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May  2020, 16(3): 1135-1148. doi: 10.3934/jimo.2018196

Performance evaluation and analysis of a discrete queue system with multiple working vacations and non-preemptive priority

School of Science, Yanshan University, Qinhuangdao 066004, China

*Corresponding author

Received  October 2017 Revised  April 2018 Published  May 2020 Early access  December 2018

In this paper, we introduce a discrete time Geo/Geo/1 queue system with non-preemptive priority and multiple working vacations. We assume that there are two types of customers in this queue system named "Customers of type-Ⅰ" and "Customers of type-Ⅱ". Customer of type-Ⅱ has a higher priority with non-preemption than Customer of type-Ⅰ. By building a discrete time four-dimensional Markov Chain which includes the numbers of customers with different priorities in the system, the state of the server and the service state, we obtain the state transition probability matrix. Using a birth-and-death chain and matrix-geometric method, we deduce the average queue length, the average waiting time of the two types of customers, and the average busy period of the system. Then, we provide some numerical results to evaluate the effect of the parameters on the system performance. Finally, we develop some benefit functions to analyse both the personal and social benefit, and obtain some optimization results within a certain range.

Citation: Zhanyou Ma, Wenbo Wang, Linmin Hu. Performance evaluation and analysis of a discrete queue system with multiple working vacations and non-preemptive priority. Journal of Industrial and Management Optimization, 2020, 16 (3) : 1135-1148. doi: 10.3934/jimo.2018196
References:
[1]

P. Bocharov, O. Pavlova and D. Puzikova, M/G/1/r retrial queueing systems with priority of primary customers, Mathematical and Computer Modelling, 30 (1999), 89-98. doi: 10.1016/S0895-7177(99)00134-X.

[2]

Y. DengQ. Wu and Z. Li, Priority queues with threshold switching and setup time, Operations Research Transactions, 4 (2000), 41-53. 

[3]

W. Feng and M. Umemura, Analysis of a finite buffer model with two servers and two nonpreemptive priority classes, European Journal of Operational Research, 192 (2009), 151-172.  doi: 10.1016/j.ejor.2007.09.021.

[4]

F. Kamoun, Performance analysis of a non-preemptive priority queuing system subjected to a correlated markovian interruption process, Operations Research, 35 (2008), 3969-3988. 

[5]

E. Kao and K. Narayanan, Computing steady-state probabilities of a nonpreemptive priority multiserver queue, ORSA Journal on Computing, 2 (1990), 211-218. 

[6]

T. Katayama and K. Kobayashi, Analysis of a nonpreemptive priority queue with exponential timer and server vacations, Performance Evaluation, 64 (2007), 495-506. 

[7]

O. Kella and U. Yechiali, Waiting times in the non-preemptive priority M/M/c queue, Communications in Statistics. Part C: Stochastic Models, 1 (1985), 257-262. doi: 10.1080/15326348508807014.

[8]

A. Krishnamoorthy, S. Babu and V. Narayanan, The MAP/(PH/PH)/1 queue with self-generation of priorities and non-preemptive service, European Journal of Operational Research, 195 (2009), 174-185. doi: 10.1016/j.ejor.2008.01.048.

[9]

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

[10]

K. Madan, A non-preemptive priority queueing system with a single server serving two queues M/G/1 and M/D/1 with optional server vacations based on exhaustive service of the priority units, Applied Mathematics, 2 (2011), 791-799. doi: 10.4236/am.2011.26106.

[11] M. Neuts, Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach, The Johns Hopkins Universit Press, Baltimore, MD, 1981. 
[12]

D. Pandey and A. Pal, Delay analysis of a discrete-time non-preemptive priority queue with priority jumps, Applied Mathematics, 9 (2014), 1-12. 

[13]

A. SleptvhenkoJ. SelenI. Adan and G. J. Houtum, Joint queue length distribution of multi-class, single-server queues with preemptive priorities, Queueing Systems, 81 (2015), 379-395.  doi: 10.1007/s11134-015-9460-z.

