Abstract

This paper deals with an unreliable server having three phases of heterogeneous service on the basis of M/G/1 queueing system. We suppose that customers arrive and join the system according to a Poisson's process with arrival rate λ. After completion of three successive phases of service the server either goes for a vacation with probablity p (0≤ p≤1) or continue to serve the next units, if any, with probablity q(=1 -p). Otherwise it remains in the system until a customer arrives. The server is supposed to be unreliable, hence when the server is working during any phase of service, it may breakdown at any instant and thus service facilty will fail for a short interval of time. Firstly, now we derive the joint probability distribution for the server. Secondly, we derive the probability generating function of the stationary queue size distribution at a departure epoch. Third, we derive Laplace Stieltijes transform of busy period distribution and waiting time distribution. Finally, we obtain some important performance measures and reliability analysis of this model.

Keywords

References

[1] D.P. Gaver, A waiting line with interrupted service including priorities, Journal of the Royal Statistical Society Ser. B 24 (1962) 73–90.
[2] B. Avi-ltzhak, P. Naor, Some queueing problems with the service station subject to breakdowns, Operation Research 11 (1963) 303–320.
[3] K. Thirurengadan, Queueing with breakdowns, Operation Research 11 (1963) 303–320.
[4] I.L. Mitrany, B. Avi-Itzhak, A many server queue with service interruptions, Operation Research 16 (1968) 628–638.
[5] W. Li, D. Shi, X. Chao, Reliability analysis of M/G/1 queueing system with server breakdowns and vacations, J. Appl. Probability 34 (1997) 546–555.
[6] B. Sengupta, A queue with service interruptions in an alternating random environment, Operation Research 38 (1990) 308–318.
[7] T. Takine, B. Sengupta, A single server queue with service interruptions, Queueing System 26 (1998) 285–300.
[8] Y. Tang, A single server M/G/1 queueing system subject to breakdowns – some reliability and queueing problems, Microelectron. Reliability 37 (1997) 315–321.
[9] J.C. Ke, C.U. Wu, Z.G. Zhang, Recent developments in vacation models: a short survey, International Journal Operation Research 7 (4) (2010) 3–8.
[10] J.C. Ke, W.L. Pearn, Optimal management policy for heterogeneous arrival queueing system with server breakdown and vacations’, Quality Techno Quantitative Manage. 1 (2004) 149–162.
[11] J.C, Ke, An M/G/1 queue under hysteretic vacation policy with an early startup and unreliable server, Mathematical Methods of Operation Research 63 (2) (2006) 357–369.
[12] J.C. Ke, On M/G/1 system under NT policies with breakdowns, startup and unreliable server, Applied Mathematical Model. 30 (2006) 49–66.
[13] J.C. Ke, Batch arrival queue under vacation policies with server breakdowns and startup/closedown, Applied Mathematical Model. 31 (2007) 1282–1292.
[14] J.C. Ke, An MX/G/1 system with startup server and J additional options for server, Applied Mathematical Modelling 32 (2008) 443–458.
[15] J.C. Ke, K.B. Huang, Analysis of an unreliable server M[X]/G/1 system with a randomized vacation policy and delayed repair, Stochastic Models 26 (2) (2010) 212–241.
[16] J.C. Ke, K.B. Huang, W.L. Pearn, The performance measures and randomized optimization for an unreliable server MX/G/1 vacation system, Applied Mathematical Computation 217 (2011) 8277–8290.
[17] J. Keilson, L.D. Servi, Oscillatory random walk models for GI/G/1 vacation systems with Bernoulli schedules, Journal of Applied Probability 23 (1986) 790–802.
[18] O. Kella, Optimal control of vacation scheme in an M/G/1 queue, Operation Research 38 (1990) 724–728.
[19] G. Choudhury, K.C. Madan, A two phases batch arrival queueing system with a vacation time under Bernoulli schedule,Applied Mathematical Computation 149 (2004) 337–349.
[20] G. Choudhury, K.C. Madan, A two-stage batch arrival queueing system with a modified Bernoulli schedule vacation under N-policy, Math. Computation Modelling 42 (2005) 71–85.
[21] G. Choudhury, M. Paul, A two phase queueing system with Bernoulli vacation schedule under multiple vacation policy, Statistical Methodology 3 (2006) 174–185.
[22] G. Choudhury, L. Tadj, M. Paul, Steady state analysis of an M/G/1 queue with two phases of service and Bernoulli vacation schedule under multiple vacation policy, Appl. Math. Model. 31 (2007) 1079–1091.
[23] D.R. Cox, The analysis of non Markovian stochastic process by the inclusion of supplementary variables, Proc. Cambridge Philosophical Society 51 (1955) 433–441.