October  2013, 9(4): 901-917. doi: 10.3934/jimo.2013.9.901

Equilibrium joining probabilities in observable queues with general service and setup times

1. 

Department of Mathematics, Beijing Jiaotong University, 100044 Beijing

2. 

Department of Mathematics, University of Northern Iowa, Cedar Falls, IA 50614-0506

Received  October 2012 Revised  March 2013 Published  August 2013

This paper analyzes an M/G/1 queue with general setup times from an economical point of view. In such a queue whenever the system becomes empty, the server is turned off. A new customer's arrival will turn the server on after a setup period. Upon arrival, the customers decide whether to join or balk the queue based on observation of the queue length and the status of the server, along with the reward-cost structure of the system. For the observable and almost observable cases, the equilibrium joining strategies of customers who wish to maximize their expected net benefit are obtained. Two numerical examples are presented to illustrate the equilibrium joining probabilities for these cases under some specific distribution functions of service times and setup times.
Citation: Feng Zhang, Jinting Wang, Bin Liu. Equilibrium joining probabilities in observable queues with general service and setup times. Journal of Industrial & Management Optimization, 2013, 9 (4) : 901-917. doi: 10.3934/jimo.2013.9.901
References:
[1]

E. Altman and R. Hassin, Non-threshold equilibrium for customers joining an M/G/1 queue,, in, (2002), 56.   Google Scholar

[2]

J. R. Artalejo, A. Economou and M. J. Lopez-Herrero, Analysis of a multiserver queue with setup times,, Queueing Systems, 51 (2005), 53.  doi: 10.1007/s11134-005-1740-6.  Google Scholar

[3]

W. Bischof, Analysis of M/G/1-queues with setup times and vacations under six different service disciplines,, Queueing Systems, 39 (2001), 265.  doi: 10.1023/A:1013992708103.  Google Scholar

[4]

A. Borthakur and G. Choudhury, A multiserver Poisson queue with a general startup time under $N$-policy,, Calcutta Statistical Association Bulletin, 49 (1999), 199.   Google Scholar

[5]

O. Boudali and A. Economou, Optimal and equilibrium balking strategies in the single server Markovian queue with catastrophes,, European Journal of Operational Research, 218 (2012), 708.  doi: 10.1016/j.ejor.2011.11.043.  Google Scholar

[6]

A. Burnetas, Customer equilibrium and optimal strategies in Markovian queues in series,, Annals of Operations Research, 208 (2013), 515.  doi: 10.1007/s10479-011-1010-4.  Google Scholar

[7]

A. Burnetas and A. Economou, Equilibrium customer strategies in a single server Markovian queue with setup times,, Queueing Systems, 56 (2007), 213.  doi: 10.1007/s11134-007-9036-7.  Google Scholar

[8]

G. Choudhury, On a batch arrival Poisson queue with a random setup and vacation period,, Computers $&$ Operations Research, 25 (1998), 1013.  doi: 10.1016/S0305-0548(98)00038-0.  Google Scholar

[9]

G. Choudhury, An $M^X$/G/1 queueing system with a setup period and a vacation period,, Queueing Systems, 36 (2000), 23.  doi: 10.1023/A:1011089403694.  Google Scholar

[10]

A. Economou and S. Kanta, On balking strategies and pricing for the single server Markovian queue with compartmented waiting space,, Queueing Systems, 59 (2008), 237.  doi: 10.1007/s11134-008-9083-8.  Google Scholar

[11]

A. Economou and S. Kanta, Equilibrium balking strategies in the observable single-server queue with breakdowns and repairs,, Operations Research Letters, 36 (2008), 696.  doi: 10.1016/j.orl.2008.06.006.  Google Scholar

[12]

A. Economou and S. Kanta, Equilibrium customer strategies and social-profit maximization in the single-server constant retrial queue,, Naval Research Logistics, 58 (2011), 107.  doi: 10.1002/nav.20444.  Google Scholar

[13]

A. Economou, A. Gomez-Corral and S. Kanta, Optimal balking strategies in single-server queues with general service and vacation times,, Performance Evaluation, 68 (2011), 967.  doi: 10.1016/j.peva.2011.07.001.  Google Scholar

[14]

A. Economou and A. Manou, Equilibrium balking strategies for a clearing queueing system in alternating environment,, Annals of Operations Research, 208 (2013), 489.  doi: 10.1007/s10479-011-1025-x.  Google Scholar

[15]

N. M. Edelson and K. Hildebrand, Congestion tolls for Poisson queueing processes,, Econometrica, 43 (1975), 81.  doi: 10.2307/1913415.  Google Scholar

