Abstract    In this paper, we study N-policy for a finite population Bernoulli feedback queueing model for machine repair problem with degraded failure. The running times of the machines between breakdowns have an exponential distribution. The repair times of the machines are independent and identically distributed random variables. If at any time a machine fails, it is sent to the repairman for repairing, the repairman restores the machine to the state as before failure. When the failed machine finds the repairman busy upon its failure, it has to wait until its turn as repairman stores only one machine at a time. When all the standby components are used, the failure of components occurs in a degraded fashion. To obtain the steady-state probabilities, the supplementary variable is introduced and a recursive method is employed. Some performance measures viz. expected number of down machines, expected number of machines waiting for repair in the queue, expected number of operating machines, expected number of spare machines, machine availability, etc. are established. Some special cases are deduced that match with the earlier existing results. To provide sensitivity analysis, numerical experiment is performed.

Keywords    M/G/1 queue, N-Policy, Machine repair, Mixed standby, Degraded failure, Bernoulli feedback

References    1. Keilson, J. and Kooharian, A., “On Time Dependent Queuing processes”, Annals. of Mathematical Statistics, Vol. 31, (1960), 104-112. 2. Takacs, L., “Delay Distributions for one Line with Poisson Input, General Holding Times and Various Orders of Service”, Bell System Technical Journal, Vol. 42, (1963a), 487-504. 3. Hokstad, P., “A Supplementary Variable Technique Applied to the M/G/1 Queue”, Scandinavian Journal of Statistics, Vol. 2, (1975), 95-98. 4. Bell, C. E., “Characterization and Computation of Optimal Policies for Operating an M/G/1 Queuing System with Removable Server”, Oper. Res., Vol. 19, (1971), 208-218. 5. Courtois, P. J. and Georges, J., “On a single-server finite queuing model with state-dependent arrival and service processes” Oper. Res., Vol. 19, (1971), 424-435. 6. Herzog, U., Woo, L. and Chandy, K. M., “Solution of Queuing Problems by a Recursive Technique”. IBM J. Res. Develop, Vol. 19, (1975), 295-300. 7. Gupta, U. C. and Srinivasa Rao, T. S. S., “A recursive Method to Compute the Steady-State Probabilities of the Machine Interference Model” (M/G/1)/K. Computers and Operations Research, Vol. 21, (1994), 597-605. 8. Zhang, Z. G. and Love, C. E., “The Threshold Policy in the M/G/1 Queue with an Exceptional First Vacation”, INFOR., Vol. 36, No. 4, (1998), 193-204. 9. Wang, K. H., Chang, K. W. and Sivazlian, B. D., “Optimal Control of a Removable and Non-Reliable Server in an Infinite and a Finite M/H2/1 Queuing System”, Appl. Math. Model, Vol. 23, (1999), 651-666. 10. Lillo, R. E. and Martin, M., “On Optimal Exhaustive Policies for the M/G/1 Queue”. Oper. Res. Lett., Vol. 27, (2000), 39-46. 11. Wang, K. H. and Ke, J. C., “A Recursive Method to the Optimal Control of an M/G/1 Queuing System with finite Capacity and Infinite Capacity”, Appl. Math. Model, Vol. 24, (2000), 899-914. 12. Yadin, M. and Naor, P., “Queuing Systems with Removable Service Stations”, Oper. Res. Quart., Vol. 14, (1963), 393-405. 13. Wang, K. H. and Huang, H. M., “Optimal Control of an M/Ek/1 Queuing System with a Removable Service Station”, OPSEARCH, Vol. 46, (1995), 1014-1022. 14. Jain, M., “N-Policy for Redundant Repairable System with Additional Repairmen”, OPSEARCH, Vol. 40, No. 2, (2003), 97-114. 15. Jain, M., Sharma, G. C. and Singh, M., “N-Policy for Degraded Machining System with Spares and Server Breakdowns”, IJOMAS., Vol. 18/19 (In press), (2003). 16. Takacs, L., “A Single-Server Queue with Feedback”, The Bell System Technical Journal, Vol. 42, No. 2, (1963b), 505-519. 17. Heyman, D. P, “Optimal Operating Policies for M/G/1 Queuing system”. Oper. Res., Vol. 16, (1968), 362-382. 18. Disney, R. L., McNickle, D. C. and Simon, B., “The M/G/1 Queue with Instantaneous Bernoulli Feedback” Naval Res. Logist. Quart., Vol. 27, (1980), 635-644. 19. Disney, R. L., “A Note on Sojourn Times in M/G/1 Queues with Instantaneous, Bernoulli Feedback” Nav. Res. Logist. Quart., Vol. 27, (1981), 679-684. 20. Fontana, B. and Berzosa, C. D., “Stationary Queue- Length Distribution in an M/G/1 Queue with Two Non- Preemptive Priorities and General Feedback. Performance of Computer-Communication Systems”, Ed. W. Bux and H. Rudin, Elsevier, North-Holland, Amsterdam, (1984), 333-347. 21. Simon, B., “Priority Queues with Feedback”, J. Assoc. Comput. Mach., Vol. 31, (1984), 134-149. 22. Takine, T. A., Takagi, H. and Hasegawa, T., “Sojourn Times in Vacation and Polling Systems with Bernoulli Feedback”, J. Appl. Prob, Vol. 28, (1991), 422-432. 23. Gong, W. B., Yan, A. and Gassandras, C. G., “The M/G/1 Queue with Queue-Length Dependent Arrival Rate”, Comm. Statist. Stochastic Models, Vol. 8, No. 4, (1992), 733-741. 24. Takagi, H., “A Note on the Response Time in M/G/1 Queue with Service in Random Order and Bernoulli Feedback”, J. Oper. Res. Soc. of Japan, Vol. 39, No. 4, (1996), 486-500. 25. Boxma, O. J. and Yechiali, U., “An M/G/1 Queue with Multiple Types of Feedback and Gated Vacations”, J. Appl. Prob., Vol. 34, (1997), 773-784. 26. Medhi, J., “Response Time in an M/G/1 Queuing System with Bernoulli Feedback”, Recent Developments in Operational Research, Ed. M.L. Agarwal and K. Sen, Narosa Publishing House, New Delhi, India, (2001), 249-259. 27. Wang, K.H. and Kuo, C.C., “Cost and Probabilistic Analysis of Series Systems with Mixed Standby Components”, Appl. Math. Model, Vol. 24, (2000), 957-967. 28. Gupta, U. C. and Srinivasa Rao, T. S. S., “On the Machine Interference Model with Spares, Euro”. J. Oper. Res, Vol. 89, (1996), 164-171.