Abstract
Abstract
This paper presents the distribution of the number of customers served during a busy period for special cases of the Geo/G/1 queue when initiated with m customers. We analyze the system under the assumptions of a late arrival system with delayed access and early arrival system policies. It is not easy to invert the functional equation for the number of customers served during a busy period except for the simple case Geo/Geo/1 queue, as stated by several researchers. Using the Lagrange inversion theorem, we give an elegant solution to this equation. We find the distribution of the number of customers served during a busy period for various service-time distributions such as geometric, deterministic, binomial, negative binomial, uniform, Delaporte, discrete phase-type and interrupted Bernoulli process. We compute the mean and variance of these distributions and also give numerical results. Due to the clarity of the expressions, the computations are very fast and robust. We also show that in the limiting case, the results tend to the analogous continuous-time counterparts.
Publisher
Cambridge University Press (CUP)
Subject
Industrial and Manufacturing Engineering,Management Science and Operations Research,Statistics, Probability and Uncertainty,Statistics and Probability
Reference29 articles.
1. A generalization of the ballot problem and its application in the theory of queues;Takács;Journal of the American Statistical Association,1962
2. A PROBLEM OF INTERFERENCE BETWEEN TWO QUEUES
3. Investigation of waiting time problems by reduction to Markov processes
4. Distribution of the number of customers served during a busy period in a discrete time single server queue;Goswami;International Journal of Information and Management Sciences,2004