Abstract
We investigate optimal load balancing strategies for a multi-class multi-server processor-sharing system with a Poisson input stream, heterogeneous service rates, and a server-dependent holding cost per unit time. Specifically, we study (i) the centralized setting in which a dispatcher routes incoming jobs based on their service time requirements so as to minimize the weighted mean sojourn time in the system; and (ii) the decentralized, distributed non-cooperative setting in which each job, aware of its service time, selects a server with the objective of minimizing its weighted mean sojourn time in the system. For the decentralized setting we show the existence of a potential function, which allows us to transform the non-cooperative game into a standard convex optimization problem.
For the two aforementioned settings, we characterize the set of optimal routing policies and obtain a closed form expression for the load on each server under any such policy. Furthermore, we show the existence of an optimal policy that routes a job independently of its service time requirement. We also show that the set of servers used in the decentralized setting is a subset of set of servers used in the centralized setting. Finally, we compare the performance perceived by jobs in the two settings by studying the so-called Price of Anarchy (PoA), that is, the ratio between the decentralized and the optimal centralized solutions. When the holding cost per unit time is the same for all servers, it is known that the PoA is upper bounded by the number of servers in the system. Interestingly, we show that the PoA for our system can be unbounded. In particular this indicates that in our system, the performance of selfish routing can be extremely inefficient.
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References
Beckmann, M., McGuire, C. B., & Winsten, C. B. (1956). Studies in the economics and transportation. New Haven: Yale University Press.
Bell, C. H., & Stidham, S. (1983). Individual versus social optimization in the allocation of customers to alternative servers. Management Science, 29, 831–839.
Borst, S. C. (1995). Optimal probabilistic allocation of customer types to servers. In Proceedings of ACM SIGMETRICS (pp. 116–125), September 1995.
Boyd, S., & Vandenberghe, L. (2004). Convex optimization. Cambridge: Cambridge University Press.
Cardellini, V., Casalicchio, E., Colajanni, M., & Yu, P. S. (2001). The state of the art in locally distributed Web-server systems. ACM Computing Surveys, 34(2), 263–311.
Chen, H. L., Marden, J., & Wierman, A. (2009). The effect of local scheduling in load balancing designs. In Proceedings of IEEE INFOCOM.
Chow, Y.-C., & Kohler, W. H. (1979). Models for dynamic load balancing in a heterogeneous multiple processor system. IEEE Transactions on Computers, 28(5), 354–361.
Czumaj, A., Krysta, P., & Vocking, B. (2002). Selfish traffic allocation for server farms. In Proceedings of STOC, 2002.
Feng, H., Misra, V., & Rubenstein, D. (2005). Optimal state-free, size-aware dispatching for heterogeneous M/G/-type systems. Performance Evaluation, 62(1–4), 36–39.
Gupta, V., Harchol-Balter, M., Sigman, K., & Whitt, W. (2007). Analysis of join-the-shortest-queue routing for web server farms. In Proceedings of performance (p. 180), 2007.
Harchol-Balter, M., Crovella, M., & Murta, C. (1999). On choosing a task assignment policy for a distributed server system. IEEE Journal of Parallel and Distributed Computing, 59(2), 204–228.
Haviv, M., & Roughgarden, T. (2007). The price of anarchy in an exponential multi-server. Operations Research Letters, 35, 421–426.
Kameda, H., Altman, E., Pourtallier, O., Li, J., & Hosokawa, Y. (2000). Paradoxes in performance optimization of distributed systems. In Proceedings of SSGRR 2000 computer and ebusiness conference.
Kameda, H., Li, J., Kim, C., & Zhang, Y. (1997). Optimal load balancing in distributed computer systems. Berlin: Springer.
Kelly, F. (1979). Stochastic networks and reversibility. Chichester: Wiley.
Koutsoupias, E., & Papadimitriou, C. H. (1999). Worst-case equilibria. In Proceedings of STACS 1999.
Monderer, D., & Shapley, L. S. (1996). Potential games. Games and Economic Behavior, 14, 124–143.
Ni, L. M., & Hwang, K. (1985). Optimal load balancing in a multiple processor with many job classes. IEEE Transactions on Software Engineering, 11(5), 491–496.
Patriksson, M. (1994). The traffic assignment problem: models and methods. The Netherlands: VSP BV.
Rosenthal, R. W. (1973). A class of games possessing pure strategy Nash equilibria. International Journal of Game Theory, 2, 65–67.
Roughgarden, T. (2005). Selfish routing and the price of anarchy. Cambridge: MIT Press.
Sandholm, W. H. (2001). Potential games with continuous player sets. Journal of Economic Theory, 97, 81–108.
Starobinski, D., & Wu, T. (2005). Performance of server selection algorithms for content replication networks. In IFIP networking.
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Altman, E., Ayesta, U. & Prabhu, B.J. Load balancing in processor sharing systems. Telecommun Syst 47, 35–48 (2011). https://doi.org/10.1007/s11235-010-9300-8
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DOI: https://doi.org/10.1007/s11235-010-9300-8