Conservation Laws

  • Hong Chen
  • David D. Yao
Part of the Stochastic Modelling and Applied Probability book series (SMAP, volume 46)


Conservation laws belong to the most fundamental principles that govern the dynamics, or law of motion, of a wide range of stochastic systems. Under conservation laws, the performance space of the system becomes a polymatroid, that is, a polytope with a matroid-like structure, with all the vertices corresponding to the performance under priority rules, and all the vertices are easily identified. Consequently, the optimal control problem can be translated into an optimization problem. When the objective is a linear function of the performance measure, the optimization problem becomes a special linear program, for which the optimal solution is a vertex that is directly determined by the relative order of the cost coefficients in the linear objective. This implies that the optimal control is a priority rule that assigns priorities according to exactly the order of the cost coefficients.


Busy Period Priority Rule Submodular Function Performance Vector Priority Index 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [1]
    Asmussen, S., Applied Probability and Queues. Wiley, Chichester, U.K., 1987.MATHGoogle Scholar
  2. [2]
    Bertsimas, D., The Achievable Region Method in the Optimal Control of Queueing Systems; Formulations, Bounds and Policies. Queueing Systems, 21 (1995), 337–389.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    Bertsimas, D. and Niivo-Mora, J., Conservation Laws, Extended Polymatroid and Multi-Armed Bandit Problems: A Unified Approach to Indexable Systems. Mathematics of Operations Research, 21 (1996), 257–306.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    Bertsimas, D. Paschalidis, I.C. and Tsitsiklis, J.N., Optimization of Multiclass Queueing Networks: Polyhedral and Nonlinear Characterization of Achievable Performance. Ann. Appl. Prob., 4 (1994), 43–75.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    Baras, J.S., Dorsey, A.J. and Makowski, A.M., Two Competing Queues with Linear Cost: the pc Rule Is Often Optimal. Adv. Appl. Prob., 17 (1985), 186–209.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    Buyukkoc, C., Varaiya, P. And Walrand, J., The cu Rule Revisited. Adv. Appl. Prob., 30 (1985), 237–238.CrossRefGoogle Scholar
  7. [7]
    Chvátal, V., Linear Programming. W.H. Freeman, New York, 1983.MATHGoogle Scholar
  8. [8]
    Coffman, E. and Mitrani, I., A Characterization of Waiting Time Performance Realizable by Single Server Queues. Operations Research, 28 (1980), 810–821.MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    Cox, D.R. and Smith, W.L., Queues. Methunen, London, 1961.Google Scholar
  10. [10]
    Dacre, K.D., Glazebrook, K.D., and Nino-Mora, J., The Achievable Region Approach to the Optimal Control of Stochastic Systems. J. Royal Statist. Soc. (1999).Google Scholar
  11. [11]
    Dunstan, F.D.J. and Welsh, D.J.A., A Greedy Algorithm for Solving a Certain Class of Linear Programmes. Math. Programming, 5 (1973), 338–353.MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    Edmonds, J., Submodular Functions, Matroids and Certain Polyhedra. Proc. Int. Conf. on Combinatorics (Calgary), Gordon and Breach, New York, 69–87, 1970.Google Scholar
  13. [13]
    Federgruen, A. And Groenevelt, H., The Greedy Procedure for Resource Allocation Problems: Necessary and Sufficient Conditions for Optimality. Operations Res., 34 (1986), 909–918.MathSciNetMATHCrossRefGoogle Scholar
  14. [14]
    Federgruen, A. and Groenevelt, H., The Impact of the Composition of the Customer Base in General Queueing Models. J. Appl. Prob., 24 (1987), 709–724.MathSciNetMATHCrossRefGoogle Scholar
