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July  2017, 13(3): 1467-1481. doi: 10.3934/jimo.2017002

## Performance analysis and optimization of a pseudo-fault Geo/Geo/1 repairable queueing system with N-policy, setup time and multiple working vacations

 1 Department of Applied Mathematics, College of Science, Yanshan University, Qinhuangdao 066004, China 2 School of Transportation Science and Engineering, Beihang University, Beijing 100191, China 3 Department of Intelligence and Informatics, Konan University, Kobe 658-8501, Japan

The reviewing process of the paper was handled by Yutaka Takahashi as Guest Editor.

Received  September 2015 Published  December 2016

In this paper, we consider a discrete time Geo/Geo/1 repairable queueing system with a pseudo-fault, setup time, $N$-policy and multiple working vacations. We assume that the service interruption is caused by pseudo-fault or breakdown, and occurs only when the server is busy. If the pseudo-fault occurs, the server will enter into a vacation period instead of a busy period. At a breakdown instant, the repair period starts immediately and after repaired the server is assumed to be as good as new. Using a quasi birth-and-death chain, we establish a two-dimensional Markov chain. We obtain the distribution of the steady-state queue length by using a matrix-geometric solution method. Moreover, we analyze the considered queueing system and provide several performance indices of the system in steady-state. According to the queueing system, we first investigate the individual and social optimal behaviors of the customer. Then we propose a pricing policy to optimize the system socially, and study the Nash equilibrium and social optimization of the proposed strategy to determine the optimal expected parameters of the system. Finally, we present some numerical results to illustrate the effect of several parameters on the systems.

Citation: Zhanyou Ma, Pengcheng Wang, Wuyi 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 & Management Optimization, 2017, 13 (3) : 1467-1481. doi: 10.3934/jimo.2017002
##### References:
 [1] A. Aissani and J. Artalejo, On the single server retrial queue subject to breakdowns, Queueing Systems, 30 (1998), 309-321.  doi: 10.1023/A:1019125323347.  Google Scholar [2] I. Atencia, I. Fortes, S. Nishimura and S. Sánchez, A discrete-time retrial queueing system with recurrent customers, Computers & Operations Research, 37 (2010), 1167-1173.  doi: MR2577277.  Google Scholar [3] I. Atencia, I. Fortes and S. Sánchez, A discrete-time retrial queueing system with starting failures, Bernoulli feedback and general retrial times, Computers & Industrial Engineering, 57 (2009), 1291-1299.  doi: 10.1016/j.cie.2009.06.011.  Google Scholar [4] I. Atencia and A. Pechinkin, A discrete-time queueing system with optional LCFS discipline, Annals of Operations Research, 202 (2013), 3-17.  doi: 10.1007/s10479-012-1097-2.  Google Scholar [5] B. Avi-Itzhak and P. Naor, Some queuing problems with the service station subject to breakdown, Operations Research, 11 (1963), 303-320.  doi: 10.1287/opre.11.3.303.  Google Scholar [6] Y. Baba, Analysis of a GI/M/1 queue with multiple working vacations, Operations Research Letters, 33 (2005), 201-209.  doi: 10.1016/j.orl.2004.05.006.  Google Scholar [7] H. Bruneel, Performance of discrete-time queueing systems, Computers & Operations Research, 20 (1993), 303-320.  doi: 10.1016/0305-0548(93)90006-5.  Google Scholar [8] G. Choudhury, On a batch arrival Poisson queue with a random setup time and vacation period, Computers & Operations Research, 25 (1998), 1013-1026.  doi: 10.1016/S0305-0548(98)00038-0.  