Abstract
We study a GI/M/c type queueing system with vacations in which all servers take vacations together when the system becomes empty. These servers keep taking synchronous vacations until they find waiting customers in the system at a vacation completion instant.The vacation time is a phase-type (PH) distributed random variable. Using embedded Markov chain modeling and the matrix geometric solution methods, we obtain explicit expressions for the stationary probability distributions of the queue length at arrivals and the waiting time. To compare the vacation model with the classical GI/M/c queue without vacations, we prove conditional stochastic decomposition properties for the queue length and the waiting time when all servers are busy. Our model is a generalization of several previous studies.
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References
X. Chao and Y. Zhao, Analysis of multi-server queues with station and server vacations, European J. Oper. Re. 110 (1990) 392–406.
B. Doshi, Queueing systems with vacations – A survey, Queueing Systems 1 (1986) 29–66.
B. Doshi, Single server queues with vacations, in: Stochastic Analysis of Computer and Communication Systems, ed. H. Takagi (North-Holland, Amsterdam, 1990) pp. 217–265.
S.W. Fuhrmann and R.B. Cooper, Stochastic decomposition in the M/G/1 queue with generalized vacations, Oper. Res. 33 (1985) 1117–1129.
D. Gross and C. Harris, Fundamentals of Queueing Theory, 2nd ed. (Wiley, New York, 1985).
C. Harris and W. Marchal, State dependence in M/G/1 server vacation models, Oper. Res. 36 (1988) 560–565.
N. Igaki, Exponential two server queue with N-policy and multiple vacations, Queueing Systems 10 (1992) 279–294.
L. Kleinrock, Queueing Systems, Vol. 1, Theory (Wiley, New York, 1975)
Y. Levy and L. Kleinrock, A queue with starter and a queue with vacations: Delay analysis by decomposition, Oper. Res. 34 (1986) 426–436.
Y. Levy and U. Yechiali, An M/M/c queue with server's vacations, INFOR 14 (1976) 153–163.
M. Miyazawa, Decomposition formulas for single server queues with vacations: A unified approach, Stochastic Models. 10 389–413.
M. Neuts, Matrix-Geometric Solutions in Stochastic Models (Johns Hopkins Univ. Baltimore MD 1981).
H. Takagi, Queueing Analysis, Vol. 1 (Elsevier Science, Amsterdam 1981).
H. Takagi, Queueing Analysis, Vol. 3, Discrete Time Systems (North-Holland, Amsterdam 1981).
J. Teghem, Control of service process in a queueing system, European J. Oper. Res. 23 (1986) 141– 158
N. Tian and Q. Li, TheM/M/c queue with PH synchronous vacations, SystemMath. Sci. 13(1) (2000) 7–16.
N. Tian, Q. Li and J. Cao, Conditional stochastic decomposition in M/M/c queue with server vacations, Stochastic Models 15(2) (1999) 367–377.
N. Tian, D. Zhang and C. Cao, The GI/M/1 queue with exponential vacations, Queueing Systems 5 (1989) 331–344.
N. Tian and Z.G. Zhang, M/M/c queue with synchronous vacations of some servers and its application to e-commerce operations, Working paper, Faculty of Business Administration, Simon Fraser University, Burnaby, Canada (2000).
B. Vinod, Exponential queue with server vacations, J. Oper. Res. Soc. 37 (1986) 1007–1014.
Z.G. Zhang and N. Tian, M/M/c queue with asynchronous vacations of partial servers, Performance Evaluation (2001) to appear.
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Tian, N., Zhang, Z.G. Stationary Distributions of GI/M/c Queue with PH Type Vacations. Queueing Systems 44, 183–202 (2003). https://doi.org/10.1023/A:1024424606007
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DOI: https://doi.org/10.1023/A:1024424606007