[14]

W. Sun, P. Guo, N. Tian and S. Li, Relative priority policies for minimizing the cost of queueing systems with service discrimination, Applied Mathematical Modelling, 33 (2009), 4241-4258. doi: 10.1016/j.apm.2009.03.012.

[15]

A. M. K. Tarabia, Two-class priority queuing system with restricted number of priority customers, AEU - International Journal of Electronics and Communications, 61 (2007), 534-539. 

[16]

R. TianD. Yue and W. Yue, Optimal balking strategies in an M/G/1 queueing system with a removable server under N-policy, Journal of Industrial and Management Optimization, 11 (2015), 715-731.  doi: 10.3934/jimo.2015.11.715.

[17]

R. Wang and W. Long, Weak convergence theorems for multiserver queueing system in nonpreemptive priority displine, Acta Mathematicae Applicatae Sinica, 17 (1994), 192-200.

[18]

G. ZhaoY. Chen and X. Xue, M/M/1 queue under nonpreemptive priority, College Mathematics, 22 (2006), 1454-1672. 

show all references

References:
[1]

P. Bocharov, O. Pavlova and D. Puzikova, M/G/1/r retrial queueing systems with priority of primary customers, Mathematical and Computer Modelling, 30 (1999), 89-98. doi: 10.1016/S0895-7177(99)00134-X.

[2]

Y. DengQ. Wu and Z. Li, Priority queues with threshold switching and setup time, Operations Research Transactions, 4 (2000), 41-53. 

[3]

W. Feng and M. Umemura, Analysis of a finite buffer model with two servers and two nonpreemptive priority classes, European Journal of Operational Research, 192 (2009), 151-172.  doi: 10.1016/j.ejor.2007.09.021.

[4]

F. Kamoun, Performance analysis of a non-preemptive priority queuing system subjected to a correlated markovian interruption process, Operations Research, 35 (2008), 3969-3988. 

[5]

E. Kao and K. Narayanan, Computing steady-state probabilities of a nonpreemptive priority multiserver queue, ORSA Journal on Computing, 2 (1990), 211-218. 

[6]

T. Katayama and K. Kobayashi, Analysis of a nonpreemptive priority queue with exponential timer and server vacations, Performance Evaluation, 64 (2007), 495-506. 

[7]

O. Kella and U. Yechiali, Waiting times in the non-preemptive priority M/M/c queue, Communications in Statistics. Part C: Stochastic Models, 1 (1985), 257-262. doi: 10.1080/15326348508807014.

[8]

A. Krishnamoorthy, S. Babu and V. Narayanan, The MAP/(PH/PH)/1 queue with self-generation of priorities and non-preemptive service, European Journal of Operational Research, 195 (2009), 174-185. doi: 10.1016/j.ejor.2008.01.048.

[9]

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

[10]

K. Madan, A non-preemptive priority queueing system with a single server serving two queues M/G/1 and M/D/1 with optional server vacations based on exhaustive service of the priority units, Applied Mathematics, 2 (2011), 791-799. doi: 10.4236/am.2011.26106.

[11] M. Neuts, Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach, The Johns Hopkins Universit Press, Baltimore, MD, 1981. 
[12]

D. Pandey and A. Pal, Delay analysis of a discrete-time non-preemptive priority queue with priority jumps, Applied Mathematics, 9 (2014), 1-12. 

[13]

A. SleptvhenkoJ. SelenI. Adan and G. J. Houtum, Joint queue length distribution of multi-class, single-server queues with preemptive priorities, Queueing Systems, 81 (2015), 379-395.  doi: 10.1007/s11134-015-9460-z.

[14]

W. Sun, P. Guo, N. Tian and S. Li, Relative priority policies for minimizing the cost of queueing systems with service discrimination, Applied Mathematical Modelling, 33 (2009), 4241-4258. doi: 10.1016/j.apm.2009.03.012.

