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Optimal routeing in two-queue polling systems

Part of: Operations research and management science Special processes

Published online by Cambridge University Press:  16 November 2018

I. J. B. F. Adan ,
V. G. Kulkarni ,
N. Lee  and
E. Lefeber
I. J. B. F. Adan*
Affiliation:
Eindhoven University of Technology
V. G. Kulkarni*
Affiliation:
University of North Carolina
N. Lee*
Affiliation:
University of North Carolina
E. Lefeber*
Affiliation:
Eindhoven University of Technology
*
* Postal address: Department of Industrial Engineering, Eindhoven University of Technology, PO Box 513, 5600MB Eindhoven, The Netherlands. Email address: i.adan@tue.nl
** Postal address: Department of Statistics and Operations Research, University of North Carolina, Chapel Hill, NC 27599, USA.
** Postal address: Department of Statistics and Operations Research, University of North Carolina, Chapel Hill, NC 27599, USA.
*** Postal address: Department of Mechanical Engineering, Eindhoven University of Technology, PO Box 513, 5600MB Eindhoven, The Netherlands.
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Abstract

We consider a polling system with two queues, exhaustive service, no switchover times, and exponential service times with rate µ in each queue. The waiting cost depends on the position of the queue relative to the server: it costs a customer c per time unit to wait in the busy queue (where the server is) and d per time unit in the idle queue (where there is no server). Customers arrive according to a Poisson process with rate λ. We study the control problem of how arrivals should be routed to the two queues in order to minimize the expected waiting costs and characterize individually and socially optimal routeing policies under three scenarios of available information at decision epochs: no, partial, and complete information. In the complete information case, we develop a new iterative algorithm to determine individually optimal policies (which are symmetric Nash equilibria), and show that such policies can be described by a switching curve. We use Markov decision processes to compute the socially optimal policies. We observe numerically that the socially optimal policy is well approximated by a linear switching curve. We prove that the control policy described by this linear switching curve is indeed optimal for the fluid version of the two-queue polling system.

Keywords

Customer routeingdynamic programmingfluid queuelinear quadratic regulatorNash equilibriumpolling systemRicatti equationindividual optimum, social optimum

MSC classification

Primary: 60K25: Queueing theory
Secondary: 90B22: Queues and service
Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 3 , September 2018 , pp. 944 - 967
Copyright
Copyright © Applied Probability Trust 2018 

