N-policy for a repairable redundant machining system with controlled rates

Jain, Madhu; Shekhar, Chandra; Shukla, Shalini

doi:10.1051/ro/2015032

Jain, Madhu ¹ ; Shekhar, Chandra ² ; Shukla, Shalini ³

¹ Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, 247667 Uttarakhand, India.
² Department of Mathematics, Birla Institute of Technology and Science, Pilani, 333031 Rajasthan, India.
³ Department of Mathematics, D.A.V. (PG) College, Dehra Dun, 248001 Uttarakhand, India.

RAIRO - Operations Research - Recherche Opérationnelle, Special issue - Advanced Optimization Approaches and Modern OR-Applications, Tome 50 (2016) no. 4-5, pp. 891-907.

Résumé

The present investigation deals with a multi-component machining system operating under $N$ -policy. There is a provision of $k$ type of mixed standbys units and maintenance crew consisting of permanent repairman as well as removable additional repairmen. The life time and repair time of the units are exponentially distributed with interdependent rates. The permanent repairman can start the repair only when $N$ failed units are accumulated in the system. As soon as a unit fails, it is replaced by an available standby unit. In case when all the standby units are exhausted, the machining system starts to function in a degraded mode due to overload. Markov model is developed by constructing the governing transient state equations which are solved by using Runge−Kutta method. Various performance measures viz. queue size distribution, expected number of failed units, cost function, etc. are evaluated. By taking numerical illustration, numerical results are computed and also depicted with the help of graphs and figures.

Reçu le : 2015-02-23
Accepté le : 2015-09-06

MR Zbl

DOI : 10.1051/ro/2015032

Classification : 90B22, 90B25, 60K25, 60K30, 68M20
Mots-clés : Markovian machine repair model, N-policy, mixed standbys, additional repairmen, interdependent rates

Affiliations des auteurs :

Jain, Madhu ¹ ; Shekhar, Chandra ² ; Shukla, Shalini ³

@article{RO_2016__50_4-5_891_0,
     author = {Jain, Madhu and Shekhar, Chandra and Shukla, Shalini},
     title = {N-policy for a repairable redundant machining system with controlled rates},
     journal = {RAIRO - Operations Research - Recherche Op\'erationnelle},
     pages = {891--907},
     publisher = {EDP-Sciences},
     volume = {50},
     number = {4-5},
     year = {2016},
     doi = {10.1051/ro/2015032},
     zbl = {1353.90040},
     mrnumber = {3570537},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ro/2015032/}
}

TY  - JOUR
AU  - Jain, Madhu
AU  - Shekhar, Chandra
AU  - Shukla, Shalini
TI  - N-policy for a repairable redundant machining system with controlled rates
JO  - RAIRO - Operations Research - Recherche Opérationnelle
PY  - 2016
SP  - 891
EP  - 907
VL  - 50
IS  - 4-5
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/ro/2015032/
DO  - 10.1051/ro/2015032
LA  - en
ID  - RO_2016__50_4-5_891_0
ER  -

%0 Journal Article
%A Jain, Madhu
%A Shekhar, Chandra
%A Shukla, Shalini
%T N-policy for a repairable redundant machining system with controlled rates
%J RAIRO - Operations Research - Recherche Opérationnelle
%D 2016
%P 891-907
%V 50
%N 4-5
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/ro/2015032/
%R 10.1051/ro/2015032
%G en
%F RO_2016__50_4-5_891_0

Jain, Madhu; Shekhar, Chandra; Shukla, Shalini. N-policy for a repairable redundant machining system with controlled rates. RAIRO - Operations Research - Recherche Opérationnelle, Special issue - Advanced Optimization Approaches and Modern OR-Applications, Tome 50 (2016) no. 4-5, pp. 891-907. doi : 10.1051/ro/2015032. http://www.numdam.org/articles/10.1051/ro/2015032/

Bibliographie
Cité par

M.I.A. Begum and D. Maheshwari, The M/M/C interdependent queueing model with controllable arrival rates. Opsearch 39 (2002) 89–110. | DOI | MR | Zbl

L. Haque and M.J. Armstrong, A survey of machine interference problem. Eur. J. Oper. Res. 179 (2007) 469–482. | DOI | Zbl

M. Jain, M/M/R machine repairmen problem with spares and additional repairmen. Indian J. Pure Appl. Math. 29 (1998) 517–524. | MR | Zbl

M. Jain, Transient analysis of machining systems with service interruption, mixed standbys and priority. Int. J. Math. Oper. Res. 5 (2013) 604–625. | DOI | MR

M. Jain, Transient analysis of a machine repair system with standby, two modes of failure, Discouragement and Switching Failure. Int. J. Oper. Res. 21 (2014) 365–390. | DOI | MR | Zbl

M. Jain and R.P. Gimire, Machine repair queueing system with non reliable service stations and heterogeneous service discipline. Int. J. Eng. 12 (1999) 271–276.

