Computational aspects of the workload distribution in the MMPP/GI/1 queue

We show how the analysis of Markov modulated rate processes can be used to address the problem of computing the distribution of W, the stationary workload in the MMPP/GI/1 queue. Using the results of papers by Anick et al. (1982); Mitra (1988); and Elwalid et al. (1991), we present the decomposition properties of the Laplace transform of W and efficient computational algorithms for computing its distribution. The techniques are also applied to compute the bounds on the distribution of W developed by Liu et al. (see JACM, vol.44, no.2, p.366-94, 1997). Numerical results illustrating the usefulness of the methods are given for the case of the superposition of independent, nonidentical sources.

[1]  Philippe Nain,et al.  Upper and Lower Bounds for the Multiplexing of Multiclass Markovian on/off Sources , 1996, Perform. Evaluation.

[2]  Donald F. Towsley,et al.  Exponential bounds with applications to call admission , 1997, JACM.

[3]  J. Keilson.  Markov Chain Models--Rarity And Exponentiality , 1979 .

[4]  Alexander Graham,et al.  Kronecker Products and Matrix Calculus: With Applications , 1981 .

[5]  Ward Whitt,et al.  The Fourier-series method for inverting transforms of probability distributions , 1992, Queueing Syst. Theory Appl..

[6]  A.I. Elwalid,et al.  Statistical multiplexing with loss priorities in rate-based congestion control of high-speed networks , 1994, IEEE Trans. Commun..

[7]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[8]  Joe Brewer,et al.  Kronecker products and matrix calculus in system theory , 1978 .

[9]  Thomas E. Stern,et al.  Analysis of separable Markov-modulated rate models for information-handling systems , 1991, Advances in Applied Probability.

[10]  Wolfgang Fischer,et al.  The Markov-Modulated Poisson Process (MMPP) Cookbook , 1993, Perform. Evaluation.

[11]  Ward Whitt,et al.  Tail probabilities with statistical multiplexing and effective bandwidths in multi-class queues , 1993, Telecommun. Syst..

[12]  D. Mitra,et al.  Stochastic theory of a data-handling system with multiple sources , 1982, The Bell System Technical Journal.

[13]  Marcel F. Neuts.  The Fundamental Period of the Queue with Markov-Modulated Arrivals , 1989 .

[14]  David E. Stewart,et al.  Meschach : matrix computations in C , 1994 .

[15]  D. Mitra.  Stochastic theory of a fluid model of producers and consumers coupled by a buffer , 1988, Advances in Applied Probability.

[16]  Debasis Mitra,et al.  Markovian Arrival and Service Communication Systems: Spectral Expansions, Separability and Kronecker-Product Forms , 1995 .

[17]  W. Whitt,et al.  The transient BMAP/G/l queue , 1994 .