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An M\(^{[x]}\)/G/1 system with startup server and \(J\) additional options for service. (English) Zbl 1162.90399
Summary: We consider an M\(^{[x]}\)/G/1 queueing system with a startup time, where all arriving customers demand first the essential service and some of them may further demand one of other optional services: Type 1, Type 2,\( \cdots \) , and Type \(J\) service. The service times of the essential service and of the Type \(i (i=1,2,\cdots ,J)\) service are assumed to be random variables with arbitrary distributions. The server is turned off each time when the system is empty. As soon as a customer or a batch of customers arrives, the server immediately performs a startup which is needed before starting each busy period. We derive the steady-state results, including system size distribution at a random epoch and at a departure epoch, the distributions of idle and busy periods, and waiting time distribution in the queue. Some special cases are also presented.

90B22 Queues and service in operations research
Full Text: DOI
[1] Doshi, B.T., A note on stochastic decomposition in a GI/G/1 queue with vacations or set-up times, J. appl. prob., 22, 419-428, (1985) · Zbl 0566.60090
[2] Doshi, B.T., Queueing systems with vacations – a survey, Queueing syst., 1, 29-66, (1986) · Zbl 0655.60089
[3] Takagi, H., Queueing analysis: A foundation of performance evaluation, vacation and priority systems, part I, vol. 1, (1991), North-Holland Amsterdam · Zbl 0744.60114
[4] Li, H.; Zhu, Y.; Yang, P., Computational analysis of M(n)/G/1/N queues with setup time, Comput. oper. res., 22, 8, 829-840, (1995) · Zbl 0838.90049
[5] Choudhury, G., An M^[x]/G/1 queueing system with a setup period and a vacation period, Queueing syst., 36, 23-38, (2000) · Zbl 0966.60100
[6] Borthakur, A.; Medhi, J.; Gohain, R., Poisson input queueing systems with startup time and under control operating policy, Comput. oper. res., 14, 33-40, (1987) · Zbl 0654.60091
[7] Medhi, J.; Templeton, J.G.C., A Poisson input queue under N-policy and with a general start up time, Comput. oper. res., 19, 1, 35-41, (1992) · Zbl 0742.60100
[8] Takagi, H., M/G/1/K queues with N-policy and setup times, Queueing syst., 14, 79-98, (1993) · Zbl 0793.60099
[9] Hur, S.; Paik, S.J., The effect of different arrival rates on the N-policy of M/G/1 with server setup, Appl. math. model., 23, 289-299, (1999) · Zbl 0957.90031
[10] Ke, J.-C., Bi-level control for batch arrival queues with an early startup and un-reliable server, Appl. math. model., 28, 469-485, (2004) · Zbl 1090.90038
[11] Madan, K.C., An M/G/1 queue with second optional service, Queueing syst., 34, 37-46, (2000) · Zbl 0942.90008
[12] Keilson, J.; Servi, L.D., Oscillating random walk models for GI/G/1 vacation systems with Bernoulli schedules, J. appl. prob., 23, 790-802, (1986) · Zbl 0612.60087
[13] Keilson, J.; Servi, L.D., Blocking probabilities for the M/G/1 vacation systems with occupancy level dependent schedules, Oper. res., 37, 137-140, (1989) · Zbl 0666.60097
[14] Arumuganathan, R.; Jeyakumar, S., Steady state analysis of a bulk queue with multiple vacations, setup times with N-policy with closedown times, Appl. math. model., 29, 10, 972-986, (2005) · Zbl 1163.90414
[15] Cox, D.R., The analysis of non-Markovian stochastic processes by the inclusion of supplementary variables, Proc. Cambridge philos. soc., 51, 433-441, (1955) · Zbl 0067.10902
[16] Medhi, J., A single server Poisson input queue with a second optional channel, Queueing syst., 42, 239-242, (2002) · Zbl 1011.60072
[17] Wolff, R.W., Poisson arrival see time average, Oper. res., 30, 223-231, (1982) · Zbl 0489.60096
[18] Boxma, O.J., Workloads and waiting times in single-server systems with multiple customer classes, Queueing syst., 5, 185-214, (1989) · Zbl 0681.60098
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