M. Jain and S. Maheshwari, Transient analysis of redundant repairable system with additional repairmen. Amer. J. Math. Manag. Sci. 23 (2003) 1–37. | MR

M. Jain and Preeti, Cost analysis of a machine repair problem with standby, working vacation and server breakdown. Int. J. Math. Oper. Res. 6 (2014) 437–451. | DOI | MR

M. Jain and S. Rani, Transient analysis of hardware and software systems with warm standbys and switching failures. Int. J. Math. Oper. Res. 6 (2014) 1–28. | DOI | MR | Zbl

M. Jain and S. Upadhyaya, Threshold N-policy for degraded machining system with multiple type spares and multiple vacations. Qual. Technol. Quant. Manag. 6 (2009) 185–203. | DOI | MR

M. Jain, G.C. Sharma and P. Sharma, Controllable multi-server queue with balking. Int. J. Eng. 18 (2005) 1–11. | Zbl

M. Jain, G.C. Sharma and N. Kumar, Controllable and interdependent MRP with additional repairman and mixed standbys. J. Rajasthan Acad. Phys. Sci. 8 (2009) 447–456. | Zbl

M. Jain, G.C. Sharma and R. Sharma, Performance modeling of state dependent system with mixed standbys and two modes of failure. Appl. Math. Mod. 32 (2008) 712–724. | DOI | MR | Zbl

M. Jain, G.C. Sharma and R. Pundhir, Some perspectives of machine repair problem. Int. J. Eng. Trans. B: Appl. 23 (2010) 253–268. | Zbl

M. Jain, C. Shekhar and S. Shukla, Queueing analysis of two unreliable servers machining system with switching and common cause failure. Int. J. Math. Oper. Res. 5 (2013) 508–536. | DOI | MR | Zbl

M. Jain, C. Shekhar and S. Shukla, Vacation queuing model for a machining system with two unreliable repairmen. Int. J. Oper. Res. 20 (2014) 469–491. | DOI | MR | Zbl

M. Jain, G.C. Sharma and V. Rani, M/M/R+r machining system with reneging, spares and interdependent controlled rates. Int. J. Math. Oper. Res. 6 (2014) 655–679. | DOI | MR | Zbl

M. Jain, V. Rani and R. Kumari, Threshold switch over policy for multi-component machining system with controllable rates. Int. J. Ind. Systems Eng. 20 (2015) 397–414.

W.M. Kempa, The transient analysis of the queue-length distribution in the batch arrival system with N-policy, multiple vacations and setup times. Proc. of Applications of Mathematics in Engineering and Economics, edited by G. Venkov, R. Kovacheva and V. Pasheva. AIP publishing, Bulgaria (2010) 235–242.

W.M. Kempa, On transient queue-size distribution in the batch arrival system with the N-policy and setup times. Math. Commun. 17 (2012) 285–302. | MR | Zbl

K. Kumar and M. Jain, Threshold F-policy and N-policy for multi-component machining system with warm standbys. J. Ind. Eng. Int. 9 (2013) 1–9. | DOI

K. Kumar and M. Jain, Threshold N-policy for (M, m) degraded machining system with K-heterogeneous servers, standby switching failure and multiple vacations. Int. J. Math. Oper. Res. 5 (2013) 423–445. | DOI | MR | Zbl

C.D. Liou, Optimization analysis of the machine repair problem with multiple vacations and working breakdowns. Management 11 (2015) 83–104. | MR | Zbl

S. Maheshwari, P. Sharma and M. Jain, Machine repair problem with k-type warm spares, multiple vacations for repairmen and reneging. Int. J. Eng. Technol. 2 (2010) 252–258.

J. Medhi and J.G.C. Templeton, A poisson input queue under N-policy and with a general start up time. Comput. Oper. Res. 19 (1992) 35–41. | DOI | Zbl

S.S. Mishra and S. Pal, A computational approach to the M/M/1/N interdependent queuing model with controllable arrival rates. J. Indian Statist. Assoc. 41 (2003) 83–94. | MR

C.R. Mitchell and A.S. Paulson, The M/M/1 queues with independent arrival and service process. Naval Res. Logist. Quart. 26 (1979) 81–98. | DOI | Zbl

P.R. Parthasarathy and R. Sudhesh, A state-dependent queue alternating between arrivals and services. Int. J. Oper. Res. 7 (2010) 16–30. | DOI | MR | Zbl

P.R. Parthasarathy and R. Sudhesh, Transient solution of a multiserver Poisson queue with N-policy. Comput. Math. Appl. 55 (2008) 550–562. | DOI | MR | Zbl

K. Rao, Z. Rao and T. Shobha, The M/M/1 independent queuing model with controllable arrival rates. Opsearch 37 (2000) 41–48. | MR | Zbl

G.C. Sharma, M. Jain and K.D. Biswas, Controlling rates in double-ended bulk queue. J. Eng. Appl. Sci. 21 (2004) 1–5.

A.I. Shawky, Single server machine interference model with balking, reneging and an additional server for longer queues. Microelectron. Reliab. 37 (1997) 355–357. | DOI

L. Shree, P. Singh, D.C. Sharma and P. Jharotia, Mathematical modeling and performance analysis of machine repairable system with hot spares. Proc. Natl. Acad. Sci., India Sect. A: Phys. Sci. 85 (2015) 127–135. | DOI | MR | Zbl

L. Tadj and G. Choudhury, Optimal design and control of queues. Top 13 (2005) 359–412. | DOI | MR | Zbl

A.J.J. Talman, A simple proof of the optimality of the best N-policy in the M/G/1 queueing control problem with removable server. Statist. Neerlandica 33 (1979) 143–150. | DOI | MR | Zbl

W. Wu, Y. Tang, M. Yu and Y. Jiang, Computation and profit analysis of a K-out-of-n: G repairable system under N-policy with multiple vacations and one replaceable repair facility. RAIRO: OR 49 (2015) 717–734. | DOI | Numdam | MR | Zbl

D.Y. Yang and C.H. Wu, Cost-minimization analysis of a working vacation queue with N-policy and server breakdowns. Comput. Ind. Eng. 82 (2015) 151–158. | DOI

D. Yue, W. Yue and H. Qi, Performance analysis and optimization of a machine repair problem with warm spares and two heterogeneous repairmen. Optim. Eng. 13 (2012) 545–562. | MR | Zbl

Cité par Sources :