This paper considers a multi-server polling system with batch service of an unlimited size, i.e., the so called "Israeli queue" with multi-server, where the service rate of each server switches between a low and a high value depending on the number of groups standing in front of the servers upon its service completion. By means of matrix geometric method and LU-type RG factorization of the infinitesimal generator in irreducible QBD process, the explicit closed-form of rate matrix $R$ and the steady state distribution of the queue length are respectively derived. In terms of the results, some stationary performance measures are obtained. In addition, some numerical examples are presented.
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O. J. Boxma, Y. van der Wal and U. Yechiali, Polling with gated batch service, in: Proceedings of the Sixth International Conference on "Analysis of Manufacturing Systems", Lunteren, Netherlands, 2007,155-159.
![]() |
|
O.J. Boxma
, Y. van der Wal
and U. Yechiali
, Polling with batch service, Stochastic Models, 24 (2008)
, 604-625.
doi: 10.1080/15326340802427497.![]() ![]() ![]() |
|
J. D. Cordeiro
and J. P. Kharoufeh
, The unreliable M/M/1 retrial queue in a random environment, Stochastic Models, 28 (2012)
, 29-48.
doi: 10.1080/15326349.2011.614478.![]() ![]() ![]() |
|
Y. Dimitrakopoulosa
and A. N. Burnetas
, Customer equilibrium and optimal strategies in an M/M/1 queue with dynamic service control, European Journal of Operational Research, 252 (2016)
, 477-486.
doi: 10.1016/j.ejor.2015.12.029.![]() ![]() ![]() |
|
E. H. Elhafsi
and M. Molle
, The solution to QBD processes with finite state space, Stochastic Analysis and Applications, 25 (2007)
, 763-779.
doi: 10.1080/07362990701419946.![]() ![]() ![]() |
|
D. P. Heyman
, The T policy for the M/G/1 queue, Management Science, 23 (1977)
, 775-778.
![]() |
|
P. Jayachitra
and A. J. Albert
, Recent developments in queueing models under N-policy: A short survey, International Journal of Mathematical Archive, 5 (2014)
, 227-233.
![]() |
|
K. Kalidass
and R. Kasturi
, A queue with working breakdowns, Computers and Industrial Engineering, 63 (2012)
, 779-783.
doi: 10.1016/j.cie.2012.04.018.![]() ![]() |
|
G. Latouche and V. Ramaswami, Introduction to Matrix Analytic Methods in Stochastic Modeling, SIAM, Philadelphia, 1999.
doi: 10.1137/1.9780898719734.![]() ![]() ![]() |
|
Q. L. Li, Constructive Computation in Stochastic Models with Applications: the RG-Factorizations, Springer, Berlin and Tsinghua University Press, Beijing, 2010.
doi: 10.1007/978-3-642-11492-2.![]() ![]() ![]() |
|
Z. Ma
, P. Wang
and W. Yue
, Performance analysis and optimization of a pseudo-fault Geo/Geo/1 repairable queueing system with N-policy, setup time and multiple working vacations, Journal of Industrial and Management Optimization, 13 (2017)
, 1467-1481.
doi: 10.3934/jimo.2017002.![]() ![]() |
|
M. F. Neuts, Matrix-Geometric Solutions in Stochastic Models: Algorithmic Approach, Johns Hopkins University Press, Baltimore, 1981.
![]() ![]() |
|
N. Perel
and U. Yechiali
, The Israeli queue with priorities, Stochastic Models, 29 (2013)
, 353-379.
doi: 10.1080/15326349.2013.808911.![]() ![]() ![]() |
|
N. Perel
and U. Yechiali
, The Israeli queue with infinite number of groups, Probability in the Engineering and Informational Sciences, 28 (2014)
, 1-19.
doi: 10.1017/S0269964813000296.![]() ![]() ![]() |
|
N. Perel
and U. Yechiali
, The Israeli Queue with retrials, Queueing Systems, 78 (2014)
, 31-56.
doi: 10.1007/s11134-013-9389-z.![]() ![]() ![]() |
|
N. Perel
and U. Yechiali
, The Israeli Queue with a general group-joining policy, Annals of Operations Research, (2015)
, 1-34.
doi: 10.1007/s10479-015-1942-1.![]() ![]() |
|
L. D. Servi
and S. G. Finn
, M/M/1 queues with working vacations (M/M/1WV), Performances Evaluation, 50 (2002)
, 41-52.
![]() |
|
N. Tian and Z. G. Zhang, Vacation Queueing Models: Theory and Applications, Springer, New York, 2006.
![]() ![]() |
|
A. Tirdad
, W. K. Grassmann
and J. Tavakoli
, Optimal policies of M(t)/M/c/c queues with two different levels of servers, European Journal of Operational Research, 249 (2016)
, 1124-1130.
doi: 10.1016/j.ejor.2015.10.040.![]() ![]() ![]() |
|
Y. van der Wal
and U. Yechiali
, Dynamic visit-order rules for batch-service polling, Probability in the Engineering and Informational Sciences, 17 (2003)
, 351-367.
doi: 10.1017/S0269964803173044.![]() ![]() ![]() |
|
T. Y. Wang
, K. H. Wang
and W. L. Pearn
, Optimization of the T-policy M/G/1 queue with server breakdowns and general start up times, Journal of Computational and Applied Mathematics, 228 (2009)
, 270-278.
doi: 10.1016/j.cam.2008.09.021.![]() ![]() ![]() |
|
Z. G. Zhang
and N. Tian
, An analysis of queueing systems with multi-task servers, European Journal of Operational Research, 156 (2004)
, 375-389.
doi: 10.1016/S0377-2217(03)00015-8.![]() ![]() ![]() |
|
X. Zhang
, J. Wang
and T. V. Do
, Threshold properties of the M/M/1 queue under T-policy with applications, Applied Mathematics and Computation, 261 (2015)
, 284-301.
doi: 10.1016/j.amc.2015.03.109.![]() ![]() ![]() |