The distribution of a random sum of exponentials with an application to a traffic problem

Author:
Frank Recker

Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom **76** (2007).

Journal:
Theor. Probability and Math. Statist. **76** (2008), 159-167

MSC (2000):
Primary 60G40, 90B20

Published electronically:
July 17, 2008

MathSciNet review:
2368748

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

**1.**Søren Asmussen,*Applied probability and queues*, Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics, John Wiley & Sons, Ltd., Chichester, 1987. MR**889893****2.**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**, 10.1080/15326349808807488**3.**E. Grycko and O. Moeschlin,*A concept of optimal control at a bottleneck with symmetric volume of traffic*, Comm. Statist. Stochastic Models**14**(1998), no. 3, 585–600. MR**1621334**, 10.1080/15326349808807489**4.**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****5.**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

DOI:
https://doi.org/10.1090/S0094-9000-08-00740-0

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