Analysis of Queuing Model for Machine Repairing System with Bernoulli Vacation Schedule

R.K. Shrivastava; Awadhesh Kumar Mishra

doi:https://doi.org/10.14445/22315373/IJMTT-V10P514

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 10 | Number 2 | Year 2014 | Article Id. IJMTT-V10P514 | DOI : https://doi.org/10.14445/22315373/IJMTT-V10P514

Analysis of Queuing Model for Machine Repairing System with Bernoulli Vacation Schedule

R.K. Shrivastava , Awadhesh Kumar Mishra

Citation :

R.K. Shrivastava , Awadhesh Kumar Mishra, "Analysis of Queuing Model for Machine Repairing System with Bernoulli Vacation Schedule," International Journal of Mathematics Trends and Technology (IJMTT), vol. 10, no. 2, pp. 85-92, 2014. Crossref, https://doi.org/10.14445/22315373/IJMTT-V10P514

Abstract

In this paper, we consider a queueing model for machine repairing system with Bernoulli vacation schedule. The failure times, repair times and vacation times are all assumed to be exponentially distributed. In congestion, the server may increase the repair rate with pressure coefficient  to reduce the queue length. We assume that the server begins the working vacation when the system is empty. The server may go for a vacation of random length with probability p or may continue to repair the next (if available) failed machine with probability q = 1  p. The whole system is modelled as a finite state Markov Chain and its steady state distribution is obtained by matrix recursive approach.

Keywords

Machine repair, Bernoulli vacation, pressure coefficient, Markov Chain, Matrix-recursive method.

References

[1] Bension. F., and Cox,D.R.(1951): The production of machine repairing attention and random intervals. Journal of the Royal Statistical Society,B, 13, pp.65-82.
[2] Choudhury, G. and Daka, K. (2009): An MX / G / 1 unreliable retrial queue with two phases of service and Bernoulli admission mechanism. Applied Mathematics and Computation, 215, pp.936-949.
[3] Choudhury,G.(2005): A two stage batch arrival queueing system with a modified Bernoulli schedule vacation under N-policy. Mathematical and Computer Modeling, 42, pp.71-85.
[4] Gross, D. and Harris, C.M. (1985): Fundamentals of queueing theory, 2nd edition John Wiley and Sons, New York.
[5] Gupta, S.M. (1997): Machine interference problem with warm spares, server vacations and exhaustive service. Performance Evaluation, 29(3), pp.195-211.
[6] Hsies, Y. C and Wang, K .H. (1995): Reliability of a repairable system with spares and removable repairmen. Microelectronics and reliability, 35, pp.197-208.
[7] Jain, M. (1998): M/M/R/ machine repair problem with spares and additional servers. Indian Journal of Pure and Applied Mathematics, Vol. 29, no. 5, pp. 517-524.
[8] Jain, M. (2003): N-policy redundant reparable system with additional repairman. OPSEARCH, 40(2), pp. 97-114.
[9] Jain, M. (2013): Transient Analysis of machining system with service interruption, mixed standbys and priority. International Journal of Mathematics in Operations Research, Vol. 5, No.5, pp.604-625. (Inderscience).
[10] Jain, M. Sharma, G.C. and Singh, M. (2002): Diffusion process for multi-repairman machining system with spares and balking. International Journal of Engineering Science 15(1), pp. 57-62.
[11] Jain, M., and Sulekha, Rani (2013): Availability analysis of repairable system with warm standby, switching failure and reboot delay. International journal of Mathematics in operations Research Vol.5, No.1, pp. 19-39, (Inderscience).
[12] Jain, M., Chandra Shekhar and Shukla, Shalini (2012): Queueing analysis of a multi-component machining system having unreliable heterogeneous servers and impatient customers, vol. 2(3), pp. 16-26.
[13] Ke, J.C., Hsu, Y.L., Liu, T.H., and Zhang Z.G. (2013): Computational analysis of machine repair problem with unreliable multi-repairmen. Computers and Operations Research, 40(3), pp.848-85.
[14] Ke, J.C., Lee, S.I. and Liou, C.H. (2009): Machine repair problem in production systems with spares and server vacations. RAIRO, Operations Research Vol. 43, No. pp. 135-154.
[15] Ke, J.C., Wu., C.H. and Pearn, W.L.(2011) : Algorithm analysis of the multi-server system with modified Bernoulli schedule. Applied Mathematical Modelling. 35, pp. 2196-2208.
[16] Ke,J.C., Chia-Huang,Wu., and Pearn, W.L.(2013): Analysis of an infinite multi-server queue with an optional service. Computers and Industrial Engineering. 62(2), pp.216-225.
[17] Keilson, J., and Servi, L.D.(1986): Oscillating random walk models for GI/G/1 Vacation systems with Bernoulli Schedule. J. Appl. Prob. Vol. 23, pp.790-802..
[18] Khorram, E. (2008): An Optimal queuing model by dynamic numbers of repairman in finite population queueing system. Quality Technology and Quantitative Management, Vol.5, No. 4, pp. 163-178.
[19] Liu, Hsin, T., and Ke, J.C. (2014): On the multi-server machine interference with modified Bernoulli vacations. Journal of Industrial and Management Optimization. vol.10, no.4, pp. 1191-1208.
[20] Madan, K.C. W. Abu-dayyeh and Taiyyan, F. (2003): A two server queue with Bernoulli schedules and a single vacation policy. Applied Mathematics and Computation.145, pp. 59-71.
[21] Maheshwari, Supriya and Ali, Shazia. (2013): Machine Repair Problem with Mixed Spares Balking and Reneging. International Journal of Theoretic and Applied Sciences, 5(1),pp.75-83.
[22] Maheshwari, Supriya, et al. (2010), Machine repair problem with K-type warm spares, multiple vacations for repairman and reneging. International Journal of Engineering and Technology, Vol. 2(4), pp. 252-258.
[23] Wang K.H. and Wu, J.D. (1995): Cost analysis of the M/M/R machine repair problem with spares and two modes of failure. Journal of the Operational Research Society, 46, 783-790.
[24] Wang, K.H. Ke, J.B. and Ke. J.C. (2007), Profit analysis of the M/M/R machine repair problem with balking, reneging and standby switching failure, Computers and Operations Research 34(3), pp. 835-847.
[25] Wang, K.H., Chen, W.L. and Yang, D.Y. (2009): Optimal management of the machine repair problem with working vacation, Newton’s method. Journal of Computational and Applied Mathematics. 233, pp. 449-458.
[26] Xu.X. and Zhang, Z.G. (2006): Analysis of multi-server queue with a single vacation (e,d)-policy. Performance evaluation, vol. 63, pp. 825-836.
[27] Ying- Lin,Hsu., Ke,J.C.,Tzu -Hsin Liu, and Chia-huang, Wu.(2014): Modeling of multi-server repair problem with switching failure and reboot delay under related profit analysis. Computers & Industrial Engineering,69,pp.21-28.
[28] Yue, D., Yue, W., and Qi. H. (2013): Performance Analysis and Optimization of a Machine Repair Problem with warm spares and two heterogeneous Repairmen. Optimization and Engineering, Vol. 3, issue 4, pp. 545-562.