Queueing Systems

, Volume 20, Issue 1–2, pp 207–254 | Cite as

Scheduling policies using marked/phantom slot algorithms

  • Christos G. Cassandras
  • Vibhor Julka
Article
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Abstract

We address the problem of schedulingM customer classes in a single-server system, with customers arriving in one ofN arrival streams, as it arises in scheduling transmissions in packet radio networks. In general,N≠M and a customer from some stream may join one of several classes. We consider a slotted time model where at each scheduling epoch the server (channel) is assigned to a particular class (transmission set) and can serve multiple customers (packets) simultaneously, one from every arrival stream (network node) that can belong to this class. The assignment is based on arandom polling policy: the current time slot is allocated to theith class with probability θi. Our objective is to determine the optimal probabilities by adjusting them on line so as to optimize some overall performance measure. We present an approach based on perturbation analysis techniques, where all customer arrival processes can be arbitrary, and no information about them is required. The basis of this approach is the development of two sensitivity estimators leading to amarked slot and aphantom slot algorithm. The algorithms determine the effect of removing/ adding service slots to an existing schedule on the mean customer waiting times by directly observing the system. The optimal slot assignment probabilities are then used to design adeterministic scheduling policy based on the Golden Ratio policy. Finally, several numerical results based on a simple optimization algorithm are included.

Keywords

Scheduling random polling optimization perturbation analysis Golden Ratio policy radio network 
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References

  1. [1]
    D. Bertsekas and R. Gallager,Data Networks (Prentice-Hall, 1992).Google Scholar
  2. [2]
    O.J. Boxma and J.A. Weststrate, Waiting times in polling systems with Markovian server routing, preprint (1989).Google Scholar
  3. [3]
    P. Bremaud and FJ. Vazquez-Abad, On the pathwise computation of derivatives with respect to the rate of a point process: The phantom RPA method, Queueing Systems 10 (1992) 249–270.Google Scholar
  4. [4]
    X.R. Cao, Convergence of parameter in a stochastic environment, IEEE Trans. Autom. Contr. AC-30 (1985) 834–843.Google Scholar
  5. [5]
    E.K.P. Chong and P.J. Ramadge, Convergence of recursive optimization algorithms using IPA derivative estimates, J. Discr. Event Dyn. Syst. 1 (1992) 339–372.Google Scholar
  6. [6]
    I. Cidon and M. Sidi, Distributed assigned algorithms for multi-hop packet-radio networks, IEEE Trans. Comp. 38 (1989) 10.Google Scholar
  7. [7]
    P. Dupuis and R. Simha, On sampling-controlled stochastic approximation, IEEE Trans. Autom. Contr. AC-36 (1991) 915–924.Google Scholar
  8. [8]
    A. Ephremides and T.V. Truong, Scheduling broadcasts in multihop radio networks, IEEE Trans. Commun. 38(4) (1990).Google Scholar
  9. [9]
    M.C. Fu and J.Q. Hu, On choosing the characterization for smoothed perturbation analysis, IEEE Trans. Autom. Contr. AC-36 (1991).Google Scholar
  10. [10]
    P. Glasserman,Gradient Estimation via Perturbation Analysis (Kluwer Academic Publ., Boston, 1991).Google Scholar
  11. [11]
    P. Glynn, Likelihood ratio gradient estimation: An overview,Proc. 1987 Winter Simulation Conf. (1987) pp. 336–375.Google Scholar
  12. [12]
    W.B. Gong and Y.C. Ho, Smoothed perturbation analysis of discrete-event dynamic systems, IEEE Trans. Autom. Contr. AC-32 (1987) 858–866.Google Scholar
  13. [13]
    W.B. Gong and H. Schulzrinne, Application of smoothed perturbation analysis to probabilistic routing, Math. Comp. Simul. 34 (1992) 467–485.Google Scholar
  14. [14]
    H. Heffes and D. Lucantoni, A Markov modulated characterization of voice and data and related statistical multiplexer performance, IEEE J. Select. Areas Commun. SAC-4 (1984) 856–867.Google Scholar
  15. [15]
    Y.C. Ho and X. Cao,Perturbation Analysis of Discrete Event Dynamic Systems (Kluwer Academic Publ., Boston, 1991).Google Scholar
  16. [16]
    A. Itai and Z. Rosberg, A golden ratio policy for a multiple-access channel, IEEE Trans. Autom. Contr. AC-29(8) (1984).Google Scholar
  17. [17]
    V. Julka, C.G. Cassandras and W.B. Gong, Sample path techniques for admission control in multiclass queueing systems with general arrival processes,Proc. Conf. on Information Science and Systems (1992) pp. 227–232.Google Scholar
  18. [18]
    J. Kiefer and J. Wolfowitz, Stochastic estimation of the maximum of a regression function, Ann. Math. Stat. 23 (1952) 462–466.Google Scholar
  19. [19]
    L. Kleinrock and H. Levy, The analysis of random polling systems, Oper. Res. 36(5) (1988).Google Scholar
  20. [20]
    H.J. Kushner and D. S. Clark,Stochastic Approximation for Constrained and Unconstrained Systems (Springer, 1978).Google Scholar
  21. [21]
    H. Levy and M. Sidi, Polling systems: Applications, modeling and optimization, IEEE Trans. Commun. 38(10) (1990).Google Scholar
  22. [22]
    B. Mohanty and CG. Cassandras, The effect of model uncertainty on some optimal routing problems, J. Optim. Th. Appl. 77 (1993) 256–290.Google Scholar
  23. [23]
    R. Nelson and L. Kleinrock, Spatial TDMA: A collision-free multihop channel access protocol, IEEE Trans. Commun. COM-33 (1985) 934–944.Google Scholar
  24. [24]
    M. Reiman and A. Weiss, Sensitivity analysis for simulations via likelihood ratios, Oper. Res. 37 (1989) 830–844.Google Scholar
  25. [25]
    H. Robbins and S. Monro, A stochastic approximation method, Ann. Math. Stat. 22 (1951) 400–407.Google Scholar
  26. [26]
    Z. Rosberg and M. Sidi, TDM policies in multistation packet radio networks, IEEE Trans. Commun. COM-37 (1989) 1.Google Scholar
  27. [27]
    Z. Rosberg and D. Towsley, Customer routing to parallel servers with different rates, IEEE Trans. Autom. Contr. AC-30 (1985) 1140–1143.Google Scholar
  28. [28]
    R. Suri and X. Cao, The phantom and marked customer methods for optimization of closed queueing networks with blocking and general service times, ACM Perform. Eval. Rev. 12 (1986) 234–256.Google Scholar
  29. [29]
    H. Takagi, Queueing analysis of polling models: An update,Stock. Anal. of Comp. and Comm. Syst., ed. H. Takagi (1990).Google Scholar
  30. [30]
    L. Tassiulas and A. Ephremides, Stability properties of constrained queueing systems and scheduling policies for maximum throughput in multihop radio networks,Proc. 29th IEEE Conf. on Decision and Control (1990).Google Scholar
  31. [31]
    F.J. Vazquez-Abad and P. L'Ecuyer, Comparing alternative methods for derivative estimation when IPA does not apply directly,Proc. 1988 Winter Simulation Conf.Google Scholar

Copyright information

© J.C. Baltzer AG, Science Publishers 1995

Authors and Affiliations

  • Christos G. Cassandras
    • 1
  • Vibhor Julka
    • 1
  1. 1.Department of Electrical and Computer EngineeringUniversity of MassachusettsAmherstUSA

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