[15]

A. M. K. Tarabia, Two-class priority queuing system with restricted number of priority customers, AEU - International Journal of Electronics and Communications, 61 (2007), 534-539. 

[16]

R. TianD. Yue and W. Yue, Optimal balking strategies in an M/G/1 queueing system with a removable server under N-policy, Journal of Industrial and Management Optimization, 11 (2015), 715-731.  doi: 10.3934/jimo.2015.11.715.

[17]

R. Wang and W. Long, Weak convergence theorems for multiserver queueing system in nonpreemptive priority displine, Acta Mathematicae Applicatae Sinica, 17 (1994), 192-200.

[18]

G. ZhaoY. Chen and X. Xue, M/M/1 queue under nonpreemptive priority, College Mathematics, 22 (2006), 1454-1672. 

Figure 1.  Schematic illustration for the service process of the non-preemptive priority queue
Figure 2.  Schematic diagram for the model description
Figure 3.  Relation of $ E(L_1) $ with $ \lambda $ and $ \alpha $
Figure 4.  Relation of $ E(L_2) $ with $ \lambda $ and $ \alpha $
Figure 5.  Relation of $ E(L) $ with $ \theta $ and $ \mu_1 $
Figure 6.  Relation of $ E(B) $ with $ \mu _1 $ and $ \mu_2 $
Figure 7.  Relation of $ B_1 $ with $ \lambda $ and $ \alpha $
Figure 8.  Relation of $ B_2 $ with $ \lambda $ and $ \alpha $
Figure 9.  Relation of $ D $ with $ \mu _1 $ and $ \alpha $
Figure 10.  Relation of $ D $ with $ \lambda $ and $ \alpha $
Table 1.  Relation of $ E(W_1) $ with $ \mu _1 $ and $ \mu _2 $
$ \mu _2 $ $ \mu _1 =0.30 $ $ \mu _1 =0.32 $ $ \mu _1 =0.34 $ $ \mu _1 =0.36 $ $ \mu _1 =0.38 $ $ \mu _1 =0.40 $
0.45 4.7081 2.9925 2.1313 1.6271 1.3023 1.0787
0.50 2.5403 1.7894 1.3538 1.0748 0.8836 0.7459
0.55 1.7483 1.2914 1.0067 0.8156 0.6800 0.5797
$ \mu _2 $ $ \mu _1 =0.30 $ $ \mu _1 =0.32 $ $ \mu _1 =0.34 $ $ \mu _1 =0.36 $ $ \mu _1 =0.38 $ $ \mu _1 =0.40 $
0.45 4.7081 2.9925 2.1313 1.6271 1.3023 1.0787
0.50 2.5403 1.7894 1.3538 1.0748 0.8836 0.7459
0.55 1.7483 1.2914 1.0067 0.8156 0.6800 0.5797
Table 2.  Relation of $ E(W_2) $ with $ \mu _1 $ and $ \mu _2 $
$ \mu _2 $ $ \mu _1 =0.30 $ $ \mu _1 =0.32 $ $ \mu _1 =0.34 $ $ \mu _1 =0.36 $ $ \mu _1 =0.38 $ $ \mu _1 =0.40 $
0.45 3.6164 2.9022 2.3961 2.0311 1.7626 1.5610
0.50 2.4961 1.9948 1.6473 1.4001 1.2199 1.0853
0.55 1.8492 1.4802 1.2263 1.0464 0.9154 0.8175
$ \mu _2 $ $ \mu _1 =0.30 $ $ \mu _1 =0.32 $ $ \mu _1 =0.34 $ $ \mu _1 =0.36 $ $ \mu _1 =0.38 $ $ \mu _1 =0.40 $
0.45 3.6164 2.9022 2.3961 2.0311 1.7626 1.5610
0.50 2.4961 1.9948 1.6473 1.4001 1.2199 1.0853
0.55 1.8492 1.4802 1.2263 1.0464 0.9154 0.8175
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