Abstract
This investigation deals with single server queueing system wherein the arrival of the units follow Poisson process with varying arrival rates in different states and the service time of the units is arbitrary (general) distributed. The server may take a vacation of a fixed duration or may continue to be available in the system for next service. Using the probability argument, we construct the set of steady state equations by introducing the supplementary variable corresponding to elapsed service time. Then, we obtain the probability generating function of the units present in the system. Various performance indices, such as expected number of units in the queue and in the system, average waiting time, etc., are obtained explicitly. Some special cases are also deduced by setting the appropriate parameter values. The numerical illustrations are provided to carry out the sensitivity analysis in order to explore the effect of different parameters on the system performance measures.
Highlights
In some daily life congestion problems, the service time of the units may not follow exponential distribution
In queueing systems with arbitrary service time distribution, the number of units in the system at time t and the length of time for which the unit is in service are sufficient to determine the future stochastic properties of these variables
Several researchers have contributed in the direction of general distributed service time queueing system
Summary
In some daily life congestion problems, the service time of the units may not follow exponential distribution. Madan (1999) discussed the steady state behavior of an arbitrary service time queue with deterministic service vacation In his investigation, he has considered that the customers arrive at the system with uniform arrival rates. The present investigation is the extension of vacation model for single server general distributed service time studied by Madan (1999) and addresses the analysis of M/ G/1 queueing system with deterministic server vacation in which the arrival rate of the units are state dependent. Performance measures we shall establish various performance measures using the probability generating function of the queue length as follows: Theorem 3 The expected number of units in the queue is λ1λ2Eðv2Þð1 This provides the average queue length of M/G/1 model with state dependent rates. 2.5 1.59 1.10 1.74 1.24 1.92 1.39 2.11 1.56 2.34 1.75 μ, service rate; p, vacation probability; k, service phases
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