Figure 1.
State Transition Diagram
-
Previous Article
Multi-period optimal investment choice post-retirement with inter-temporal restrictions in a defined contribution pension plan
- JIMO Home
- This Issue
-
Next Article
Analysis of the queue lengths in a priority retrial queue with constant retrial policy
November
2020, 16(6): 2843-2856.
doi: 10.3934/jimo.2019083
Transient analysis of N-policy queue with system disaster repair preventive maintenance re-service balking closedown and setup times
| 1. | Department of Mathematics, St. Anne's College of Engineering and Technology, Anna University, Panruti, Tamilnadu - 607 110, India |
| 2. | Department of Mathematics, Idhaya College of Arts and Science for Women, Pondicherry University, Pakkamudayanpet, Puducherry - 605 008, India |
This paper investigates the transient behavior of a $ M/M/1 $ queueing model with N-policy, system disaster, repair, preventive maintenance, balking, re-service, closedown and setup times. The server stays dormant (off state) until N customers accumulate in the queue and then starts an exhaustive service (on state). After the service, each customer may either leave the system or get immediate re-service. When the system becomes empty, the server resumes closedown work and then undergoes preventive maintenance. After that, it comes to the idle state and waits N accumulate for service. When the $ N^{th} $ one enters the queue, the server commences the setup work and then starts the service. Meanwhile, the system suffers disastrous breakdown during busy period. It forced the system to the failure state and all the customers get eliminated. After that, the server gets repaired and moves to the idle state. The customers may either join the queue or balk when the size of the system is less than N. The probabilities of the proposed model are derived by the method of generating function for the transient case. Some system performance indices and numerical simulations are also presented.
Keywords:
Markovian queue,
N-policy,
disaster and repair,
closedown and setup times,
preventive maintenance,
balking and re-service,
transient probabilities.
Mathematics Subject Classification:
Primary: 60K25.
Citation:
A. Azhagappan, T. Deepa. Transient analysis of N-policy queue with system disaster repair preventive maintenance re-service balking closedown and setup times. Journal of Industrial & Management Optimization,
2020, 16
(6)
: 2843-2856.
doi: 10.3934/jimo.2019083
![]()
References:
| [1] |
S. I. Ammar,
Transient analysis of an $M/M/1$ queue with impatient behavior and multiple vacations, Applied Mathematics and Computation, 260 (2015), 97-105.
doi: 10.1016/j.amc.2015.03.066. |
| [2] |
R. Arumuganathan and S. Jeyakumar,
Steady state analysis of a bulk queue with multiple vacations, setup times with N-policy and closedown times, Applied Mathematical Modeling, 29 (2005), 972-986.
doi: 10.1016/j.apm.2005.02.013. |
| [3] |
S. R. Chakravarthy,
A catastrophic queueing model with delayed action, Applied Mathematical Modeling, 46 (2017), 631-649.
doi: 10.1016/j.apm.2017.01.089. |
| [4] |
F. Chang, T. Liu and J. Ke,
On an unreliable-server retrial queue with customer feedback and impatience, Applied Mathematical Modeling, 55 (2018), 171-182.
doi: 10.1016/j.apm.2017.10.025. |
| [5] |
D. I. Choi and T. S. Kim,
Analysis of a two-phase queueing system with vacations and Bernoulli feedback, Stochastic Analysis and Applications, 21 (2003), 1009-1019.
doi: 10.1081/SAP-120024702. |
| [6] |
F. A. Haight,
Queueing with balking, Biometrika, 44 (1957), 360-369.
doi: 10.1093/biomet/44.3-4.360. |
| [7] |
M. Jain, C. Shekhar and S. Shukla,
N-policy for a repairable redundant machining system with controlled rates, RAIRO-Operations Research, 50 (2016), 891-907.
doi: 10.1051/ro/2015032. |
| [8] |
M. Jain,
Priority queue with batch arrival, balking, threshold recovery, unreliable server and optimal service, RAIRO-Operations Research, 51 (2017), 417-432.
