Geodesic
The Optimal Control in Batch Arrival Queue With Server Vacations, Startup and Breakdowns
Yugoslav journal of operations research, Tome 14 (2004) no. 1, p. 41 .

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This paper studies the N policy $M^[x]/G/1$ queue with server vacations; startup and breakdowns, where arrivals form a compound Poisson process and service times are generally distributed. The server is turned off and takes a vacation whenever the system is empty. If the number of customers waiting in the system at the instant of a vacation completion is less than N, the server will take another vacation. If the server returns from a vacation and finds at least N customers in the system, he is immediately turned on and requires a startup time before providing the service until the system is empty again. It is assumed that the server breaks down according to a Poisson process whose repair time has a general distribution. The system characteristics of such a model are analyzed and the total expected cost function per unit time is developed to determine the optimal threshold of N at a minimum cost.
Classification : 60K25 90B22
Keywords: Breakdowns, cost model, queue, startup, vacation. 
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     author = {Jau-Chuan Ke},
     title = {The {Optimal} {Control} in {Batch} {Arrival} {Queue} {With} {Server} {Vacations,} {Startup} and {Breakdowns}},
     journal = {Yugoslav journal of operations research},
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     year = {2004},
     zbl = {1055.60087},
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Jau-Chuan Ke. The Optimal Control in Batch Arrival Queue With Server Vacations, Startup and Breakdowns. Yugoslav journal of operations research, Tome 14 (2004) no. 1, p. 41 . http://geodesic.mathdoc.fr/item/YJOR_2004_14_1_a4/