Characterizing Linear Birth And Death Processes


L. L. Hoffman

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J. Am. Stat. Assoc.


Markov Process; Simulation; Steady State; Statistics & Probability


This research determined the manner of convergence of certain Markov processes to their steady state limiting distributions. This article looks at linear birth and death processes with birth rate at each state determined by the immigration constant a and the natural growth multiplier b; with death rate at each state determined by fixed execution constant c and the natural declination multiplier d. All parameters are nonnegative. There is a reflective barrier at state 0. It is shown that when the natural growth multiplier is less than the declination parameter a limiting distribution exists, that is, when the multiplier difference is negative. We define a modal indicator as the ratio of the sum of the death parameters c and d diminished by immigration a to the multiplier difference. It is shown that when the modal indicator is negative then the mode occurs at state 0. When the indicator is an integer then the process is bimodal with the mode at that integral value and at the next larger integer. When the indicator is not an integer then the mode occurs at the first integral value greater than the modal indicator. Additionally, bounds for the birth probabilities and the tail probabilities are derived. These equations are applied to an example in the area of computer performance analysis. The objective of studying the characteristics of the limiting distribution is to understand the difficulties involved when simulating these processes. The most extensively studied of these types of Markov processes is the M/M/1 process (Poisson arrivals to one server having exponential service times). This is a simple case of a linear birth and death process where b = d = 0. Researchers have suggested speeding convergence by initializing the process in a state other than 0. This article reveals that zero is a good choice for the M/M/1 process, but it is not the best choice for the general linear birth and death process. Also, practitioners have devised empirical methods to decide when the number of iterations N is sufficient to declare convergence. This article presents bounds on the tail probabilities in order to guide the selection of N at the onset of simulation. The sense of the theorems proved below can be captured by these statements: 1. The natural growth multiplier b must be less than the declination parameter d to insure convergence. 2. If the modal indicator is relatively large then intitializing the process in the modal state might be wise. 3. Large values of the declination parameter d relative to the growth multiplier b will result in a better behaved process.

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Journal of the American Statistical Association





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