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. Google Scholar

[2]

O.J. BoxmaY. van der Wal and U. Yechiali, Polling with batch service, Stochastic Models, 24 (2008), 604-625.  doi: 10.1080/15326340802427497.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

D. P. Heyman, The T policy for the M/G/1 queue, Management Science, 23 (1977), 775-778.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[9]

G. Latouche and V. Ramaswami, Introduction to Matrix Analytic Methods in Stochastic Modeling, SIAM, Philadelphia, 1999. doi: 10.1137/1.9780898719734.  Google Scholar

[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.  Google Scholar

[11]

Z. MaP. 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.  Google Scholar

[12]

M. F. Neuts, Matrix-Geometric Solutions in Stochastic Models: Algorithmic Approach, Johns Hopkins University Press, Baltimore, 1981.  Google Scholar

[13]

N. Perel and U. Yechiali, The Israeli queue with priorities, Stochastic Models, 29 (2013), 353-379.  doi: 10.1080/15326349.2013.808911.  Google Scholar

[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.  Google Scholar

[15]

N. Perel and U. Yechiali, The Israeli Queue with retrials, Queueing Systems, 78 (2014), 31-56.  doi: 10.1007/s11134-013-9389-z.  Google Scholar

[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.  Google Scholar

[17]

L. D. Servi and S. G. Finn, M/M/1 queues with working vacations (M/M/1WV), Performances Evaluation, 50 (2002), 41-52.   Google Scholar

[18]

N. Tian and Z. G. Zhang, Vacation Queueing Models: Theory and Applications, Springer, New York, 2006.  Google Scholar

[19]

A. TirdadW. 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.  Google Scholar

[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.  Google Scholar

[21]

T. Y. WangK. 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.  Google Scholar

[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.  Google Scholar

[23]

X. ZhangJ. 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.  Google Scholar

show all references

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. Google Scholar

[2]

O.J. BoxmaY. van der Wal and U. Yechiali, Polling with batch service, Stochastic Models, 24 (2008), 604-625.  doi: 10.1080/15326340802427497.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

D. P. Heyman, The T policy for the M/G/1 queue, Management Science, 23 (1977), 775-778.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[9]

G. Latouche and V. Ramaswami, Introduction to Matrix Analytic Methods in Stochastic Modeling, SIAM, Philadelphia, 1999. doi: 10.1137/1.9780898719734.  Google Scholar

[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.  Google Scholar

[11]

Z. MaP. 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.  Google Scholar

[12]

M. F. Neuts, Matrix-Geometric Solutions in Stochastic Models: Algorithmic Approach, Johns Hopkins University Press, Baltimore, 1981.  Google Scholar

[13]

N. Perel and U. Yechiali, The Israeli queue with priorities, Stochastic Models, 29 (2013), 353-379.  doi: 10.1080/15326349.2013.808911.  Google Scholar

[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.  Google Scholar

[15]

N. Perel and U. Yechiali, The Israeli Queue with retrials, Queueing Systems, 78 (2014), 31-56.  doi: 10.1007/s11134-013-9389-z.  Google Scholar

[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.  Google Scholar

[17]

L. D. Servi and S. G. Finn, M/M/1 queues with working vacations (M/M/1WV), Performances Evaluation, 50 (2002), 41-52.   Google Scholar

[18]

N. Tian and Z. G. Zhang, Vacation Queueing Models: Theory and Applications, Springer, New York, 2006.  Google Scholar

[19]

A. TirdadW. 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.  Google Scholar

[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.  Google Scholar

[21]

T. Y. WangK. 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.  Google Scholar

[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.  Google Scholar

[23]

X. ZhangJ. 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.  Google Scholar

Figure 1.  $L_q$ versus $\lambda$ ($p = 0.6, \theta= 0.2, \mu_1=3$)
Figure 2.  $L_q$ versus $\mu_1$ ($\lambda=3, p = 0.6, \theta= 0.2, $)
Figure 3.  $L_q$ versus $p$ ($\lambda=3, \mu_1=3, \theta= 0.2, $)
Figure 4.  $L_q$ versus $p$ ($\lambda=3, \mu_1=3, p= 0.6, $)
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