doi: 10.1051/ro/2016032. |
| [9] |
K. Kalidass, S. Gopinath, J. Gnanaraj and K. Ramanath,
Time-dependent analysis of an $M/M/1/N$ queue with catastrophes and a repairable server, Opsearch, 49 (2012), 39-61.
doi: 10.1007/s12597-012-0065-6. |
| [10] |
J. C. Ke,
Batch arrival queues under vacation policies with server breakdowns and startup/closedown times, Applied Mathematical Modelling, 31 (2007), 1282-1292.
doi: 10.1016/j.apm.2006.02.010. |
| [11] |
B. K. Kumar, P. R. Parthasarthy and M. Sharafali,
Transient solution of an $M/M/1$ queue with balking, Queueing Systems, 13 (1993), 441-447.
doi: 10.1007/BF01149265. |
| [12] |
B. K. Kumar and D. Arivudainambi,
Transient solution of an $M/M/1$ queue with catastrophes, Computers and Mathematics with Applications, 40 (2000), 1233-1240.
doi: 10.1016/S0898-1221(00)00234-0. |
| [13] |
B. K. Kumar and S. P. Madheswari,
Transient analysis of an $M/M/1$ queue subject to catastrophes and server failures, Stochastic Analysis and Applications, 23 (2005), 329-340.
doi: 10.1081/SAP-200050101. |
| [14] |
B. K. Kumar, S. P. Madheswari and K. S. Venkatakrishnan,
Transient solution of an $M/M/2$ queue with heterogeneous servers subject to catastrophes, International Journal of Information and Management Sciences, 18 (2007), 63-80.
|
| [15] |
B. K. Kumar, A. Krishnamoorthy, S. P. Madheswari and S. S. Basha,
Transient analysis of a single server queue with catastrophes, failures and repairs, Queueing Systems, 56 (2007), 133-141.
doi: 10.1007/s11134-007-9014-0. |
| [16] |
B. K. Kumar, S. Anbarasu and S. R. A. Lakshmi,
Performance analysis for queueing systems with close down periods and server under maintenance, International Journal of Systems Science, 46 (2015), 88-110.
doi: 10.1080/00207721.2013.775384. |
| [17] |
P. R. Parthasarathy,
A transient solution to an $M/M/1$ queue: A simple approach, Advances in Applied Probability, 19 (1987), 997-998.
doi: 10.2307/1427113. |
| [18] |
P. R. Parthasarathy and R. Sudhesh,
Transient solution of a multiserver Poisson queue with N-policy, Computers & Mathematics with Applications, 55 (2008), 550-562.
doi: 10.1016/j.camwa.2007.04.024. |
| [19] |
R. Sudhesh,
Transient analysis of a queue with system disasters and customer impatience, Queueing Systems, 66 (2010), 95-105.
doi: 10.1007/s11134-010-9186-x. |
| [20] |
R. Sudhesh, R. Sebasthi Priya and R. B. Lenin,
Analysis of N-policy queues with disastrous breakdown, TOP, 24 (2016), 612-634.
doi: 10.1007/s11750-016-0411-6. |
| [21] |
R. Sudhesh, A. Azhagappan and S. Dharmaraja,
Transient analysis of $M/M/1$ queue with working vacation heterogeneous service and customers' impatience, RAIRO-Operations Research, 51 (2017), 591-606.
doi: 10.1051/ro/2016046. |
| [22] |
R. Sudhesh and A. Azhagappan,
Transient analysis of an $M/M/1$ queue with variant impatient behavior and working vacations, Opsearch, 55 (2018), 787-806.
doi: 10.1007/s12597-018-0339-8. |
| [23] |
L. Takacs,
A single server queue with feedback, Bell System Technical Journal, 42 (1963), 505-519.
