The distribution of a random sum of exponentials with an application to a traffic problem
Author:
Frank Recker
Journal:
Theor. Probability and Math. Statist. 76 (2008), 159-167
MSC (2000):
Primary 60G40, 90B20
DOI:
https://doi.org/10.1090/S0094-9000-08-00740-0
Published electronically:
July 17, 2008
MathSciNet review:
2368748
Full-text PDF Free Access
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Abstract: We study a random sum of exponentially distributed random variables. The stopping time is defined to be the first realization that is greater than or equal to a given constant. We will derive an expression for the distribution function of this sum. This has applications in determining the waiting time for a large gap in a Poisson process. As an example, we will give a traffic problem, where such a waiting time occurs.
References
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- E. Grycko and O. Moeschlin, A criterion for the occurrence or non-occurrence of a traffic collapse at a bottleneck, Comm. Statist. Stochastic Models 14 (1998), no. 3, 571–584. MR 1621330, DOI https://doi.org/10.1080/15326349808807488
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- S. P. Meyn and R. L. Tweedie, Markov chains and stochastic stability, Communications and Control Engineering Series, Springer-Verlag London, Ltd., London, 1993. MR 1287609
- F. Recker, On the asymptotical queue length in vehicular traffic confluence (2005). (to appear)
References
- S. Asmussen, Applied Probability and Queues, Wiley, New York, 1987. MR 889893 (89a:60208)
- E. Grycko and O. Moeschlin, A criterion for the occurrence or non-occurrence of a traffic collapse at a bottleneck, Commun. Statist. Stochastic Models 14 (1998), no. 3, 571–584. MR 1621330 (99g:90042)
- E. Grycko and O. Moeschlin, A concept of optimal control at a bottleneck with symmetric volume of traffic, Commun. Stat. Stochastic Models 14 (1998), no. 3, 585–600. MR 1621334 (99g:90043)
- S. P. Meyn and R. L. Tweedie, Markov Chains and Stochastic Stability, 2nd. Ed., Springer, London, 1993. MR 1287609 (95j:60103)
- F. Recker, On the asymptotical queue length in vehicular traffic confluence (2005). (to appear)
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Additional Information
Frank Recker
Affiliation:
Department of Mathematics, University of Hagen, D-58084 Hagen, Germany
Email:
frank.recker@fernuni-hagen.de
Keywords:
Poisson process,
stopping time,
queuing theory,
traffic problems
Received by editor(s):
October 3, 2005
Published electronically:
July 17, 2008
Article copyright:
© Copyright 2008
American Mathematical Society