  15. [15]
    Federgruen, A. and Groenevelt, H., M/G/c Queueing Systems with Multiple Customer Classes: Characterization and Control of Achievable Performance under Non-Preemptive Priority Rules. Management Science, 34 (1988), 1121–1138.MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    Federgruen, A. and Groenevelt, H., Characterization and Optimization of Achievable Performance in Queueing Systems. Operations Res., 36 (1988), 733–741.MathSciNetMATHCrossRefGoogle Scholar
  17. [17]
    Fong, L.L. and Squillante, M.S., Time-Function Scheduling: A General Approach to Controllable Resource Management. Ibm Research Report RC-20155, Ibm Research Division, T.J. Watson Research Center, Yorktown Hts., New York, NY 10598, 1995.Google Scholar
  18. [18]
    Franaszek, P.A. and Nelson, R.D., Properties of Delay Cost Scheduling in Timesharing Systems. Ibm Research Report RC-13777, Ibm Research Division, T.J. Watson Research Center, Yorktown Hts., New York, NY 10598, 1990.Google Scholar
  19. [19]
    Glazebrook, K.D. and Garbe, R., Almost Optimal Policies for Stochastic Systems which Almost Satisfy Conservation Laws. Preprint, 1996.Google Scholar
  20. [20]
    Gelenbe, E. and Mitrani, I., Analysis and Synthesis of Computer Systems. Academic Press, London, 1980.MATHGoogle Scholar
  21. [21]
    Gittins, J.C., Bandit Processes and Dynamic Allocation Indices (with discussions). J. Royal Statistical Society, Ser. B,41 (1979), 148–177.Google Scholar
  22. [22]
    Gittins, J.C., Multiarmed Bandit Allocation Indices. Wiley, Chichester, 1989.Google Scholar
  23. [23]
    Gittins, J.C. and Jones, D.M., A Dynamic Allocation Index for the Sequential Design of Experiments. In: Progress in Statistics: European Meeting of Statisticians, Budapest, 1972, J. Gani, K. Sarkadi and I. Vince (eds.), North-Holland, Amsterdam, 1974, 241–266.Google Scholar
  24. [24]
    Glasserman, P. and Yao, D.D., Monotone Structure in Discrete-Event Systems. Wiley, New York, 1994.MATHGoogle Scholar
  25. [25]
    Glasserman, P. and Yao, D.D., Monotone Optimal Control of Permutable Gsmp’s. Mathematics of Operations Research, 19 (1994), 449–476.MathSciNetMATHCrossRefGoogle Scholar
  26. [26]
    Grötschel, M., Lovasz, L., and Schrijver, A., Geometric Algorithms and Combinatorial Optimization, second corrected edition. Springer-Verlag, Berlin, 1993.MATHCrossRefGoogle Scholar
  27. [27]
    Harrison, J.M., Dynamic Scheduling of a Multiclass Queue: Discount Optimality. Operations Res., 23 (1975), 270–282.MathSciNetMATHCrossRefGoogle Scholar
  28. [28]
    Harrison, J.M., The Bigstep Approach to Flow Management in Stochastic Processing Networks. In: Stochastic Networks: Theory and Applications, Kelly, Zachary, and Ziedens (eds.), Royal Statistical Society Lecture Note Series, #4, 1996, 57–90.Google Scholar
  29. [29]
    Harrison, J.M. and Wein, L., Scheduling Networks of Queues: Heavy Traffic Analysis of a Simple Open Network. Queueing Systems, 5 (1989), 265–280.MathSciNetMATHCrossRefGoogle Scholar
  30. [30]
    Kleinrock, L., A Delay Dependent Queue Discipline. Naval Research Logistics Quarterly, 11 (1964), 329–341.MathSciNetMATHCrossRefGoogle Scholar
  31. [31]
    Kleinrock, L., Queueing Systems, Vol. 2. Wiley, New York, 1976.MATHGoogle Scholar
  32. [32]
    Klimov, G.P., Time Sharing Service Systems, Theory of Probability and Its Applications,19(1974), 532–551 (Part I) and 23(1978), 314–321 (Part II).Google Scholar
  33. [33]
    Lai, T.L. and Ying, Z., Open Bandit Processes and Optimal Scheduling of Queueing Networks. Adv. Appl. Prob.,20(1988), 447–472.Google Scholar
  34. [34]
