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View all- Tian NZhang Z(2019)The Discrete-Time GI/Geo/1 Queue with Multiple VacationsQueueing Systems: Theory and Applications10.1023/A:101471152974040:3(283-294)Online publication date: 1-Jun-2019
Vacations in GI^X/M^Y/1 systems and Riemann boundary value problems
Pages 351 - 366
Abstract
We consider systems of GI/M/1 type with bulk arrivals, bulk service and exponential server vacations. The generating functions of the steady-state probabilities of the embedded Markov chain are found in terms of Riemann boundary value problems, a necessary and sufficient condition of ergodicity is proved. Explicit formulas are obtained for the case where the generating function of the arrival group size is rational. Resonance between the vacation rate and the system is studied. Complete formulas are given for the cases of single and geometric arrivals.
References
[1]
{1} B. T. Doshi, Queueing systems with vacations - A survey, Queueing Systems 1 (1986) 29-66.
[2]
{2} A. M. Dukhovny, Markov chains with quasitoeplitz transition matrix, J. Appl. Math. Simul. 2(1) (1989) 71-82.
[3]
{3} A. M. Dukhovny, Applications of vector Riemann boundary value problems to analysis of queueing systems, in: Advances in Queueing: Theory, Methods, and Open Problems , ed. J. H. Dshalalow (CRC Press, Boca Raton, FL, 1995).
[4]
{4} A. M. Dukhovny, GI X/M Y /1 systems with resident server and generally distributed arrival and service groups, J. Appl. Math. Stoch. Analysis 9(2) (1996), 159-170.
[5]
{5} F. D. Gakhov, Boundary Value Problems (Fizmatgiz, Moscow, 1977).
[6]
{6} N. Tian, D. Zhang and C. Cao, The GI/M /1 queue with exponential vacations, Queueing Systems 5 (1989) 331-344.
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Copyright © Copyright © 1997 Kluwer Academic Publishers.
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J. C. Baltzer AG, Science Publishers
United States
Publication History
Published: 30 December 1997
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View all- Tian NZhang Z(2019)The Discrete-Time GI/Geo/1 Queue with Multiple VacationsQueueing Systems: Theory and Applications10.1023/A:101471152974040:3(283-294)Online publication date: 1-Jun-2019