Extreme values of birth and death processes and queues. (English) Zbl 0637.60098
Consider a birth and death process and the maxima in a time interval that can increase indefinitely. In general, even with linear transformations, these maxima do not have a limiting non-degenerate distribution, although in some cases they have a limiting degenerate distribution. Under certain conditions not only Fréchet distributions (with shape parameter 1) or Gumbel distributions can be the limiting distributions but also a new limiting distribution appears to fit better to the data. The results, under the conditions stated, are valid for transient and recurrent birth and death processes and connected queues.
Reviewer: J.Tiago de Oliveira
MSC:
| 60J80 | Branching processes (Galton-Watson, birth-and-death, etc.) |
| 60K25 | Queueing theory (aspects of probability theory) |
| 60F99 | Limit theorems in probability theory |
Keywords:
birth and death process; Fréchet distributions; Gumbel distributions; transient and recurrent birth and death processes; queuesReferences:
| [1] | Anderson, C. W., Extreme value theory for a class of discrete distributions with applications to some stochastic processes, J. Appl. Prob., 7, 99-113 (1970) · Zbl 0192.54202 |
| [2] | Chung, K. L., Markov Chains with Stationary Transition Probabilities (1967), Springer-Verlag: Springer-Verlag New York · Zbl 0146.38401 |
| [3] | Cohen, J. W., The Single Server Queue (1982), North-Holland: North-Holland Amsterdam · Zbl 0481.60003 |
| [4] | Galambos, J., The Asymptotic Theory of Extreme Order Statistics (1978), Wiley: Wiley New York · Zbl 0381.62039 |
| [5] | Heyde, C., On the growth of the maximum queue length in a stable queue, Operations Res., 19, 447-452 (1971) · Zbl 0218.60096 |
| [6] | Iglehart, D. L., Extreme values in the GI/G/1 queue, Ann. Math. Statist., 43, 627-635 (1972) · Zbl 0238.60072 |
| [7] | Leadbetter, M. R.; Lindgren, G.; Rootzén, R., Extremes and Related Properties of Random Sequences and Processes (1983), Springer-Verlag: Springer-Verlag New York · Zbl 0518.60021 |
| [8] | Serfozo, R. F., Extreme values of queue lengths in M/G/1 and GI/M1 systems, Mathematics of Operations Research (1988), to appear in · Zbl 0651.60031 |
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