Google Scholar [9] C. Cramer, A seismic hazard uncertainty analysis for the New Madrid seismic zone, Engineering Geology, 62 (2001), 251-266.  doi: 10.1016/S0013-7952(01)00064-3.  Google Scholar [10] S. Gao and J. Wang, On a discrete-time GI$^X$/Geo/1/N-G queue with randomized working vacations and at most $J$ vacations, Journal of Industrial and Management Optimization, 11 (2015), 779-806.  doi: 10.3934/jimo.2015.11.779.  Google Scholar [11] V. Goswami and G. Mund, Analysis of discrete-time batch service renewal input queue with multiple working vacations, Computers & Industrial Engineering, 61 (2011), 629-636.  doi: 10.1016/j.cie.2011.04.018.  Google Scholar [12] R. Hassin and M. Haviv, Queue or not to Queue: Equilibrium Behavior in Queueing Systems Kluwer Academic Publishers, 2003. doi: 10.1007/978-1-4615-0359-0.  Google Scholar [13] D. Heyman, The T-policy for the M/G/1 queue, Management Science, 23 (1977), 775-778.   Google Scholar [14] K. Kalidass, J. Gnanaraj, S. Gopinath and R. Kasturi, Transient analysis of an M/M/1 queue with a repairable server and multiple vacations, International Journal of Mathematics in Operational Research, 6 (2014), 193-216.  doi: 10.1504/IJMOR.2014.059522.  Google Scholar [15] V. Kulkarni and B. Choi, Retrial queues with server subject to breakdowns and repairs, Queueing Systems, 7 (1990), 191-208.  doi: 10.1007/BF01158474.  Google Scholar [16] P. Laxmi and S. Demie, Performance analysis of renewal input $(a, c, b)$ policy queue with multiple working vacations and change over times, Journal of Industrial and Management Optimization, 10 (2014), 839-857.  doi: 10.3934/jimo.2014.10.839.  Google Scholar [17] P. Laxmi, S. Indira and K. Jyothsna, Ant colony optimization for optimum service times in a Bernoulli schedule vacation interruption queue with balking and reneging, Journal of Industrial and Management Optimization, 12 (2016), 1199-1214.  doi: 10.3934/jimo.2016.12.1199.  Google Scholar [18] J. Li, N. Tian and M. Liu, Discrete-time GI/Geo/1 queue with multiple working vacations, Queueing Systems, 56 (2007), 53-63.  doi: 10.1007/s11134-007-9030-0.  Google Scholar [19] J. Li, N. Tian and Z. Ma, Performance analysis of GI/M/1 queue with working vacations and vacation interruption, Applied Mathematical Modelling, 32 (2008), 2715-2730.  doi: 10.1016/j.apm.2007.09.017.  Google Scholar [20] D. Lim, D. Lee, W. Yang and K. Chae, Analysis of the GI/Geo/1 queue with $N$-policy, Applied Mathematical Modelling, 37 (2013), 4643-4652.  doi: 10.1016/j.apm.2012.09.037.  Google Scholar [21] Z. Ma, P. Wang, G. Cui and Y. Hao, The discrete time Geom/Geom/1 repairable queuing system with pseudo-fault and multiple vacations, Journal of Information and Computational Science, 11 (2014), 4667-4678.   Google Scholar [22] Z. Ma, P. Wang and W. Yue, The pseudo-fault Geo/Geo/1 queue with setup time and multiple working vacation, Proceedings of the 10th International Conference on Queueing Theory and Network Applications, 383 (2015), 105-112.  doi: 10.1007/978-3-319-22267-7_10.  Google Scholar [23] T. Meisling, Discrete-time queuing theory, Operations Research, 6 (1958), 96-105.  doi: 10.1287/opre.6.1.96.  Google Scholar [24] S. Ndreca and B. Scoppola, Discrete time GI/Geom/1 queueing system with priority, European Journal of Operational Research, 189 (2008), 1403-1408.  doi: 10.1016/j.ejor.2007.02.056.  Google Scholar [25] M. Neuts, Matrix-geometric Solution in Stochastic Model: An Algorithmic Application, The Johns Hopkins University Press, 1981. Google Scholar [26] E. Papatheou, G. Manson, R. J. Barthorpe and K. Worden, The use of pseudo-faults for damage location in SHM: An experimental investigation on a Piper Tomahawk aircraft wing, Journal of Sound and Vibration, 333 (2014), 971-990.  