doi: 10.1002/j.1538-7305.1963.tb00510.x. |
| [24] |
P. Vijayalaxmi and K. Jyothsna,
Analysis of finite buffer renewal input queue with balking and multiple working vacations, Opsearch, 50 (2013), 548-565.
doi: 10.1007/s12597-013-0123-8. |
| [25] |
U. Yechiali,
Queues with system disasters and impatient customers when system is down, Queueing Systems, 56 (2007), 195-202.
doi: 10.1007/s11134-007-9031-z. |
show all references
References:
| [1] |
S. I. Ammar,
Transient analysis of an $M/M/1$ queue with impatient behavior and multiple vacations, Applied Mathematics and Computation, 260 (2015), 97-105.
doi: 10.1016/j.amc.2015.03.066. |
| [2] |
R. Arumuganathan and S. Jeyakumar,
Steady state analysis of a bulk queue with multiple vacations, setup times with N-policy and closedown times, Applied Mathematical Modeling, 29 (2005), 972-986.
doi: 10.1016/j.apm.2005.02.013. |
| [3] |
S. R. Chakravarthy,
A catastrophic queueing model with delayed action, Applied Mathematical Modeling, 46 (2017), 631-649.
doi: 10.1016/j.apm.2017.01.089. |
| [4] |
F. Chang, T. Liu and J. Ke,
On an unreliable-server retrial queue with customer feedback and impatience, Applied Mathematical Modeling, 55 (2018), 171-182.
doi: 10.1016/j.apm.2017.10.025. |
| [5] |
D. I. Choi and T. S. Kim,
Analysis of a two-phase queueing system with vacations and Bernoulli feedback, Stochastic Analysis and Applications, 21 (2003), 1009-1019.
doi: 10.1081/SAP-120024702. |
| [6] |
F. A. Haight,
Queueing with balking, Biometrika, 44 (1957), 360-369.
doi: 10.1093/biomet/44.3-4.360. |
| [7] |
M. Jain, C. Shekhar and S. Shukla,
N-policy for a repairable redundant machining system with controlled rates, RAIRO-Operations Research, 50 (2016), 891-907.
doi: 10.1051/ro/2015032. |
| [8] |
M. Jain,
Priority queue with batch arrival, balking, threshold recovery, unreliable server and optimal service, RAIRO-Operations Research, 51 (2017), 417-432.
doi: 10.1051/ro/2016032. |
| [9] |
K. Kalidass, S. Gopinath, J. Gnanaraj and K. Ramanath,
Time-dependent analysis of an $M/M/1/N$ queue with catastrophes and a repairable server, Opsearch, 49 (2012), 39-61.
doi: 10.1007/s12597-012-0065-6. |
| [10] |
J. C. Ke,
Batch arrival queues under vacation policies with server breakdowns and startup/closedown times, Applied Mathematical Modelling, 31 (2007), 1282-1292.
doi: 10.1016/j.apm.2006.02.010. |
| [11] |
B. K. Kumar, P. R. Parthasarthy and M. Sharafali,
Transient solution of an $M/M/1$ queue with balking, Queueing Systems, 13 (1993), 441-447.
doi: 10.1007/BF01149265. |
| [12] |
B. K. Kumar and D. Arivudainambi,
Transient solution of an $M/M/1$ queue with catastrophes, Computers and Mathematics with Applications, 40 (2000), 1233-1240.
doi: 10.1016/S0898-1221(00)00234-0. |
| [13] |
B. K. Kumar and S. P. Madheswari,
Transient analysis of an $M/M/1$ queue subject to catastrophes and server failures, Stochastic Analysis and Applications, 23 (2005), 329-340.
doi: 10.1081/SAP-200050101. |
| [14] |
B. K. Kumar, S. P. Madheswari and K. S. Venkatakrishnan,
Transient solution of an $M/M/2$ queue with heterogeneous servers subject to catastrophes, International Journal of Information and Management Sciences, 18 (2007), 63-80.
