# American Institute of Mathematical Sciences

April 2018, 14(2): 743-757. doi: 10.3934/jimo.2017073

## Analysis of a batch service multi-server polling system with dynamic service control

 1 College of Economics and Management, Shandong University of Science and Technology, Qingdao, Shandong, 266590, China 2 Department of Mathematics, School of Science, Nanjing University of Science and Technology, Nanjing, Jiangsu, 210094, China

* Corresponding author: Tao Jiang

Received  April 2016 Revised  September 2016 Published  September 2017

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.

Citation: Tao Jiang, Liwei Liu. Analysis of a batch service multi-server polling system with dynamic service control. Journal of Industrial & Management Optimization, 2018, 14 (2) : 743-757. doi: 10.3934/jimo.2017073
##### References:
 [1] 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. [2] O.J. Boxma, Y. van der Wal and U. Yechiali, Polling with batch service, Stochastic Models, 24 (2008), 604-625. doi: 10.1080/15326340802427497. [3] 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. [4] 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. [5] 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. [6] D. P. Heyman, The T policy for the M/G/1 queue, Management Science, 23 (1977), 775-778. [7] 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. [8] 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. [9] G. Latouche and V. Ramaswami, Introduction to Matrix Analytic Methods in Stochastic Modeling, SIAM, Philadelphia, 1999. doi: 10.1137/1.9780898719734. [10] 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. [11] 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. [12] M. F. Neuts, Matrix-Geometric Solutions in Stochastic Models: Algorithmic Approach, Johns Hopkins University Press, Baltimore, 1981. [13] N. Perel and U. Yechiali, The Israeli queue with priorities, Stochastic Models, 29 (2013), 353-379. doi: 10.1080/15326349.2013.808911. [14] 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. [15] N. Perel and U. Yechiali, The Israeli Queue with retrials, Queueing Systems, 78 (2014), 31-56. doi: 10.1007/s11134-013-9389-z. [16] 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. [17] L. D. Servi and S. G. Finn, M/M/1 queues with working vacations (M/M/1WV), Performances Evaluation, 50 (2002), 41-52. [18] N. Tian and Z. G. Zhang, Vacation Queueing Models: Theory and Applications, Springer, New York, 2006. [19] 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. [20] 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. [21] 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. [22] 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. [23] 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.

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##### References:
 [1] 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. [2] O.J. Boxma, Y. van der Wal and U. Yechiali, Polling with batch service, Stochastic Models, 24 (2008), 604-625. doi: 10.1080/15326340802427497. [3] 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. [4] 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. [5] 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. [6] D. P. Heyman, The T policy for the M/G/1 queue, Management Science, 23 (1977), 775-778. [7] 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. [8] 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. [9] G. Latouche and V. Ramaswami, Introduction to Matrix Analytic Methods in Stochastic Modeling, SIAM, Philadelphia, 1999. doi: 10.1137/1.9780898719734. [10] 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. [11] 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. [12] M. F. Neuts, Matrix-Geometric Solutions in Stochastic Models: Algorithmic Approach, Johns Hopkins University Press, Baltimore, 1981. [13] N. Perel and U. Yechiali, The Israeli queue with priorities, Stochastic Models, 29 (2013), 353-379. doi: 10.1080/15326349.2013.808911. [14] 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. [15] N. Perel and U. Yechiali, The Israeli Queue with retrials, Queueing Systems, 78 (2014), 31-56. doi: 10.1007/s11134-013-9389-z. [16] 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. [17] L. D. Servi and S. G. Finn, M/M/1 queues with working vacations (M/M/1WV), Performances Evaluation, 50 (2002), 41-52. [18] N. Tian and Z. G. Zhang, Vacation Queueing Models: Theory and Applications, Springer, New York, 2006. [19] 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. [20] 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. [21] 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. [22] 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. [23] 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.
$L_q$ versus $\lambda$ ($p = 0.6, \theta= 0.2, \mu_1=3$)
$L_q$ versus $\mu_1$ ($\lambda=3, p = 0.6, \theta= 0.2,$)
$L_q$ versus $p$ ($\lambda=3, \mu_1=3, \theta= 0.2,$)
$L_q$ versus $p$ ($\lambda=3, \mu_1=3, p= 0.6,$)
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