On the analysis of the virtual waiting time in open queueing networks
An open network of single server stations with waiting rooms of unlimited capacities and operating under first come first serve (FCFS) discipline is considered. The external arrival streams of customers at the stations are independent renewal processes and the service times at the nodes are generally i.i.d. distributed and independent of the external arrival processes. The routing of the customers after finishing service at the stations is Markovian and governed by a transition matrix. The main result of the paper is that under proper overload of all stations and further moment conditions the vector of the virtual waiting time processes of the stations -- associated to a considered sequence of arrival and service processes as well as routing processes -- converges under an appropriate scaling to the vector process of independent standard Wiener processes. The proof uses results from \textit{L. L. Sakalauskas} and \textit{S. Minkevičius} [Eur. J. Oper. Res. 120, No. 3, 632--640 (2000; Zbl 0981.60093)]. Further an application to a reliability model is given. Further the authors state that the law of iterated logarithm for the virtual waiting time of the stations can be proved similarly as their main theorem of the paper.
scientific article; zbMATH DE number 5549560
scientific article