The distribution of a random sum of exponentials with an application to a traffic problem. (English) Zbl 1199.60144

Recker, Frank

Teor. Jmovirn. Mat. Stat. 76, 142-149 (2007) and Theory Probab. Math. Stat. 76, 159-167 (2008).

Let \(\lambda,\tau\in\mathbb R\), \(\lambda>0,\tau>0\), and let \(\{X_n\}_{n\in \mathbb N}\) be a sequence of i.i.d. \(\text{exp}(\lambda)\)-distributed random variables. Further let \(T\) be the stopping time \(T:=\min\{n\in{\mathbb N}| X_n\geq\tau\}\). Then the waiting time for intensity \(\lambda\) and minimal gap \(\tau\) is the random variable \(Z:=\sum_{n=1}^{T-1}X_n\). In renewal theory, the intensity is defined as the reciprocal of the expected inter-arrival time. In this case the renewal process is a Poisson process and hence this value is exactly that of the parameter \(\lambda\). An expression is given for the distribution function of this waiting time. It has applications in determining the waiting time for a large gap in a Poisson process. As an example, a traffic problem is considered, where such a waiting time occurs.

Reviewer: Mikhail P. Moklyachuk (Kyïv)

MSC:

60G40	Stopping times; optimal stopping problems; gambling theory
60K35	Interacting random processes; statistical mechanics type models; percolation theory
90B20	Traffic problems in operations research
90B22	Queues and service in operations research

Keywords:

Poisson process; random sum of exponentially distributed random variables; stopping time; waiting time; traffic problem