[16]

P. Guo and R. Hassin, Strategic behavior and social optimization in Markovian vacation queues,, Operations Research, 59 (2011), 986.  doi: 10.1287/opre.1100.0907.  Google Scholar

[17]

P. Guo and R. Hassin, Strategic behavior and social optimization in Markovian vacation queues: The case of heterogeneous customers,, European Journal of Operational Research, 222 (2012), 278.  doi: 10.1016/j.ejor.2012.05.026.  Google Scholar

[18]

R. Hassin and M. Haviv, Equilibrium threshold strategies: the case of queues with priorities,, Operations Research, 45 (1997), 966.  doi: 10.1287/opre.45.6.966.  Google Scholar

[19]

R. Hassin and M. Haviv, "To Queue or Not to Queue: Equilibrium Behavior in Queueing Systems,", International Series in Operations Research & Management Science, 59 (2003).  doi: 10.1007/978-1-4615-0359-0.  Google Scholar

[20]

M. Haviv and Y. Kerner, On balking from an empty queue,, Queueing Systems, 55 (2007), 239.  doi: 10.1007/s11134-007-9020-2.  Google Scholar

[21]

Q. M. He and E. Jewkes, Flow time in the $M AP$/G/1 queue with customer batching and setup times,, Stochastic Models, 11 (1995), 691.  doi: 10.1080/15326349508807367.  Google Scholar

[22]

Y. Kerner, The conditional distribution of the residual service time in the $M_n$/G/1 queue,, Stochastic Models, 24 (2008), 364.  doi: 10.1080/15326340802232210.  Google Scholar

[23]

Y. Kerner, Equilibrium joining probabilities for an M/G/1 queue,, Game and Economic Behavior, 71 (2011), 521.  doi: 10.1016/j.geb.2010.06.002.  Google Scholar

[24]

W. Liu, Y. Ma and J. Li, Equilibrium threshold strategies in observable queueing systems under single vacation policy,, Applied Mathematical Modelling, 36 (2012), 6186.  doi: 10.1016/j.apm.2012.02.003.  Google Scholar

[25]

P. Naor, The regulation of queue size by levying tolls,, Econometrica, 37 (1969), 15.  doi: 10.2307/1909200.  Google Scholar

[26]

S. Stidham, Jr., "Optimal Design of Queueing Systems,", CRC Press, (2009).  doi: 10.1201/9781420010008.  Google Scholar

[27]

W. Sun, P. Guo and N. Tian, Equilibrium threshold strategies in observable queueing systems with setup/closedown times,, Central European Journal of Operational Research, 18 (2010), 241.  doi: 10.1007/s10100-009-0104-4.  Google Scholar

[28]

H. Takagi, "Queueing Analysis: A Foundation of Performance Evaluation. Vol. 1. Vacation and Priority Systems. Part I,", North-Holland, (1991).   Google Scholar

[29]

N. Tian and Z.G. Zhang, "Vacation Queueing Models. Theory and Applications,", International Series in Operations Research & Management Science, 93 (2006).   Google Scholar

[30]

J. Wang and F. Zhang, Equilibrium analysis of the observable queues with balking and delayed repairs,, Applied Mathematics and Computation, 218 (2011), 2716.  doi: 10.1016/j.amc.2011.08.012.  Google Scholar

[31]

F. Zhang, J. Wang and B. Liu, On the optimal and equilibrium retrial rates in an unreliable retrial queue with vacations,, Journal of Industrial and Management Optimization, 8 (2012), 861.  doi: 10.3934/jimo.2012.8.861.  Google Scholar

show all references

References:
[1]

E. Altman and R. Hassin, Non-threshold equilibrium for customers joining an M/G/1 queue,, in, (2002), 56.   Google Scholar

[2]

J. R. Artalejo, A. Economou and M. J. Lopez-Herrero, Analysis of a multiserver queue with setup times,, Queueing Systems, 51 (2005), 53.  doi: 10.1007/s11134-005-1740-6.  Google Scholar

[3]

W. Bischof, Analysis of M/G/1-queues with setup times and vacations under six different service disciplines,, Queueing Systems, 39 (2001), 265.  doi: 10.1023/A:1013992708103.  Google Scholar

[4]

A. Borthakur and G. Choudhury, A multiserver Poisson queue with a general startup time under $N$-policy,, Calcutta Statistical Association Bulletin, 49 (1999), 199.   Google Scholar

[5]

O. Boudali and A. Economou, Optimal and equilibrium balking strategies in the single server Markovian queue with catastrophes,, European Journal of Operational Research, 218 (2012), 708.  doi: 10.1016/j.ejor.2011.11.043.  Google Scholar

