Bounds for performance characteristics : a systematic approach via cost structures

In this paper we present a systematic approach to the construction of bounds for the average costs in Markov chains with possibly infinitely many states. The technique used to prove the bounds is based on dynamic programming. Most performance characteristics of Markovian systems can be represented by the average costs for some appropriately chosen cost structure. Therefore, the approach can be used to generate bounds for relevant performance characteristics. The approach is demonstrated for the shortest queue model. It is shown how for this model several bounds for the mean waiting time can be constructed. We include numerical results to demonstrate the quality of these bounds

[1]  J. Kingman.  Two Similar Queues in Parallel , 1961 .

[2]  J. Hunter.  Two Queues in Parallel , 1969 .

[3]  G. J. Foschini,et al.  A Basic Dynamic Routing Problem and Diffusion , 1978, IEEE Trans. Commun..

[4]  Wk Grassmann,et al.  Transient and steady state results for two parallel queues , 1980 .

[5]  Marcel F. Neuts,et al.  Matrix-Geometric Solutions in Stochastic Models , 1981 .

[6]  B. W. Conolly,et al.  THE AUTOSTRADA QUEUEING PROBLEM , 1984 .

[7]  Ilya Gertsbakh,et al.  The shorter queue problem: A numerical study using the matrix-geometric solution☆ , 1984 .

[8]  Shlomo Halfin.  The shortest queue problem , 1985 .

[9]  Nico M. van Dijk.  A formal proof for the insensitivity of simple bounds for finite multi-server non-exponential tandem queues based on monotonicity results , 1987 .

[10]  M. J. M. Posner,et al.  Algorithmic and approximation analyses of the shorter queue model , 1987 .

[11]  Bernard F. Lamond,et al.  Simple Bounds for Finite Single-Server Exponential Tandem Queues , 1988, Oper. Res..

[12]  Nico M. van Dijk,et al.  Simple Bounds for Queueing Systems with Breakdowns , 1988, Perform. Evaluation.

[13]  Randolph D. Nelson,et al.  An approximation to the response time for shortest queue routing , 1989, SIGMETRICS '89.

[14]  Jan van der Wal,et al.  Simple bounds and monotonicity results for finite multi-server exponential tandem queues , 1989, Queueing Syst. Theory Appl..

[15]  J. van der Wal.  Monotonicity of the throughput of a closed exponential queueing network in the number of jobs , 1989 .

[16]  Nico M. Van Dijk,et al.  The Importance of Bias Terms for Error Bounds and Comparison Results , 1989 .

[17]  Nico M. van Dijk,et al.  Transient Error Bound Analysis for Continuous-Time Markov Reward Structures , 1991, Perform. Evaluation.

[18]  Ivo J. B. F. Adan,et al.  Analysis of the symmetric shortest queue problem , 1990 .

[19]  Nico M. van Dijk,et al.  Truncation of Markov Chains with Applications to Queueing , 1991, Oper. Res..

[20]  John C. S. Lui,et al.  Algorithmic approach to bounding the mean response time of a minimum expected delay routing system , 1992, SIGMETRICS '92/PERFORMANCE '92.

[21]  Ger Koole,et al.  On the Assignment of Customers to Parallel Queues , 1992, Probability in the Engineering and Informational Sciences.

[22]  J. P. C. Blanc.  The Power-Series Algorithm Applied to the Shortest-Queue Model , 1992, Oper. Res..

[23]  Vaidyanathan Ramaswami,et al.  A logarithmic reduction algorithm for quasi-birth-death processes , 1993, Journal of Applied Probability.

[24]  Ivo J. B. F. Adan,et al.  Matrix-geometric analysis of the shortest queue problem with threshold jockeying , 1993, Oper. Res. Lett..

[25]  Ravindra K. Ahuja,et al.  Network Flows: Theory, Algorithms, and Applications , 1993 .

[26]  Nico M. van Dijk.  Queueing networks and product forms - a systems approach , 1993, Wiley-Interscience series in systems and optimization.

[27]  Randolph D. Nelson,et al.  An Approximation for the Mean Response Time for Shortest Queue Routing with General Inerarrival and Service Times , 1993, Perform. Evaluation.

[28]  Ivo J. B. F. Adan,et al.  Upper and lower bounds for the waiting time in the symmetric shortest queue system , 1994, Ann. Oper. Res..

[29]  Geert-Jan van Houtum,et al.  The symmetric longest queue system , 1994 .

[30]  Donald F. Towsley,et al.  Bounding the Mean Response Time of the Minimum Expected Delay Routing Policy: An Algorithmic Approach , 1995, IEEE Trans. Computers.

[31]  Yiqiang Zhao,et al.  Queueing Analysis of a Jockeying Model , 1995, Oper. Res..

[32]  van Gjjan Geert-Jan Houtum,et al.  New approaches for multi-dimensional queueing systems , 1995 .

[33]  Geert-Jan van Houtum,et al.  Performance analysis of parallel machines with a generalized shortest queue arrival mechanism , 1997 .

[34]  Sheldon M. Ross.  Introduction to probability models , 1998 .

[35]  Ivo J. B. F. Adan,et al.  The M/M/c with critical jobs , 1998, Math. Methods Oper. Res..