# Analysis of an M/G/1 Queue with Multiple Vacations, N-policy, Unreliable Service Station and Repair Facility Failures

Document Type: Research Paper

Authors

1 School of Mathematics & Software Science, Sichuan Normal University, Chengdu, Sichuan, 610068, China

2 School of Science, Sichuan University of Science and Engineering, Zigong, Sichuan, 643000, China

Abstract

This paper studies an M/G/1 repairable queueing system with multiple vacations and N-policy, in which the service station is subject to occasional random breakdowns. When the service station breaks down, it is repaired by a repair facility. Moreover, the repair facility may fail during the repair period of the service station. The failed repair facility resumes repair after completion of its replacement. Under these assumptions, applying a simple method, the probability that the service station is broken, the rate of occurrence of breakdowns of the service station, the probability that the repair facility is being replaced and the rate of occurrence of failures of the repair facility along with other performance measures are obtained. Following the construction of the long-run expected cost function per unit time, the direct search method is implemented for determining the optimum threshold N* that minimises the cost function.

Keywords

### References

Alfa A.S. and Frigui I. (1996). Discrete NT-policy single server queue with Markov arrival process and phase type service. European Journal of Operational Research, Vol. 88(8), pp. 599-613.

Alfa A.S. and Li W. (2000). Optimal (N,T)-policy for M/G/1 system with cost structures. Performance Evaluation, Vol. 42(4), pp. 265-277.

Baek J.W., Lee H.W., Lee S.W. and Ahn S. (2014). A workload factorization for BMAP/G/1 vacation queues under variable service speed. Operations Research Letters, Vol. 42(1), pp. 58-63.

Cao J.H. and Cheng K. (1982). Analysis of M/G/1 queueing system with repairable service station. Acta Mathematicae Applicattae Sinica, Vol. 5(2), pp. 113-127.

Cao J.H. and Cheng K. Mathematical Theory of Reliability (Revised). Beijing: Higher Education Press. 2006.

Choudhury G., and Deka K. (2008). An M/G/1 retrial queueing system with two phases of service subject to the server breakdown and repair. Performance Evaluation, Vol. 65(10), pp. 714-724.

Gakis K.G., Rhee H.K. and Sivazlian B.D. (1995). Distributions and first moments of the busy and idle periods in controllable M/G/1 queueing models with simple and dyadic policies. Stochastic Analysis and Applications, Vol. 13(1), pp. 47-81.

Gao S. and Wang J.T. (2014). Performance and reliability analysis of an M/G/1-G retrial queue with orbital search and non-persistent customers. European Journal of Operational Research, Vol. 236(2), pp. 561-572.

Hur S., Kim J. and Kang C. (2003). An analysis of the M/G/1 system with N and T policy. Applied Mathematical Modelling, Vol. 27(8), pp. 665-675.

Kella O. (1989). The threshold policy in the M/G/1 queue with server vacation. Naval Research Logistics, Vol. 36(1), pp.111-123.

Lee H.W., Lee S.S., Park J.O. and Chae K.C. (1994). Analysis of / /1 X M G queue with N policy and multiple vacations. Journal of Applied Probability, Vol. 31(2), pp. 467-496.

Reddy G.V.K., Nadarajan R. and Arumuganathan R. (1998). Analysis of a bulk queue with N-policy, multiple vacations and setup times. Computers & Operations Research, Vol. 25(11), pp. 957-967.

Tang Y.H. (1997). A single server M/G/1 queueing system subject to breakdowns--some reliability and queueing problem. Microelectronics & Reliability, Vol. 37(2), pp. 315-321.

Tang Y.H. (2010). Revisiting the model of servicing machines with repairable service facility--a new analyzing idea and some new results. Acta Mathematicae Applicatae Sinica, Vol. 26(4), pp. 557-566.

Wang J.T., Liu B. and Li J.H. (2008). Transient analysis of an M/G/1 retrial queue subject to disasters and server failure, European Journal of Operational Research, Vol. 189(3), pp. 1118-1132.

Yu M.M., Tang Y.H., Liu L.P. and Cheng J. (2013). A phase-type geometric process repair model with spare device procurement and repairman's multiple vacations. European Journal of Operational Research, Vol. 225(2), pp. 310-323.

Zhang Y.L. and Wu S.M. (2009). Reliability analysis for a k/n(F) system with repairable repair-equipment, Applied Mathematical Modelling, Vol. 33(7), pp. 3052-3067.