[6]

A. Burnetas, Customer equilibrium and optimal strategies in Markovian queues in series,, Annals of Operations Research, 208 (2013), 515.  doi: 10.1007/s10479-011-1010-4.  Google Scholar

[7]

A. Burnetas and A. Economou, Equilibrium customer strategies in a single server Markovian queue with setup times,, Queueing Systems, 56 (2007), 213.  doi: 10.1007/s11134-007-9036-7.  Google Scholar

[8]

G. Choudhury, On a batch arrival Poisson queue with a random setup and vacation period,, Computers $&$ Operations Research, 25 (1998), 1013.  doi: 10.1016/S0305-0548(98)00038-0.  Google Scholar

[9]

G. Choudhury, An $M^X$/G/1 queueing system with a setup period and a vacation period,, Queueing Systems, 36 (2000), 23.  doi: 10.1023/A:1011089403694.  Google Scholar

[10]

A. Economou and S. Kanta, On balking strategies and pricing for the single server Markovian queue with compartmented waiting space,, Queueing Systems, 59 (2008), 237.  doi: 10.1007/s11134-008-9083-8.  Google Scholar

[11]

A. Economou and S. Kanta, Equilibrium balking strategies in the observable single-server queue with breakdowns and repairs,, Operations Research Letters, 36 (2008), 696.  doi: 10.1016/j.orl.2008.06.006.  Google Scholar

[12]

A. Economou and S. Kanta, Equilibrium customer strategies and social-profit maximization in the single-server constant retrial queue,, Naval Research Logistics, 58 (2011), 107.  doi: 10.1002/nav.20444.  Google Scholar

[13]

A. Economou, A. Gomez-Corral and S. Kanta, Optimal balking strategies in single-server queues with general service and vacation times,, Performance Evaluation, 68 (2011), 967.  doi: 10.1016/j.peva.2011.07.001.  Google Scholar

[14]

A. Economou and A. Manou, Equilibrium balking strategies for a clearing queueing system in alternating environment,, Annals of Operations Research, 208 (2013), 489.  doi: 10.1007/s10479-011-1025-x.  Google Scholar

[15]

N. M. Edelson and K. Hildebrand, Congestion tolls for Poisson queueing processes,, Econometrica, 43 (1975), 81.  doi: 10.2307/1913415.  Google Scholar

[16]

P. Guo and R. Hassin, Strategic behavior and social optimization in Markovian vacation queues,, Operations Research, 59 (2011), 986.  doi: 10.1287/opre.1100.0907.  Google Scholar

[17]

P. Guo and R. Hassin, Strategic behavior and social optimization in Markovian vacation queues: The case of heterogeneous customers,, European Journal of Operational Research, 222 (2012), 278.  doi: 10.1016/j.ejor.2012.05.026.  Google Scholar

[18]

R. Hassin and M. Haviv, Equilibrium threshold strategies: the case of queues with priorities,, Operations Research, 45 (1997), 966.  doi: 10.1287/opre.45.6.966.  Google Scholar

[19]

R. Hassin and M. Haviv, "To Queue or Not to Queue: Equilibrium Behavior in Queueing Systems,", International Series in Operations Research & Management Science, 59 (2003).  doi: 10.1007/978-1-4615-0359-0.  Google Scholar

[20]

M. Haviv and Y. Kerner, On balking from an empty queue,, Queueing Systems, 55 (2007), 239.  doi: 10.1007/s11134-007-9020-2.  Google Scholar

[21]

Q. M. He and E. Jewkes, Flow time in the $M AP$/G/1 queue with customer batching and setup times,, Stochastic Models, 11 (1995), 691.  doi: 10.1080/15326349508807367.  Google Scholar

[22]

Y. Kerner, The conditional distribution of the residual service time in the $M_n$/G/1 queue,, Stochastic Models, 24 (2008), 364.  doi: 10.1080/15326340802232210.  Google Scholar

[23]

Y. Kerner, Equilibrium joining probabilities for an M/G/1 queue,, Game and Economic Behavior, 71 (2011), 521.  doi: 10.1016/j.geb.2010.06.002.  Google Scholar

[24]

W. Liu, Y. Ma and J. Li, Equilibrium threshold strategies in observable queueing systems under single vacation policy,, Applied Mathematical Modelling, 36 (2012), 6186.  doi: 10.1016/j.apm.2012.02.003.  Google Scholar

[25]