    Lu, Y., Dynamic Scheduling of Stochastic Networks with Side Constraints. Ph.D. thesis, Columbia University, 1998.Google Scholar
  35. [35]
    Meilijson, I. and Weiss, G., Multiple Feedback at a Single-Server Station. Stochastic Proc. and Appl., 5 (1977), 195–205.MathSciNetMATHCrossRefGoogle Scholar
  36. [36]
    Ross, K.W. And Yao, D.D., Optimal Dynamic Scheduling in Jackson Networks. Ieee Transactions on Automatic Control, 34 (1989), 47–53.MATHCrossRefGoogle Scholar
  37. [37]
    Ross, K.W. and Yao, D.D., Optimal Load Balancing and Scheduling in a Distributed Computer System. Journal of the Association for Computing Machinery, 38 (1991), 676–690.MATHCrossRefGoogle Scholar
  38. [38]
    Shanthikumar, J.G. and Sumita, U., Convex Ordering of Sojourn Times in Single-Server Queues: Extremal Properties of Fifo and Lifo Service Disciplines. J. Appl. Prob., 24 (1987), 737–748.MathSciNetMATHCrossRefGoogle Scholar
  39. [39]
    Shanthikumar J.G. and Yao D.D., Multiclass queueing systems: polymatroid structure and optimal scheduling control. Operation Research, 40 (1992), Supplement 2, S293–299.Google Scholar
  40. [40]
    Smith, W.L., Various Optimizers for Single-Stage Production. Naval Research Logistics Quarterly, 3 (1956), 59–66.MathSciNetCrossRefGoogle Scholar
  41. [41]
    Tcha, D. and Pliska, S.R., Optimal Control of Single-Server Queueing Networks and Multiclass M/G/1 Queues with Feedback. Operations Research, 25 (1977), 248–258.MathSciNetMATHCrossRefGoogle Scholar
  42. [42]
    Tsoucas, P., The Region of Achievable Performance in a Model of Klimov. Ibm Research Report RC-16543, Ibm Research Division, T.J. Watson Research Center, Yorktown Hts., New York, NY 10598, 1991.Google Scholar
  43. [43]
    Varaiya, P., Walrand, J., and Buyyokoc, C., Extensions of the Multiarmed Bandit Problem: The Discounted Case. IEEE Trans. Automatic Control, 30 (1985), 426–439.MATHCrossRefGoogle Scholar
  44. [44]
    Weber, R., On the Gittins Index for Multiarmed Bandits. Annals of Applied Probability, (1992), 1024–1033.Google Scholar
  45. [45]
    Weber, R. and Stidham, S., JR., Optimal Control of Service Rates in Networks of Queues. Adv. Appl. Prob., 19 (1987), 202–218.MathSciNetMATHCrossRefGoogle Scholar
  46. [46]
    Weiss, G., Branching Bandit Processes. Probability in the Engineering and Informational Sciences, 2 (1988), 269–278.MATHCrossRefGoogle Scholar
  47. [47]
    Welsh, D., Matroid Theory, (1976), Academic Press, London.MATHGoogle Scholar
  48. [48]
    Whittle, P., Multiarmed Bandits and the Gittins Index. J. Royal Statistical Society, Ser. B, 42 (1980), 143–149.MathSciNetMATHGoogle Scholar
  49. [49]
    Whittle, P., Optimization over Time: Dynamic Programming and Stochastic Control, vols. I, II, Wiley, Chichester, 1982.MATHGoogle Scholar
  50. [50]
    Yao, D.D. and Shanthikumar, J.G., Optimal Scheduling Control of a Flexible Machine. IEEE Trans. on Robotics and Automation, 6 (1990), 706–712.CrossRefGoogle Scholar
  51. [51]
    Yao, D.D. and Zhang, L., Stochastic Scheduling and Polymatroid Optimization, Lecture Notes in Applied Mathematics, 33, G. Ying and Q. Zhang (eds.), Springer-Verlag, 1997, 333–364.Google Scholar
  52. [52]
    Zhang, L., Reliability and Dynamic Scheduling in Stochastic Networks. Ph.D. thesis, Columbia University, 1997.Google Scholar

Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • Hong Chen
    • 1
  • David D. Yao
    • 2
  1. 1.Faculty of Commerce and Business AdministrationUniversity of British ColumbiaVancouverCanada
  2. 2.Department of Operations Research and Industrial EngineeringColumbia UniversityNew YorkUSA

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