# American Institute of Mathematical Sciences

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

* Corresponding author: T. Deepa, deepatmaths@gmail.com

Received  October 2018 Revised  March 2019 Published  July 2019

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.

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, 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [6] F. A. Haight, Queueing with balking, Biometrika, 44 (1957), 360-369.  doi: 10.1093/biomet/44.3-4.360.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [6] F. A. Haight, Queueing with balking, Biometrika, 44 (1957), 360-369.  doi: 10.1093/biomet/44.3-4.360.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
State Transition Diagram
Transient probabilities for the off state of the server
Transient probabilities for the on state of the server
Mean system size with different values of $\sigma$
Variance of system size for various values of $\sigma$
Mean system size with different values of $\eta$
Variance of system size for various values of $\eta$
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