P. Naor, The regulation of queue size by levying tolls,, Econometrica, 37 (1969), 15.  doi: 10.2307/1909200.  Google Scholar

[26]

S. Stidham, Jr., "Optimal Design of Queueing Systems,", CRC Press, (2009).  doi: 10.1201/9781420010008.  Google Scholar

[27]

W. Sun, P. Guo and N. Tian, Equilibrium threshold strategies in observable queueing systems with setup/closedown times,, Central European Journal of Operational Research, 18 (2010), 241.  doi: 10.1007/s10100-009-0104-4.  Google Scholar

[28]

H. Takagi, "Queueing Analysis: A Foundation of Performance Evaluation. Vol. 1. Vacation and Priority Systems. Part I,", North-Holland, (1991).   Google Scholar

[29]

N. Tian and Z.G. Zhang, "Vacation Queueing Models. Theory and Applications,", International Series in Operations Research & Management Science, 93 (2006).   Google Scholar

[30]

J. Wang and F. Zhang, Equilibrium analysis of the observable queues with balking and delayed repairs,, Applied Mathematics and Computation, 218 (2011), 2716.  doi: 10.1016/j.amc.2011.08.012.  Google Scholar

[31]

F. Zhang, J. Wang and B. Liu, On the optimal and equilibrium retrial rates in an unreliable retrial queue with vacations,, Journal of Industrial and Management Optimization, 8 (2012), 861.  doi: 10.3934/jimo.2012.8.861.  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]

Liangliang Ma. Stability of hydrostatic equilibrium to the 2D fractional Boussinesq equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021068

[4]

Jan Prüss, Laurent Pujo-Menjouet, G.F. Webb, Rico Zacher. Analysis of a model for the dynamics of prions. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 225-235. doi: 10.3934/dcdsb.2006.6.225

[5]

Nikolaz Gourmelon. Generation of homoclinic tangencies by $C^1$-perturbations. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 1-42. doi: 10.3934/dcds.2010.26.1

[6]

Braxton Osting, Jérôme Darbon, Stanley Osher. Statistical ranking using the $l^{1}$-norm on graphs. Inverse Problems & Imaging, 2013, 7 (3) : 907-926. doi: 10.3934/ipi.2013.7.907

[7]

Sohana Jahan. Discriminant analysis of regularized multidimensional scaling. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 255-267. doi: 10.3934/naco.2020024

[8]

Zaihong Wang, Jin Li, Tiantian Ma. An erratum note on the paper: Positive periodic solution for Brillouin electron beam focusing system. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1995-1997. doi: 10.3934/dcdsb.2013.18.1995

[9]

Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521

[10]

Changpin Li, Zhiqiang Li. Asymptotic behaviors of solution to partial differential equation with Caputo–Hadamard derivative and fractional Laplacian: Hyperbolic case. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021023

[11]

Jianping Gao, Shangjiang Guo, Wenxian Shen. Persistence and time periodic positive solutions of doubly nonlocal Fisher-KPP equations in time periodic and space heterogeneous media. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2645-2676. doi: 10.3934/dcdsb.2020199

[12]

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

[13]

Bernold Fiedler, Carlos Rocha, Matthias Wolfrum. Sturm global attractors for $S^1$-equivariant parabolic equations. Networks & Heterogeneous Media, 2012, 7 (4) : 617-659. doi: 10.3934/nhm.2012.7.617

[14]

Qiang Guo, Dong Liang. An adaptive wavelet method and its analysis for parabolic equations. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 327-345. doi: 10.3934/naco.2013.3.327

[15]

Vieri Benci, Sunra Mosconi, Marco Squassina. Preface: Applications of mathematical analysis to problems in theoretical physics. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : i-i. doi: 10.3934/dcdss.2020446

[16]

Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825

[17]

Cécile Carrère, Grégoire Nadin. Influence of mutations in phenotypically-structured populations in time periodic environment. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3609-3630. doi: 10.3934/dcdsb.2020075

[18]

Paula A. González-Parra, Sunmi Lee, Leticia Velázquez, Carlos Castillo-Chavez. A note on the use of optimal control on a discrete time model of influenza dynamics. Mathematical Biosciences & Engineering, 2011, 8 (1) : 183-197. doi: 10.3934/mbe.2011.8.183

[19]

Guillermo Reyes, Juan-Luis Vázquez. Long time behavior for the inhomogeneous PME in a medium with slowly decaying density. Communications on Pure & Applied Analysis, 2009, 8 (2) : 493-508. doi: 10.3934/cpaa.2009.8.493

[20]

Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189

2019 Impact Factor: 1.366

Metrics

  • PDF downloads (41)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]