doi: 10.1016/j.jsv.2013.10.013.  Google Scholar [27] L. Servi and S. Finn, M/M/1 queues with working vacations (M/M/1/WV), Performance Evaluation, 50 (2002), 41-52.  doi: 10.1016/S0166-5316(02)00057-3.  Google Scholar [28] W. Sun, H. Zhang and N. Tian, The discrete-time Geom/G/1 queue with multiple adaptive vacations and server setup/closedown times, International Journal of Management Science and Engineering Management, 2 (2007), 289-296.   Google Scholar [29] N. Tian and Z. Zhang, The discrete-time GI/Geo/1 queue with multiple vacations, Queueing Systems, 40 (2002), 283-294.  doi: 10.1023/A:1014711529740.  Google Scholar [30] R. Tian, D. 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.  Google Scholar [31] D. Towsley and S. Tripathi, A single server priority queue with server failures and queue flushing, Operations Research Letters, 10 (1991), 353-362.  doi: 10.1016/0167-6377(91)90008-D.  Google Scholar [32] C. Wei, Z. Zou and Y. Qin, Discrete time Geom/G/1 queue with second optional service and server breakdowns, Fuzzy Engineering and Operations Research, 147 (2012), 557-568.  doi: 10.1007/978-3-642-28592-9_59.  Google Scholar [33] M. Yadin and P. Naor, Queueing systems with a removable service station, Operations Research Quarterly, 14 (1963), 393-405.   Google Scholar [34] Z. Zhang and N. Tian, Discrete time Geo/G/1 queue with multiple adaptive vacations, Queueing Systems, 38 (2001), 419-429.  doi: 10.1023/A:1010947911863.  Google Scholar [35] Z. Zhang and N. Tian, The $N$-threshold policy for the GI/M/1 queue, Operations Research Letters, 32 (2004), 77-84.  doi: 10.1016/S0167-6377(03)00067-1.  Google Scholar

show all references

##### References:
 [1] A. Aissani and J. Artalejo, On the single server retrial queue subject to breakdowns, Queueing Systems, 30 (1998), 309-321.  doi: 10.1023/A:1019125323347.  Google Scholar [2] I. Atencia, I. Fortes, S. Nishimura and S. Sánchez, A discrete-time retrial queueing system with recurrent customers, Computers & Operations Research, 37 (2010), 1167-1173.  doi: MR2577277.  Google Scholar [3] I. Atencia, I. Fortes and S. Sánchez, A discrete-time retrial queueing system with starting failures, Bernoulli feedback and general retrial times, Computers & Industrial Engineering, 57 (2009), 1291-1299.  doi: 10.1016/j.cie.2009.06.011.  Google Scholar [4] I. Atencia and A. Pechinkin, A discrete-time queueing system with optional LCFS discipline, Annals of Operations Research, 202 (2013), 3-17.  doi: 10.1007/s10479-012-1097-2.  Google Scholar [5] B. Avi-Itzhak and P. Naor, Some queuing problems with the service station subject to breakdown, Operations Research, 11 (1963), 303-320.  doi: 10.1287/opre.11.3.303.  Google Scholar [6] Y. Baba, Analysis of a GI/M/1 queue with multiple working vacations, Operations Research Letters, 33 (2005), 201-209.  doi: 10.1016/j.orl.2004.05.006.  Google Scholar [7] H. Bruneel, Performance of discrete-time queueing systems, Computers & Operations Research, 20 (1993), 303-320.  doi: 10.1016/0305-0548(93)90006-5.  Google Scholar [8] G. Choudhury, On a batch arrival Poisson queue with a random setup time and vacation period, Computers & Operations Research, 25 (1998), 1013-1026.  doi: 10.1016/S0305-0548(98)00038-0.  Google Scholar [9] C. Cramer, A seismic hazard uncertainty analysis for the New Madrid seismic zone, Engineering Geology, 62 (2001), 251-266.  doi: 10.1016/S0013-7952(01)00064-3.  