Abstract
We consider a single server Markovian queue with setup times. Whenever this system becomes empty, the server is turned off. Whenever a customer arrives to an empty system, the server begins an exponential setup time to start service again. We assume that arriving customers decide whether to enter the system or balk based on a natural reward-cost structure, which incorporates their desire for service as well as their unwillingness to wait.
We examine customer behavior under various levels of information regarding the system state. Specifically, before making the decision, a customer may or may not know the state of the server and/or the number of present customers. We derive equilibrium strategies for the customers under the various levels of information and analyze the stationary behavior of the system under these strategies. We also illustrate further effects of the information level on the equilibrium behavior via numerical experiments.
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References
Adan, I., van der Wal, J.: Difference and differential equations in stochastic operations research, Online Notes, URL: http://www.win.tue.nl/~iadan/ (1998)
Artalejo, J.R., Lopez-Herrero, M.J.: On the M/M/m queue with removable servers. In: Srinivasan, S.K., Vijayakumar, A. (eds.) Stochastic Point Processes, pp. 124–143. Narosa Publishing House (2003)
Artalejo, J.R., Economou, A., Lopez-Herrero, M.J.: Analysis of a multiserver queue with setup times. Queueing Syst. 52, 53–76 (2005)
Bischof, W.: Analysis of M/G/1-queues with setup times and vacations under six different service disciplines. Queueing Syst. 39, 265–301 (2001)
Borthakur, A., Choudhury, G.: A multiserver Poisson queue with a general startup time under N-Policy. Calcutta Stat. Assoc. Bull. 49, 199–213 (1999)
Chen, H., Frank, M.: State dependent pricing with a queue. IIE Trans. 33, 847–860 (2001)
Chen, H., Frank, M.: Monopoly pricing when customers queue. IIE Trans. 36, 569–581 (2004)
Choudhury, G.: On a batch arrival Poisson queue with a random setup and vacation period. Comput. Oper. Res. 25, 1013–1026 (1998)
Choudhury, G.: An MX/G/1 queueing system with a setup period and a vacation period. Queueing Syst. 36, 23–38 (2000)
Edelson, N.M., Hildebrand, K.: Congestion tolls for Poisson queueing processes. Econometrica 43, 81–92 (1975)
Elaydi, S.N.: An Introduction to Difference Equations. Mathematics. Springer, New York (1999)
Hassin, R.: Consumer information in markets with random products quality: the case of queues and balking. Econometrica 54, 1185–1195 (1986)
Hassin, R., Haviv, M.: Equilibrium threshold strategies: the case of queues with priorities. Oper. Res. 45, 966–973 (1997)
Hassin, R., Haviv, M.: Nash equilibrium and subgame perfection in observable queues. Ann. Oper. Res. 113, 15–26 (2002)
Hassin, R., Haviv, M.: To Queue or Not to Queue: Equilibrium Behavior in Queueing Systems. Kluwer Academic, Boston (2003)
He, Q.M., Jewkes, E.: Flow time in the MAP/G/1 queue with customer batching and setup times. Stoch. Models 11, 691–711 (1995)
Naor, P.: The regulation of queue size by levying tolls. Econometrica 37, 15–24 (1969)
Neuts, M.F.: Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach. Johns Hopkins University Press, Baltimore (1981). Reprinted by Dover Publications Inc., New York (1994).
Takagi, H.: Vacation and Priority Systems. Queueing Analysis—A Foundation of Performance Evaluation, vol. 1. North-Holland, New York (1991)
Tian, N., Zhang, Z.G.: Vacation Queueing Models: Theory and Applications. Springer, New York (2006)
Yechiali, U.: On optimal balking rules and toll charges in the GI/M/1 queue. Oper. Res. 19, 349–370 (1971)
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Burnetas, A., Economou, A. Equilibrium customer strategies in a single server Markovian queue with setup times. Queueing Syst 56, 213–228 (2007). https://doi.org/10.1007/s11134-007-9036-7
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DOI: https://doi.org/10.1007/s11134-007-9036-7
Keywords
- Queueing
- Setup times
- Vacations
- Balking
- Continuous time Markov chain
- Equilibrium strategies
- Individual optimization
- Pricing
- Stationary distribution
- Difference equations
- Matrix analytic methods