Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-10T12:07:48.051Z Has data issue: false hasContentIssue false
  • English
  • Français

On the optimality and the asymptotic optimality of the smallest weighted available buffer policy

Part of: Special processes Stochastic systems and control

Published online by Cambridge University Press:  01 July 2016

Duan-Shin Lee
Duan-Shin Lee*
Affiliation:
National Tsing Hua University
*
∗ Postal address: Department of Computer Science, National Tsing Hua University, Hsinchu, Taiwan 300. Email address: lds@cs.nthu.edu.tw
Get access Rights & Permissions [Opens in a new window]

Abstract

A major design challenge of Asynchronous Transfer Mode (ATM) networks is to efficiently provide the quality of service (QOS) specified by users with different demands. We classify sources so that sources in one class join the same buffer and have the same requirement for the ATM cell loss ratio. It is important to search for the service discipline that minimizes the accumulated cell loss under the constraint that the cell loss ratios of the sources are proportional to their QOS requirements. In this paper we consider a model that has N finite buffers and a single server. Buffer i, of size Bi, is assigned a positive number wi. The server serves from one of the non-empty buffers whose indices are equal to argmin wi(Bi-Qi), where Qi is the queue length of buffer i. This scheduling policy is called the smallest weighted available buffer policy (SWAB). We show that in a completely symmetric setting, the SWAB policy minimizes the discounted expected loss of cells under some technical conditions. For asymmetric models, we show that the accumulated loss of cells of the SWAB service discipline is asymptotically optimal under heavy traffic conditions in the diffusion limit. Finally, we obtain the expression of wi so that the cell loss ratios of the sources in the diffusion limit are proportional to their QOS requirements.

Keywords

Smallest weighted available buffer scheduling disciplineoptimal controlMarkov decision processweak convergenceheavy trafficreflected Brownian motionquality of serviceATM networks

MSC classification

Primary: 60K25: Queueing theory
Secondary: 93E20: Optimal stochastic control
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 31 , Issue 4 , December 1999 , pp. 1118 - 1150
Copyright
Copyright © Applied Probability Trust 1999 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Part of the work was done when the author was affiliated with C&C Research Lab, NEC USA, NJ, USA.

