Computational geometric approach to submodular function minimization for multiclass queueing systems

This paper presents an efficient algorithm for minimizing a certain class of submodular functions that arise in analysis of multiclass queueing systems. In particular, the algorithm can be used for testing whether a given multiclass M/M/1 system achieves a required average performance by an appropriate control policy. With the aid of the topological sweeping method for line arrangement, our algorithm runs in O(n) time, where n is the cardinality of the ground set. This is much faster than direct applications of general submodular function minimization algorithms.

[1]  Edward G. Coffman,et al.  A Characterization of Waiting Time Performance Realizable by Single-Server Queues , 1980, Oper. Res..

[2]  Erol Gelenbe,et al.  Analysis and Synthesis of Computer Systems , 1980 .

[3]  Martin Grötschel,et al.  The ellipsoid method and its consequences in combinatorial optimization , 1981, Comb..

[4]  L. Lovász,et al.  Geometric Algorithms and Combinatorial Optimization , 1981 .

[5]  Leonidas J. Guibas,et al.  Topologically sweeping an arrangement , 1986, STOC '86.

[6]  Herbert Edelsbrunner,et al.  Algorithms in Combinatorial Geometry , 1987, EATCS Monographs in Theoretical Computer Science.

[7]  A. Federgruen,et al.  M / G / c queueing systems with multiple customer classes: characterization and control of achievable performance under nonpreemptive priority rules , 1988 .

[8]  Awi Federgruen,et al.  Characterization and Optimization of Achievable Performance in General Queueing Systems , 1988, Oper. Res..

[9]  Leonidas J. Guibas,et al.  Topological Sweeping in Three Dimensions , 1990, SIGAL International Symposium on Algorithms.

[10]  Leonidas J. Guibas,et al.  Corrigendum: Topologically Sweeping an Arrangement , 1991, J. Comput. Syst. Sci..

[11]  John N. Tsitsiklis,et al.  Optimization of multiclass queuing networks: polyhedral and nonlinear characterizations of achievable performance , 1994 .

[12]  S. Sushanth Kumar,et al.  Performance bounds for queueing networks and scheduling policies , 1993, Proceedings of 32nd IEEE Conference on Decision and Control.

[13]  G. Ziegler.  Lectures on Polytopes , 1994 .

[14]  M. Dacre,et al.  The achievable region approach to the optimal control of stochastic systems , 1999 .

[15]  Alexander Schrijver,et al.  A Combinatorial Algorithm Minimizing Submodular Functions in Strongly Polynomial Time , 2000, J. Comb. Theory B.

[16]  Satoru Iwata,et al.  A combinatorial strongly polynomial algorithm for minimizing submodular functions , 2001, JACM.

[17]  C. SIAMJ..  A FASTER SCALING ALGORITHM FOR MINIMIZING SUBMODULAR FUNCTIONS∗ , 2001 .

[18]  Diane L. Souvaine,et al.  Topological Sweep in Degenerate Cases , 2002, ALENEX.

[19]  S. Thomas McCormick,et al.  Submodular Function Minimization , 2005 .

[20]  James B. Orlin,et al.  A faster strongly polynomial time algorithm for submodular function minimization , 2007, Math. Program..