Google Scholar [10] S. Gao and J. Wang, On a discrete-time GI$^X$/Geo/1/N-G queue with randomized working vacations and at most $J$ vacations, Journal of Industrial and Management Optimization, 11 (2015), 779-806.  doi: 10.3934/jimo.2015.11.779.  Google Scholar [11] V. Goswami and G. Mund, Analysis of discrete-time batch service renewal input queue with multiple working vacations, Computers & Industrial Engineering, 61 (2011), 629-636.  doi: 10.1016/j.cie.2011.04.018.  Google Scholar [12] R. Hassin and M. Haviv, Queue or not to Queue: Equilibrium Behavior in Queueing Systems Kluwer Academic Publishers, 2003. doi: 10.1007/978-1-4615-0359-0.  Google Scholar [13] D. Heyman, The T-policy for the M/G/1 queue, Management Science, 23 (1977), 775-778.   Google Scholar [14] K. Kalidass, J. Gnanaraj, S. Gopinath and R. Kasturi, Transient analysis of an M/M/1 queue with a repairable server and multiple vacations, International Journal of Mathematics in Operational Research, 6 (2014), 193-216.  doi: 10.1504/IJMOR.2014.059522.  Google Scholar [15] V. Kulkarni and B. Choi, Retrial queues with server subject to breakdowns and repairs, Queueing Systems, 7 (1990), 191-208.  doi: 10.1007/BF01158474.  Google Scholar [16] P. Laxmi and S. Demie, Performance analysis of renewal input $(a, c, b)$ policy queue with multiple working vacations and change over times, Journal of Industrial and Management Optimization, 10 (2014), 839-857.  doi: 10.3934/jimo.2014.10.839.  Google Scholar [17] P. Laxmi, S. Indira and K. Jyothsna, Ant colony optimization for optimum service times in a Bernoulli schedule vacation interruption queue with balking and reneging, Journal of Industrial and Management Optimization, 12 (2016), 1199-1214.  doi: 10.3934/jimo.2016.12.1199.  Google Scholar [18] J. Li, N. Tian and M. Liu, Discrete-time GI/Geo/1 queue with multiple working vacations, Queueing Systems, 56 (2007), 53-63.  doi: 10.1007/s11134-007-9030-0.  Google Scholar [19] J. Li, N. Tian and Z. Ma, Performance analysis of GI/M/1 queue with working vacations and vacation interruption, Applied Mathematical Modelling, 32 (2008), 2715-2730.  doi: 10.1016/j.apm.2007.09.017.  Google Scholar [20] D. Lim, D. Lee, W. Yang and K. Chae, Analysis of the GI/Geo/1 queue with $N$-policy, Applied Mathematical Modelling, 37 (2013), 4643-4652.  doi: 10.1016/j.apm.2012.09.037.  Google Scholar [21] Z. Ma, P. Wang, G. Cui and Y. Hao, The discrete time Geom/Geom/1 repairable queuing system with pseudo-fault and multiple vacations, Journal of Information and Computational Science, 11 (2014), 4667-4678.   Google Scholar [22] Z. Ma, P. Wang and W. Yue, The pseudo-fault Geo/Geo/1 queue with setup time and multiple working vacation, Proceedings of the 10th International Conference on Queueing Theory and Network Applications, 383 (2015), 105-112.  doi: 10.1007/978-3-319-22267-7_10.  Google Scholar [23] T. Meisling, Discrete-time queuing theory, Operations Research, 6 (1958), 96-105.  doi: 10.1287/opre.6.1.96.  Google Scholar [24] S. Ndreca and B. Scoppola, Discrete time GI/Geom/1 queueing system with priority, European Journal of Operational Research, 189 (2008), 1403-1408.  doi: 10.1016/j.ejor.2007.02.056.  Google Scholar [25] M. Neuts, Matrix-geometric Solution in Stochastic Model: An Algorithmic Application, The Johns Hopkins University Press, 1981. Google Scholar [26] E. Papatheou, G. Manson, R. J. Barthorpe and K. Worden, The use of pseudo-faults for damage location in SHM: An experimental investigation on a Piper Tomahawk aircraft wing, Journal of Sound and Vibration, 333 (2014), 971-990.  doi: 10.1016/j.jsv.2013.10.013.  Google Scholar [27] L. Servi and S. Finn, M/M/1 queues with working vacations (M/M/1/WV), Performance Evaluation, 50 (2002), 41-52.  