THE QUEUE WITH POISSON INPUT AND GENERAL SERVICE TIMES, TREATED AS A BRANCHING PROCESS,
Abstract
The M/G/1 queue is treated as a sequence of branching processes, the duration of which constitutes a busy period. The first generation in each branching process consists of the customers present at the beginning of the busy period, the second generation consists of all customers, who arrive during the service time of the first generation, etc. When the queue becomes idle, the branching process becomes extinct. This approach permits a more elementary treatment of the M/G/1 queue, without use of Rouche's theorem. It provides a natural sequence of approximants to the distributions, which we consider and it provides a simple derivation of the virtual waitingtime. The paper also considers two random variables of interest, which have not been considered hitherto. One is the total number of customers, served in the interval o < t = or < t, the other is the virtual age or the time already spent in the queue, by the customer in service at time t. A new imbedded semi-Markov process is examined and its asymptotic behavior is studied. (Author)
Document Details
- Document Type
- Technical Report
- Publication Date
- Apr 01, 1968
- Accession Number
- AD0668980
Entities
People
- Marcel F. Neuts
Organizations
- Purdue University