Abstract
We investigate a Markovian tandem queueing model in which service to the first queue is provided in batches. The main goal is to choose the batch sizes so as to minimize a linear cost function of the mean queue lengths. This model can be formulated as a Markov Decision Process (MDP) for which the optimal strategy has nice structural properties. In principle we can numerically compute the optimal decision in each state, but doing so can be computationally very demanding. A previously obtained approximation is computationally efficient for low and moderate loads, but for high loads also suffers from long computation times. In this paper, we exploit the structure of the optimal strategy and develop heuristic policies motivated by the analysis of a related controlled fluid problem. The fluid approach provides excellent approximations, and thus understanding, of the optimal MDP policy. The computational effort to determine the heuristic policies is much lower and, more importantly, hardly affected by the system load. The heuristic approximations can be extended to models with general service distributions, for which we numerically illustrate the accuracy.
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References
Avram, F.: Optimal control of fluid limits of queueing networks and stochasticity corrections. Lect. Appl. Math. 33, 1–36 (1997)
Bellman, R.E.: Dynamic Programming. Princeton University Press, Princeton (2003)
Bortolussi, L., Tribastone, M.: Fluid limits of queueing networks with batches. In: Proceedings of 3rd ACM/SPEC International Conference on Performance Engineering (2012)
Foss, S., Kovalevskii, A.: A stability criterion via fluid limits and its application to a polling model. Queueing Syst. 32, 131168 (1999)
Koole, G.: Convexity in tandem queues. Probab. Eng. Inf. Sci. 18(01), 13–31 (2004)
Kurtz, T.G.: Solutions of ordinary differential equations as limits of pure Markov processes. J. Appl. Prob. 7, 4958 (1970)
Larrañaga, M., Boxma, O.J., Núñez Queija, R., Squillante, M.S.: Efficient content delivery in the presence of impatient jobs. In: Teletraffic Congress (ITC 27), 2015 27th International. IEEE, (2015)
Lippman, S.A.: Applying a new device in the optimisation of exponential queueing systems. Oper. Res. 23, 687–710 (1975)
Puterman, M.L.: Markov Decision Processes: Discrete Stochastic Dynamic Programming. Wiley, Hoboken (1994)
Rajat, D.K., Serfozo, R.F.: Optimal control of batch service queues. Adv. Appl. Probab. 5(2), 340–361 (1973)
Rajat, D.K.: Optimal control of batch service queues with switching costs. Adv. Appl. Probab. 8(1), 177–194 (1976)
Robert, P.: Stochastic Networks and Queues, vol. 52. Springer, Berlin (2013)
Rosberg, Z., Varaiya, P.P., Walrand, J.: Optimal control of service in tandem queues. IEEE Trans. Autom. Control 27(3), 600 (1982)
Silva, D.F., Zhang, B., Ayhan, H.: Optimal admission control for tandem loss systems with two stations. Oper. Res. Lett. 41(4), 351–356 (2013)
van Leeuwen, D., Núñez-Queija, R.: Near-Optimal Switching Strategies for a Tandem Queue. In: Boucherie, R.J., van Dijk, N.M. (eds.) Markov Decision Processes in Practice, pp. 439–459. Springer, Berlin (2017)
Veatch, M.H., Lawrence, M.W.: Optimal control of a two-station tandem production/inventory system. Oper. Res. 42(2), 337–350 (1994)
Weber, R.R., Stidham, S.: Optimal control of service rates in networks of queues. Adv. Appl. Probab. 19, 202–218 (1987)
Zhang, R., Phillis, Y.A.: Fuzzy control of arrivals to tandem queues with two stations. IEEE Trans. Fuzzy Syst. 7(3), 361–367 (1999)
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van Leeuwen, D., Núñez Queija, R. Optimal dispatching in a tandem queue. Queueing Syst 87, 269–291 (2017). https://doi.org/10.1007/s11134-017-9554-x
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DOI: https://doi.org/10.1007/s11134-017-9554-x