doi: 10.1016/S0166-5316(02)00057-3.  Google Scholar [28] W. Sun, H. Zhang and N. Tian, The discrete-time Geom/G/1 queue with multiple adaptive vacations and server setup/closedown times, International Journal of Management Science and Engineering Management, 2 (2007), 289-296.   Google Scholar [29] N. Tian and Z. Zhang, The discrete-time GI/Geo/1 queue with multiple vacations, Queueing Systems, 40 (2002), 283-294.  doi: 10.1023/A:1014711529740.  Google Scholar [30] R. Tian, D. 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.  Google Scholar [31] D. Towsley and S. Tripathi, A single server priority queue with server failures and queue flushing, Operations Research Letters, 10 (1991), 353-362.  doi: 10.1016/0167-6377(91)90008-D.  Google Scholar [32] C. Wei, Z. Zou and Y. Qin, Discrete time Geom/G/1 queue with second optional service and server breakdowns, Fuzzy Engineering and Operations Research, 147 (2012), 557-568.  doi: 10.1007/978-3-642-28592-9_59.  Google Scholar [33] M. Yadin and P. Naor, Queueing systems with a removable service station, Operations Research Quarterly, 14 (1963), 393-405.   Google Scholar [34] Z. Zhang and N. Tian, Discrete time Geo/G/1 queue with multiple adaptive vacations, Queueing Systems, 38 (2001), 419-429.  doi: 10.1023/A:1010947911863.  Google Scholar [35] Z. Zhang and N. Tian, The $N$-threshold policy for the GI/M/1 queue, Operations Research Letters, 32 (2004), 77-84.  doi: 10.1016/S0167-6377(03)00067-1.  Google Scholar
The schematic diagram for the model description
The relation of $P_B$ to $\mu_v$ and $N$
The relation of $E[L]$ to $p$ and $N$
The relation of $E[L_q]$ to $\mu_b$ and $\beta$
The relation of $P_{q2}$ to $p$ and $\alpha$
Individual benefit $U_I$ versus arrival rate $p$
Social benefit $U_S$ versus arrival rate $p$
The relation of $E[W]$ to $q$ and $\theta$
 $\theta$ The expected waiting time $E[W]$ $q=0$ $q=0.05$ $q=0.1$ $q=0.15$ $q=0.2$ $q=0.25$ $q=0.3$ 0.3 20.207 22.379 24.622 26.885 29.148 31.430 33.798 0.5 19.942 22.019 24.140 26.250 28.318 30.351 32.396 0.7 19.843 21.884 23.958 26.009 28.006 29.947 31.875
 $\theta$ The expected waiting time $E[W]$ $q=0$ $q=0.05$ $q=0.1$ $q=0.15$ $q=0.2$ $q=0.25$ $q=0.3$ 0.3 20.207 22.379 24.622 26.885 29.148 31.430 33.798 0.5 19.942 22.019 24.140 26.250 28.318 30.351 32.396 0.7 19.843 21.884 23.958 26.009 28.006 29.947 31.875
The relation of $E[L]$ to $\mu_b$ and $\gamma$
 $\gamma$ The expected queue length $E[L]$ $\mu_b=0.6$ $\mu_b=0.65$ $\mu_b=0.7$ $\mu_b=0.75$ $\mu_b=0.8$ $\mu_b=0.85$ $\mu_b=0.9$ 0.4 32.139 17.645 14.184 12.449 11.387 10.673 10.165 0.6 15.069 12.820 11.560 10.757 10.207 9.809 9.508 0.8 13.100 11.697 10.830 10.250 9.838 9.529 9.288
 $\gamma$ The expected queue length $E[L]$ $\mu_b=0.6$ $\mu_b=0.65$ $\mu_b=0.7$ $\mu_b=0.75$ $\mu_b=0.8$ $\mu_b=0.85$ $\mu_b=0.9$ 0.4 32.139 17.645 14.184 12.449 11.387 10.673 10.165 0.6 15.069 12.820 11.560 10.757 10.207 9.809 9.508 0.8 13.100 11.697 10.830 10.250 9.838 9.529 9.288
Comparison of individually and socially optimal arrival rate
 Vacation parameter $\theta$ Individually optimal arrival rate $p^e$ Socially optimal arrival rate $p^*$ 0.2 0.348 0.204 0.4 0.378 0.228 0.8 0.392 0.240
 Vacation parameter $\theta$ Individually optimal arrival rate $p^e$ Socially optimal arrival rate $p^*$ 0.2 0.348 0.204 0.4 0.378 0.228 0.8 0.392 0.240
 Vacation parameter $\theta$ Socially maximum benefit $U_S^*$ Admission fee $f$ 0.2 28.726 140.813 0.4 30.796 135.071 0.8 33.101 133.754
 Vacation parameter $\theta$ Socially maximum benefit $U_S^*$ Admission fee $f$ 0.2 28.726 140.813 0.4 30.796 135.071 0.8 33.101 133.754
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