Abstract
The epochs in which successive increases—or decreases—in the levels of a quasi birth-and-death process QBD process occur are of special interest. In queueing models they usually correspond to the times of successive increments or decrements. We derive general formulas for the variances of the times between successive increments or decrements in a stationary QBD process. In the case of the decrements we also obtain the variance of the time between an arbitrary decrement and the n-th subsequent decrement. These variances provide tractable descriptors of the relative variability of the input and output processes of queues that can be modeled as QBD processes.
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References
A. T. Andersen and B. F. Nielsen, “On the use of second order descriptions to predict quieting behaviour of MAPs” Submitted for publication, 1998.
G. R. Britran and S. Dasu, “Analysis of the ? Ph i/Ph/1 Queue,” Operations Research vol. 42 pp. 158-174, 1994.
D. R. Cox and P. A. W. Lewis, The Statistical Analysis of Series of Events: Methuen: 1966.
D. J. Daley, “Queueing output processes,” Advances in Applied Probability vol. 8(2) pp. 395-415, 1976.
G. Latouche and V. Ramaswami, Introduction to Matrix Analytic Methods in Stochastic Modeling, SIAM Publications, 1999.
D. M. Lucantoni, “The BMAP/G/1 queue: A Tutorial,” In Performance Evaluation of Computer and Communication systems: Joint Tutorial Papers of Performance '93 and Sigmetrics '93, edited by Lorenzo Donatiello and Randolph Nelson, (Springer-Verlag, Berlin, 1993), pp. 330-358.
M. F. Neuts, Matrix-geometric Solutions in Stochastic Model: An Algorithmic Approach, The Johns Hopkins University Press, 1981.
A. Pacheco and N. U. Prabhu, “Markov-additive processes of arrivals.” In Advances in Queueing: Theory, Methods and Open Problems, CRC Press, 1995, edited by Jewgeni H. Dshalalow, pp. 167-194.
D. H. Shi, J. Guo, and L. Liu, “SPH-Distributions and the Rectangle-Iterative Algorithm,” In Matrix-Analytic Methods in Stochastic Models, Lecture Notes in Pure and Applied Mathematics, Vol. 183, Marcel Dekker Inc.: 1996, edited by Srinivas R. Chakravarthy and Attahiru S. Alfa, pp. 207-224.
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Andersen, A.T., Neuts, M.F. & Nielsen, B.F. Lower Order Moments of Inter-transition Times in the Stationary QBD Process. Methodology and Computing in Applied Probability 2, 339–357 (2000). https://doi.org/10.1023/A:1010037400622
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DOI: https://doi.org/10.1023/A:1010037400622