References

Barlow, R. E. and Proschan, F. (1975). Statistical Theory of Reliability and Life Testing. Holt, Rinehart and Winston, New York.Google Scholar
Berger, A. W. and Whitt, W. (1992). The Brownian approximation for rate-control throttles and the G/G/1/C queue. Discrete Event Dynamic Systems: Theory and Applications 2, 7–60.Google Scholar
Billingsley, P. (1968). Convergence of Probability Measures. John Wiley, New York.Google Scholar
Birman, A., Gail, H. R., Hantler, S. L. and Rosberg, Z. (1995). An optimal service policy for buffer systems. J. Assoc. Computing Machinery 42, 641–657.Google Scholar
Buyukkoc, C., Varaiya, P. and Walrand, J. (1985). The cμ-rule revisited. Adv. Appl. Prob. 17, 237–238.Google Scholar
Chen, H. and Mandelbaum, A. (1991). Leontief systems, RVB's and RBM's. In Proc. Imperial College Workshop on Applied Stochastic Processes, eds. Davis, M. H. A. and Elliott, R. J.. Gordon and Breach, New York.Google Scholar
Chen, H. and Mandelbaum, A. (1991). Stochastic discrete flow networks: diffusion approximations and bottlenecks. Ann. Prob. 19, 1463–1519.Google Scholar
Cruz, R. L. (1991). A calculus for network delay, part I: Network element in isolation. IEEE Trans. Inf. Theory 37, 114–131.Google Scholar
Cruz, R. L. (1991). A calculus for network delay, part II: Network analysis. IEEE Trans. Inf. Theory 37, 131–141.Google Scholar
Demers, A., Keshav, S. and Shenker, S. (1990). Analysis and simulation of a fair queueing algorithm. J. Internetworking: Res. Exper. 1, 3–26.Google Scholar
Duffield, N. G., Lakshman, T. V. and Stiliadis, D. (1998). On adaptive bandwidth sharing with rate guarantees. In Proc. IEEE INFOCOM'98. IEEE Comp. Soc., New York, pp. 1122–1130.Google Scholar
Gail, H. R., Grover, G., Guerin, R., Hantler, S. L., Rosberg, Z. and Sidi, M. (1992). Buffer size requirements under longest queue first. In Proc. IFIP'92. Elsevier Science, Amsterdam, pp. 413–424.Google Scholar
Golestani, S. J. (1991). Congestion-free communication in high-speed packet networks. IEEE Trans. Communications 39, 1802–1812.Google Scholar
Golestani, S. J. (1994). A self-clocked fair queueing scheme for broadband applications. In Proc. IEEE INFOCOM'94. IEEE Comp. Soc., New York, pp. 636–646.Google Scholar
Harrison, J. M. (1985). Brownian Motion and Stochastic Flow Systems. John Wiley, New York.Google Scholar
Harrison, J. M. and Nguyen, V. (1993). Brownian models of multiclass queueing networks: current status and open problems. Queueing Systems 13, 5–40.CrossRefGoogle Scholar
Hofri, M. and Ross, K. W. (1985). On the optimal control of two queues with server setup times and its analysis. SIAM J. Computing 16, 399–420.CrossRefGoogle Scholar
Karlin, S. (1968). Total Positivity, Vol. 1. Stanford University Press, Stanford, CA.Google Scholar
Kim, E. and Van Oyen, M. P. (1997). Beyond the cμ rule: dynamic scheduling of a two-class loss queue. Math. Meth. Operat. Res. 48,Google Scholar
Lee, D.-S. (1997). Weighted longest queue first: an adaptive scheduling discipline for ATM networks. In IEEE INFOCOM'97. IEEE Comp. Soc., New York, pp. 318–325.Google Scholar
Liu, Z. and Nain, P. (1992). On optimal polling policies. Queueing Systems 11, 59–83.Google Scholar
Marshall, A. W. and Olkin, I. (1979). Inequalities: Theory of Majorization and Its Applications. Academic Press, New York.Google Scholar
Mithal, S. (1997). The asymptotic equivalence between the segregated and shared buffer schemes in high speed networks. Preprint.Google Scholar
Parekh, A. K. and Gallager, R. G (1993). A generalized processor sharing approach to flow control in integrated services networks: the single-node case. IEEE Trans. Networking 1, 344–357.Google Scholar
Prohorov, Y. V. (1956). Convergence of random processes and limit theorems in probability theory. Theor. Prob. Appl. 1, 157–214.CrossRefGoogle Scholar
Puterman, M. L. (1994). Markov Decision Processes, Discrete Stochastic Dynamic Programming. John Wiley, New York.Google Scholar
Reiman, M. I. (1983). Some diffusion approximations with state space collapse. In Proc. Int. Seminar on Modeling and Performance Evaluation Methodology, eds. Baccelli, F. and Fayolle, G.. Springer, Berlin, pp. 209–240.Google Scholar
Ross, S. M. (1983). Stochastic Processes. John Wiley, New York.Google Scholar
Varaiya, P., Walrand, J. and Buyukkoc, C. (1985). Extensions of the multi-armed bandit problem. IEEE Trans. Automat. Control AC30, 426–439.Google Scholar
Weber, R. R. (1978). On the optimal assignment of customers to parallel servers. J. Appl. Prob. 15, 406–413.Google Scholar
Whitt, W. (1969). Weak convergence theorems for queues in heavy traffic. , Cornell University, New York.Google Scholar
Winston, W. (1977). Optimality of the shortest line discipline. J. Appl. Prob. 14, 181–189.Google Scholar
Yang, Tao (1996). An optimal service scheduling policy for packet networks with quality of service quarantees. In Proc. IEEE GLOBECOM'96. IEEE, New York, pp. 1657–1663.Google Scholar
Yaron, O. and Sidi, M. (1994). Generalized processor sharing networks with exponentially bounded burstiness arrivals. In IEEE INFOCOM'94. IEEE Comp. Soc., New York, pp. 628–634.Google Scholar
Zhang, H., Hsu, G.-H. and Wang, R. (1995). Heavy traffic limit theorems for a sequence of shortest queueing systems. Queueing Systems 21, 217–238.Google Scholar
Zhang, L. (1991). A new traffic control algorithm for packet switched networks. ACM Trans. Computer Systems 9, 101–124.CrossRefGoogle Scholar
Zhang, Z.-L., Towsley, D. and Kurose, J. (1995). Statistical analysis of the generalized processor sharing scheduling discipline. IEEE J. Sel. Areas Commun. 13, 1071–1080.Google Scholar