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Introduction to Probability Models.

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Lectu re Notes in Economics and Mathematical Systems

We study the performance of a statistical multiplexer whose inputs consist of a superposition of packetized voice sources and data. The performance analysis predicts voice packet delay distributions, which usually have a stringent requirement, as well as data packet delay distributions. The superposition is approximated by a correlated Markov modulated Poisson process (MMPP), which is chosen such that several of its statistical characteristics identically match those of the superposition. Matrix analytic methods are then used to evaluate system performance measures. In particular, we obtain moments of voice and data delay distributions and queue length distributions. We also obtain Laplace-Stieitjes transforms of the voice and data packet delay distributions, which are numerically inverted to evaluate tails of delay distributions. It is shown how the matrix analytic methodology can incorporate practical system considerations such as finite buffers and a class of overload control mechanisms discussed in the literature. Comparisons with simulation show the methods to be accurate. The numerical results for the tails of the voice packet delay distribution show the dramatic effect of traffic variability and correlations on performance.

A Markov Modulated Characterization of Packetized Voice and Data Traffic and Related Statistical Multiplexer Performance

The versatile Markovian point process was introduced by M. F. Neuts in 1979. This is a rich class of point processes whichcontains many familiar arrival process as very special cases. Recently, the Batch Markovian Arrival Process, a class of point processes which was subsequently shown to be equivalent to Neuts’ point process, has been studied using a more transparent notation. Recent results in the matrix-analytic approach to queueing theory have substantially reduced the computational complexity of the algorithmic solution of single server queues with a general Markovian arrival process. We generalize these results to the single server queue with the batch arrival process and emphasize the resulting simplifications. Algorithms for the special cases of the PH/G/l and MMPP/G/1 queues are highlighted as these models are receiving renewed attention in the literature and the new algorithms proposed here are simpler than existing ones. In particular, the PH/G/1 queue has additional structure which further enh...

New results on the single server queue with a batch markovian arrival process

In this article we shall present a summary of the various stochastic approaches and applications to chemical reaction kinetics, but before discussing these we first briefly introduce the basic ideas and definitions of classical or deterministic chemical kinetics. One of the basic questions to which chemists address themselves is the rate of chemical reactions, or in other words, how long it takes for a chemical reaction to attain completion, or equilibrium. Apparently the first significant quantitative investigation was made in 1850 by L. Wilhelmy [93]. He studied the inversion of sucrose (cane sugar) in aqueous solutions of acids, whose reaction is He found empirically that the rate of decrease of concentration of sucrose was simply proportional to the concentration remaining unconverted, i.e., if S(t) is the concentration of sucrose, then The constant of proportionality is called the rate constant of the reaction. If S o is the initial concentration of sucrose, then Since then an enormous number of reactions has been studied and the field of chemical kinetics is now one of the largest areas of chemical research. The importance of the field lies in the fact that it yields concise expressions for the time dependence of reactions, predicts yields, optimum economic conditions, and gives one much insight into the actual molecular processes involved. The detailed molecular picture of a reaction process is called the mechanism of the reaction.

G. F. Newell

Papers

Introduction to the Theory of Queues.