Analysis of operating characteristics for the heterogeneous batch arrival queue with server startup and breakdowns

Ke, Jau-Chuan; Wang, Kuo-Hsiung

RAIRO - Operations Research - Recherche Opérationnelle, Tome 37 (2003) no. 3, pp. 157-177.

In this paper we consider a like-queue production system in which server startup and breakdowns are possible. The server is turned on (i.e. begins startup) when $N$ units are accumulated in the system and off when the system is empty. We model this system by an M${}^{\left[\mathrm{x}\right]}$/M/1 queue with server breakdowns and startup time under the $N$ policy. The arrival rate varies according to the server’s status: off, startup, busy, or breakdown. While the server is working, he is subject to breakdowns according to a Poisson process. When the server breaks down, he requires repair at a repair facility, where the repair time follows the negative exponential distribution. We study the steady-state behaviour of the system size distribution at stationary point of time as well as the queue size distribution at departure point of time and obtain some useful results. The total expected cost function per unit time is developed to determine the optimal operating policy at a minimum cost. This paper provides the minimum expected cost and the optimal operating policy based on assumed numerical values of the system parameters. Sensitivity analysis is also provided.

[1] K.R. Baker, A note on operating policies for the queue M/M/1 with exponential startup. INFOR 11 (1973) 71-72. | MR | Zbl

[2] A. Borthakur, J. Medhi and R. Gohain, Poisson input queueing systems with startup time and under control operating policy. Comput. Oper. Res. 14 (1987) 33-40. | MR | Zbl

[3] M.L. Chaudhry and J.G.C. Templeton, A first course in bulk queues. Wiley, New York (1983). | MR | Zbl

[4] G. Choudhury, On a Poisson queue with general setup time and vacation period. Indian J. Pure Appl. Math. 27 (1996) 1199-1211. | MR | Zbl

[5] R.B. Cooper, Introduction to queueing theory. 3rd edn., CEE press Books (1990). | MR | Zbl

[6] S.W. Fuhrmann and R.B. Cooper, Stochastic decompositions in the M/G/1 queue with generalized vacation. RAIRO Oper. Res. 33 (1985) 1117-1129. | MR | Zbl

[7] D. Gross and C.M. Harris, Fundamentals of queueing theory. John Wiley and Sons, New York, 2nd edn. (1985). | MR | Zbl

[8] S. Hur and S.J. Paik, The effect of different arrival rates on the N-policy of M/G/1 with server setup. Appl. Math. Modelling 23 (1999) 289-299. | Zbl

[9] H.S. Lee and M.M. Srinivasan, Control policies for the M${}^{\left[\mathrm{x}\right]}$/G/1 Queueing System. Manag. Sci. 35 (1989) 708-721. | MR | Zbl

[10] H.W. Lee, S.S. Lee and K.C. Chae, Operating characteristics of the M${}^{\left[\mathrm{x}\right]}$/G/1 queue with N-policy. Queueing Syst. 15 (1994) 387-399. | MR | Zbl

[11] H.W. Lee, S.S. Lee, J.O. Park and K.C. Chae, Analysis of the M${}^{\left[\mathrm{x}\right]}$/G/1 queue with N-policy and multiple vacations. J. Appl. Prob. 31 (1994) 476-496. | MR | Zbl

[12] S.S. Lee, H.W. Lee, S.H. Yoon and K.C. Chae, Batch arrival queue with N-policy and single vacation. Comput. Oper. Res. 22 (1995) 173-189. | Zbl

[13] H.W. Lee and J.O. Park, Optimal strategy in N-policy production system with early set-up. J. Oper. Res. Soc. 48 (1997) 306-313. | Zbl

[14] J. Medhi and J.G.C. Templeton, A Poisson input queue under N-policy and with a general start-up time. Comput. Oper. Res. 19 (1992) 35-41. | Zbl

[15] D.L. Minh, Transient solutions for some exhaustive M/G/1 queues with generalized independent vacations. Eur. J. Oper. Res. 36 (1988) 197-201. | MR | Zbl

[16] B.D. Sivazlian and L.E. Stanfel, Analysis of systems in operations research. Englewood Cliffs, New Jersey (1975).

[17] H. Takagi, Queueing analysis: A foundation of performance evaluation, vacation and priority systems 1. North Holland, Amsterdam (1991). | MR | Zbl

[18] H. Takagi, M/G/1/K Queues with N-policy and setup times. Queueing Syst. 14 (1993) 79-98. | MR | Zbl

[19] K.-H. Wang, Infinite source M/M/1 queue with breakdown. J. Chinese Inst. Engrs. 7 (1990) 47-55.

[20] K.-H. Wang, Optimal operation of a Markovian queueing system with a removable and non-reliable server. Microelectron. Reliab. 35 (1995) 1131-1136.

[21] K.-H. Wang, Optimal control of an M/E${}_k$/1 queueing system with removable service station subject to breakdowns. J. Oper. Res. Soc. 48 (1997) 936-942. | Zbl

[22] K.-H. Wang, K.-W. Chang and B.D. Sivazlian, Optimal control of a removable and non-reliable server in an infinite and a finite M/H${}_2$/1 queueing system. Appl. Math. Modelling 23 (1999) 651-666. | Zbl

[23] M. Yadin and P. Naor, Queueing systems with a removable service station. Oper. Res. Quart. 14 (1963) 393-405.