Analysis of M/G/ 1 queueing model with state dependent arrival and vacation

Subject Areas : Mathematical Optimization

Charan Jeet Singh ¹ , Madhu Jain ² , Binay Kumar ³

1 - Department of Mathematics Guru Nanak Dev University, Amritsar, Punjab, 143005, India
2 - Department of Mathematics, Indian Institute of Technology, Roorkee, Roorkee, Uttarakhand, 247667, India
3 - Department of Mathematics, M.L.U. DAV College, Phagwara, Punjab, 144402, India

Received: 2020-10-01 Accepted : 2020-10-01 Published : 2012-01-01

Keywords: Queue, Supplementary variable, State dependent, Average Queue Length, Arbitrary service time, Vacation,

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.