AN M/G/1 QUEUE WITH REGULAR AND OPTIONAL PHASE VACATION AND WITH STATE DEPENDENT ARRIVAL RATE

Subject Areas : International Journal of Mathematical Modelling & Computations

Rathinasabapathy Kalyanaraman ¹ , Shanthi R ²

1 - Professor of Mathematics, Annamalai University India Professor of Mathematics 2nd
2 - Assistant professor Annamalai University India Research Scholar

Received: 2016-04-15 Accepted : 2016-04-15 Published : 2015-03-21

Keywords: Performance Measures, Supplementary variable, Vacation queue, probability generating function,

Abstract :

We consider an M/G/1 queue with regular and optional phase vacation and withstate dependent arrival rate. The vacation policy is after completion of service if there areno customers in the system, the server takes vacation consisting of K -phases, each phaseis generally distributed. Here the first phase is compulsory where as the other phases areoptional. For this model the supplementary variable technique has been applied to obtainthe probability generating functions of number of customers in the queue at the different server states. Some particular models are obtained and a numerical study is also carriedout.

References:

Chen, H. Y., Ji-Hong Li and Nai-Shuo Tian, The GI/M/1 queue with phase-type working vacations and vacation interruptions, J. Appl. Math. Comput., 30, 121-141, 2009.

Doshi, B. T., Queueing systems with vacations - a survey, Queueing System, 1, 29-66, 1986.

Doshi, B. T., Single server queues with vacations, In: Takagi, H.(ed.), Stochastic Analysis of the computer and communication systems, 217-264, North- Holland/Elsevier, Amsterdam, 1990.

Gross, D. and Harris, C. M., Fundamentals of queueing theory, 3rd edn, Wiley, New York, 1998.

Ke, J. C., The analysis of general input queue with N policy and exponential vacations, Queueing Syst., 45, 135-16, 1986.

Ke, J. C., The optimal control of an M/G/1 queueing system with server vacations, startup and breakdowns, Computers and Industrial Engineering, Vol. 44, 567-579, 2003.

Ke, J. C., and Wang, K. H., Analysis of operating characteristics for the heterogeneous batch arrival queue with server startup and breakdowns, RAIRO-Oper. Res., Vol.37, 157-177, 2003.

Keilson, J. and Servi, L. D., Dynamic of the M/G/1 vacation model, Operations Research, Vol.35 (4),575-582, 1987.

Takagi, H., Vacation and Priority Systems Part1. Queueing Analysis: A Foundation of Performance Evaluation, Vol.1., North-Holland/Elsevier, Amsterdam, 1991.

Tian, N., and Zhang, Z. G., The discrete time GI/Geo/1 queue with multiple vacations, Queueing Syst., 40, 283-294, 2002.

Tian, N., and Zhang, Z. G., A note on GI/M/1 queues with phase-type setup times or server vacations, INFOR, 41,341-351, 2003.

Tian, N., and Zhang, Z. G., Vacation queueing models: Theory and Applications, Springer, New York, 2006.