TY - JOUR

T1 - The performance measures and randomized optimization for an unreliable server M[x]/G/1 vacation system

AU - Ke, Jau Chuan

AU - Huang, Kai Bin

AU - Pearn, W.l.

PY - 2011/7/1

Y1 - 2011/7/1

N2 - This paper examines an M[x]/G/1 queueing system with a randomized vacation policy and at most J vacations. Whenever the system is empty, the server immediately takes a vacation. If there is at least one customer found waiting in the queue upon returning from a vacation, the server will be immediately activated for service. Otherwise, if no customers are waiting for service at the end of a vacation, the server either remains idle with probability p or leaves for another vacation with probability 1 - p. This pattern continues until the number of vacations taken reaches J. If the system is empty by the end of the Jth vacation, the server becomes idle in the system. Whenever one or more customers arrive at server idle state, the server immediately starts providing service for the arrivals. Assume that the server may meet an unpredictable breakdown according to a Poisson process and the repair time has a general distribution. For such a system, we derive the distributions of important system characteristics, such as system size distribution at a random epoch and at a departure epoch, system size distribution at busy period initiation epoch, the distributions of idle period, busy period, etc. Finally, a cost model is developed to determine the joint suitable parameters (p*, J*) at a minimum cost, and some numerical examples are presented for illustrative purpose.

AB - This paper examines an M[x]/G/1 queueing system with a randomized vacation policy and at most J vacations. Whenever the system is empty, the server immediately takes a vacation. If there is at least one customer found waiting in the queue upon returning from a vacation, the server will be immediately activated for service. Otherwise, if no customers are waiting for service at the end of a vacation, the server either remains idle with probability p or leaves for another vacation with probability 1 - p. This pattern continues until the number of vacations taken reaches J. If the system is empty by the end of the Jth vacation, the server becomes idle in the system. Whenever one or more customers arrive at server idle state, the server immediately starts providing service for the arrivals. Assume that the server may meet an unpredictable breakdown according to a Poisson process and the repair time has a general distribution. For such a system, we derive the distributions of important system characteristics, such as system size distribution at a random epoch and at a departure epoch, system size distribution at busy period initiation epoch, the distributions of idle period, busy period, etc. Finally, a cost model is developed to determine the joint suitable parameters (p*, J*) at a minimum cost, and some numerical examples are presented for illustrative purpose.

KW - Batch arrival queue

KW - Randomized control

KW - Reliability

KW - Server breakdown

KW - Supplementary variable technique

KW - Vacations

UR - http://www.scopus.com/inward/record.url?scp=79956062783&partnerID=8YFLogxK

U2 - 10.1016/j.amc.2011.03.008

DO - 10.1016/j.amc.2011.03.008

M3 - Article

AN - SCOPUS:79956062783

VL - 217

SP - 8277

EP - 8290

JO - Applied Mathematics and Computation

JF - Applied Mathematics and Computation

SN - 0096-3003

IS - 21

ER -