|
| [15] |
B. K. Kumar, A. Krishnamoorthy, S. P. Madheswari and S. S. Basha,
Transient analysis of a single server queue with catastrophes, failures and repairs, Queueing Systems, 56 (2007), 133-141.
doi: 10.1007/s11134-007-9014-0. |
| [16] |
B. K. Kumar, S. Anbarasu and S. R. A. Lakshmi,
Performance analysis for queueing systems with close down periods and server under maintenance, International Journal of Systems Science, 46 (2015), 88-110.
doi: 10.1080/00207721.2013.775384. |
| [17] |
P. R. Parthasarathy,
A transient solution to an $M/M/1$ queue: A simple approach, Advances in Applied Probability, 19 (1987), 997-998.
doi: 10.2307/1427113. |
| [18] |
P. R. Parthasarathy and R. Sudhesh,
Transient solution of a multiserver Poisson queue with N-policy, Computers & Mathematics with Applications, 55 (2008), 550-562.
doi: 10.1016/j.camwa.2007.04.024. |
| [19] |
R. Sudhesh,
Transient analysis of a queue with system disasters and customer impatience, Queueing Systems, 66 (2010), 95-105.
doi: 10.1007/s11134-010-9186-x. |
| [20] |
R. Sudhesh, R. Sebasthi Priya and R. B. Lenin,
Analysis of N-policy queues with disastrous breakdown, TOP, 24 (2016), 612-634.
doi: 10.1007/s11750-016-0411-6. |
| [21] |
R. Sudhesh, A. Azhagappan and S. Dharmaraja,
Transient analysis of $M/M/1$ queue with working vacation heterogeneous service and customers' impatience, RAIRO-Operations Research, 51 (2017), 591-606.
doi: 10.1051/ro/2016046. |
| [22] |
R. Sudhesh and A. Azhagappan,
Transient analysis of an $M/M/1$ queue with variant impatient behavior and working vacations, Opsearch, 55 (2018), 787-806.
doi: 10.1007/s12597-018-0339-8. |
| [23] |
L. Takacs,
A single server queue with feedback, Bell System Technical Journal, 42 (1963), 505-519.
doi: 10.1002/j.1538-7305.1963.tb00510.x. |
| [24] |
P. Vijayalaxmi and K. Jyothsna,
Analysis of finite buffer renewal input queue with balking and multiple working vacations, Opsearch, 50 (2013), 548-565.
doi: 10.1007/s12597-013-0123-8. |
| [25] |
U. Yechiali,
Queues with system disasters and impatient customers when system is down, Queueing Systems, 56 (2007), 195-202.
doi: 10.1007/s11134-007-9031-z. |
Figure 2.
Transient probabilities for the off state of the server
Figure 3.
Transient probabilities for the on state of the server
Figure 4.
Mean system size with different values of $ \sigma $
Figure 5.
Variance of system size for various values of $ \sigma $
Figure 6.
Mean system size with different values of $ \eta $
Figure 7.