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References

[1]Altman, E. and Shimkin, N. (1998). Individual equilibrium and learning in processor sharing systems. Operat. Res. 46, 776–784.Google Scholar
[2]Bertsekas, D. P. (2005). Dynamic Programming and Optimal Control, Vol. 1, 3rd edn. Athena Scientific, Belmont, MA.Google Scholar
[3]Boon, M. A. A. (2011). Polling models: from theory to traffic intersections. Doctoral thesis. Eindhoven University of Technology.Google Scholar
[4]Boon, M. A. A., van der Mei, R. D. and Winands, E. M. M. (2011). Applications of polling systems. Surveys Operat. Res. Manag. Sci. 16, 67–82.Google Scholar
[5]Boon, M. A. A., van der Mei, R. D. and Winands, E. M. M. (2013). Waiting times in queueing networks with a single shared server. Queueing Systems 74, 403–429.Google Scholar
[6]Boon, M. A. A., van Wijk, A. C. C., Adan, I. J. B. F. and Boxma, O. J. (2010). A polling model with smart customers. Queueing Systems 66, 239–274.Google Scholar
[7]Boxma, O. J., Levy, H. and Weststrate, J. A. (1991). Efficient visit frequencies for polling tables: minimization of waiting cost. Queueing Systems 9, 133–162.Google Scholar
[8]BrysonA. E., Jr. A. E., Jr. and Ho, Y. C. (1975). Applied Optimal Control: Optimization, Estimation, and Control. Hemisphere, Washington, DC.Google Scholar
[9]Burnetas, A. and Economou, A. (2007). Equilibrium customer strategies in a single server Markovian queue with setup times. Queueing Systems 56, 213–228.Google Scholar
[10]Haijema, R. and van der Wal, J. (2008). An MDP decomposition approach for traffic control at isolated signalized intersections. Prob. Eng. Inf. Sci. 22, 587–602.Google Scholar
[11]Hassin, R. and Haviv, M. (2003). To Queue or not to Queue: Equilibrium Behavior in Queueing Systems. Kluwer, Boston, MA.Google Scholar
[12]Klimov, G. P. (1975). Time-sharing service systems. I. Theory Prob. Appl. 19, 532–551.Google Scholar
[13]Klimov, G. P. (1979). Time-sharing service systems. II. Theory Prob. Appl. 23, 314–321.Google Scholar
[14]Kulkarni, V. G. (2016). Modeling and Analyisis of Stochastic Systems, 3rd edn. CRC, Boca Raton, FL.Google Scholar
[15]Kurtz, T. G. (1970). Solutions of ordinary differential equations as limits of pure jump Markov processes. J. Appl. Prob. 7, 49–58.Google Scholar
[16]Lefeber, E., Lämmer, S. and Rooda, J. E. (2011). Optimal control of a deterministic multiclass queuing system for which several queues can be served simultaneously. Systems Control Lett. 60, 524–529.Google Scholar
[17]Levy, H. and Sidi, M. (1990). Polling systems: applications, modeling, and optimization. IEEE Trans. Commun. 38, 1750–1760.Google Scholar
[18]Lippman, S. A. and StidhamS., Jr. S., Jr. (1977). Individual versus social optimization in exponential congestion systems. Operat. Res. 25, 233–247.Google Scholar
[19]Ross, S. (1983). Introduction to Stochastic Dynamic Programming. Academic Press, New York.Google Scholar
[20]Sharafali, M., Co, H. C. and Goh, M. (2004). Production scheduling in a flexible manufacturing system under random demand. Europ. J. Operat. Res. 158, 89–102.Google Scholar
[21]Sidi, M., Levy, H. and Fuhrmann, S. W. (1992). A queueing network with a single cyclically roving server. Queueing Systems 11, 121–144.Google Scholar
[22]Takagi, H. (1986). Analysis of Polling Systems. MIT Press, Cambridge, MA.Google Scholar
[23]Takine, T., Takagi, H. and Hasegawa, T. (1991). Sojourn times in vacation and polling systems with Bernoulli feedback. J. Appl. Prob. 28, 422–432.Google Scholar
[24]Van der Wal, J. and Yechiali, U. (2003). Dynamic visit-order rules for batch-service polling. Prob. Eng. Inf. Sci. 17, 351–367.Google Scholar
[25]Vishnevskiĭ, V. M. and Semenova, O. V. (2006). Mathematical methods for investigating polling systems. Autom. Remote Control 67, 173–220.Google Scholar
[26]Weiss, G. (1999). Scheduling and control of manufacturing systems—a fluid approach. In Proceedings of the 37th Allerton Conference, pp. 577–586.Google Scholar
[27]Winands, E. M. M. (2007). Polling, production & priorities. Doctoral thesis. Eindhoven University of Technology.Google Scholar
[28]Winands, E. M. M., Adan, I. J. B. F. and van Houtum, G. J. (2006). Mean value analysis for polling systems. Queueing Systems 54, 35–44.Google Scholar
[29]Yechiali, U. (1993). Analysis and control of polling systems. In Performance Evaluation of Computer and Communication Systems, Springer, Berlin, pp. 630–650.Google Scholar
[30]Van Zwieten, D. A. J., Lefeber, E. and Adan, I. J. B. F. (2016). Optimal steady-state and transient trajectories of multi-queue switching servers with a fixed service order of queues. Performance Evaluation 97, 16–35.Google Scholar