game Theory,", Oxford University Press, (2004).   Google Scholar

[12]

M. Rubinovitch, The slow server problem,, Journal of Applied Probability, 22 (1985), 205.  doi: 10.2307/3213760.  Google Scholar

[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.  Google Scholar

[14]

F. Zhang, "Coordination of Lead Times in Supply Chains,", Dissertation, (2004).   Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

J. Laffont and D. Martimort, "The Theory of Incentives: The Principal-Agent Model,", Princeton University Press, (2002).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

M. Osborne, "An Introduction to Game Theory,", Oxford University Press, (2004).   Google Scholar

[12]

M. Rubinovitch, The slow server problem,, Journal of Applied Probability, 22 (1985), 205.  doi: 10.2307/3213760.  Google Scholar

[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.  Google Scholar

[14]

F. Zhang, "Coordination of Lead Times in Supply Chains,", Dissertation, (2004).   Google Scholar

[1]

Junichi Minagawa. On the uniqueness of Nash equilibrium in strategic-form games. Journal of Dynamics & Games, 2020, 7 (2) : 97-104. doi: 10.3934/jdg.2020006

[2]

Enkhbat Rentsen, Battur Gompil. Generalized nash equilibrium problem based on malfatti's problem. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 209-220. doi: 10.3934/naco.2020022

[3]

Alexandr Mikhaylov, Victor Mikhaylov. Dynamic inverse problem for Jacobi matrices. Inverse Problems & Imaging, 2019, 13 (3) : 431-447. doi: 10.3934/ipi.2019021

[4]

Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271

[5]

Hildeberto E. Cabral, Zhihong Xia. Subharmonic solutions in the restricted three-body problem. Discrete & Continuous Dynamical Systems - A, 1995, 1 (4) : 463-474. doi: 10.3934/dcds.1995.1.463

[6]

M. Grasselli, V. Pata. Asymptotic behavior of a parabolic-hyperbolic system. Communications on Pure & Applied Analysis, 2004, 3 (4) : 849-881. doi: 10.3934/cpaa.2004.3.849

[7]

Elena Bonetti, Pierluigi Colli, Gianni Gilardi. Singular limit of an integrodifferential system related to the entropy balance. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1935-1953. doi: 10.3934/dcdsb.2014.19.1935

[8]

Dmitry Treschev. A locally integrable multi-dimensional billiard system. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5271-5284. doi: 10.3934/dcds.2017228

[9]

Nizami A. Gasilov. Solving a system of linear differential equations with interval coefficients. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2739-2747. doi: 10.3934/dcdsb.2020203

[10]

Michel Chipot, Mingmin Zhang. On some model problem for the propagation of interacting species in a special environment. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020401

[11]

Fritz Gesztesy, Helge Holden, Johanna Michor, Gerald Teschl. The algebro-geometric initial value problem for the Ablowitz-Ladik hierarchy. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 151-196. doi: 10.3934/dcds.2010.26.151

[12]

Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure & Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261

[13]

Hailing Xuan, Xiaoliang Cheng. Numerical analysis and simulation of an adhesive contact problem with damage and long memory. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2781-2804. doi: 10.3934/dcdsb.2020205

[14]

Marco Ghimenti, Anna Maria Micheletti. Compactness results for linearly perturbed Yamabe problem on manifolds with boundary. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1757-1778. doi: 10.3934/dcdss.2020453

[15]

Dugan Nina, Ademir Fernando Pazoto, Lionel Rosier. Controllability of a 1-D tank containing a fluid modeled by a Boussinesq system. Evolution Equations & Control Theory, 2013, 2 (2) : 379-402. doi: 10.3934/eect.2013.2.379

[16]

Yanqin Fang, Jihui Zhang. Multiplicity of solutions for the nonlinear Schrödinger-Maxwell system. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1267-1279. doi: 10.3934/cpaa.2011.10.1267

[17]

Xu Zhang, Xiang Li. Modeling and identification of dynamical system with Genetic Regulation in batch fermentation of glycerol. Numerical Algebra, Control & Optimization, 2015, 5 (4) : 393-403. doi: 10.3934/naco.2015.5.393

[18]

Guo-Bao Zhang, Ruyun Ma, Xue-Shi Li. Traveling waves of a Lotka-Volterra strong competition system with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 587-608. doi: 10.3934/dcdsb.2018035

[19]

Dan Wei, Shangjiang Guo. Qualitative analysis of a Lotka-Volterra competition-diffusion-advection system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2599-2623. doi: 10.3934/dcdsb.2020197

[20]

Manoel J. Dos Santos, Baowei Feng, Dilberto S. Almeida Júnior, Mauro L. Santos. Global and exponential attractors for a nonlinear porous elastic system with delay term. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2805-2828. doi: 10.3934/dcdsb.2020206

2019 Impact Factor: 1.366

Metrics

  • PDF downloads (51)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]