Variance of system size for various values of $ \eta $
| [1] |
Yen-Luan Chen, Chin-Chih Chang, Zhe George Zhang, Xiaofeng Chen. Optimal preventive "maintenance-first or -last" policies with generalized imperfect maintenance models. Journal of Industrial & Management Optimization, 2021, 17 (1) : 501-516. doi: 10.3934/jimo.2020149 |
| [2] |
Jian Zhang, Tony T. Lee, Tong Ye, Liang Huang. An approximate mean queue length formula for queueing systems with varying service rate. Journal of Industrial & Management Optimization, 2021, 17 (1) : 185-204. doi: 10.3934/jimo.2019106 |
| [3] |
Hanyu Gu, Hue Chi Lam, Yakov Zinder. Planning rolling stock maintenance: Optimization of train arrival dates at a maintenance center. Journal of Industrial & Management Optimization, 2020 doi: 10.3934/jimo.2020177 |
| [4] |
Philipp Harms. Strong convergence rates for markovian representations of fractional processes. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020367 |
| [5] |
Yiling Chen, Baojun Bian. Optimal dividend policy in an insurance company with contagious arrivals of claims. Mathematical Control & Related Fields, 2021, 11 (1) : 1-22. doi: 10.3934/mcrf.2020024 |
| [6] |
Juhua Shi, Feida Jiang. The degenerate Monge-Ampère equations with the Neumann condition. Communications on Pure & Applied Analysis, 2021, 20 (2) : 915-931. doi: 10.3934/cpaa.2020297 |
| [7] |
Tien-Yu Lin, Bhaba R. Sarker, Chien-Jui Lin. An optimal setup cost reduction and lot size for economic production quantity model with imperfect quality and quantity discounts. Journal of Industrial & Management Optimization, 2021, 17 (1) : 467-484. doi: 10.3934/jimo.2020043 |
| [8] |
Zonghong Cao, Jie Min. Selection and impact of decision mode of encroachment and retail service in a dual-channel supply chain. Journal of Industrial & Management Optimization, 2020 doi: 10.3934/jimo.2020167 |
| [9] |
Hui Lv, Xing'an Wang. Dissipative control for uncertain singular markovian jump systems via hybrid impulsive control. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 127-142. doi: 10.3934/naco.2020020 |
| [10] |
Sujit Kumar Samanta, Rakesh Nandi. Analysis of $GI^{[X]}/D$-$MSP/1/\infty$ queue using $RG$-factorization. Journal of Industrial & Management Optimization, 2021, 17 (2) : 549-573. doi: 10.3934/jimo.2019123 |
| [11] |
Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, 2021, 20 (1) : 301-317. doi: 10.3934/cpaa.2020267 |
| [12] |
Agnaldo José Ferrari, Tatiana Miguel Rodrigues de Souza. Rotated $ A_n $-lattice codes of full diversity. Advances in Mathematics of Communications, 2020 doi: 10.3934/amc.2020118 |
| [13] |
Ivan Bailera, Joaquim Borges, Josep Rifà. On Hadamard full propelinear codes with associated group $ C_{2t}\times C_2 $. Advances in Mathematics of Communications, 2021, 15 (1) : 35-54. doi: 10.3934/amc.2020041 |
| [14] |
Yu Zhou, Xinfeng Dong, Yongzhuang Wei, Fengrong Zhang. A note on the Signal-to-noise ratio of $ (n, m) $-functions. Advances in Mathematics of Communications, 2020 doi: 10.3934/amc.2020117 |
| [15] |
Manuel Friedrich, Martin Kružík, Jan Valdman. Numerical approximation of von Kármán viscoelastic plates. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 299-319. doi: 10.3934/dcdss.2020322 |
| [16] |
Mathew Gluck. Classification of solutions to a system of $ n^{\rm th} $ order equations on $ \mathbb R^n $. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5413-5436. doi: 10.3934/cpaa.2020246 |
| [17] |
Yueh-Cheng Kuo, Huan-Chang Cheng, Jhih-You Syu, Shih-Feng Shieh. On the nearest stable $ 2\times 2 $ matrix, dedicated to Prof. Sze-Bi Hsu in appreciation of his inspiring ideas. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020358 |
| [18] |
Shengbing Deng, Tingxi Hu, Chun-Lei Tang. $ N- $Laplacian problems with critical double exponential nonlinearities. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 987-1003. doi: 10.3934/dcds.2020306 |
| [19] |
Michiyuki Watanabe. Inverse $N$-body scattering with the time-dependent hartree-fock approximation. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021002 |
| [20] |
Pengyan Ding, Zhijian Yang. Well-posedness and attractor for a strongly damped wave equation with supercritical nonlinearity on $ \mathbb{R}^{N} $. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021006 |
2019 Impact Factor: 1.366
Tools
Metrics
Other articles
by authors
[Back to Top]