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Sensitivity Analysis of Markovian Queue with Discouragement, Additional Servers and Threshold Policy


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1 World College of Technology and Management, Gurugram, Haryana, India
     

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In the present paper, an attempt has been made to study the optimal threshold policy for Markovian queueing model having additional servers along with permanent server. The incorporation of customer's balking and reneging behavior has been done. The customers arrive in Poisson fashion and their service times are exponentially distributed. The first server starts service when N (≥1) or more customers are accumulated and turns off when the system is empty. The (j+1)th (j=1, 2,...., r-1) server turns on when there are Nj+1 customers in the system and will be removed as soon as the number of customers drops to threshold level Nj. We use Laplace transform technique to derive transient probabilities and some other system characteristics such as the expected number of jobs in the system, throughput, and probability that jth (j=1,2,3,…,r) server being busy in rendering the service, etc.. The effects of system parameters on the performance characteristics have been examined by taking numerical illustrations.

Keywords

Threshold Policy, Markovian Queue, Balking, Reneging, Additional Servers, Transient Analysis, Throughput, Delay Time.
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  • M. O. Abou El Ata, and A. I. Shawky, “The single server Markovian overflow queue with balking, reneging and additional server for longer queues,” Microelectronics and Reliability, vol. 32, no. 12, pp. 1389-1394, 1992.

  • R. O. Al-Seedy, A. A. El-Sherbiny, S. A. El-Shehawy, and S. I. Ammar, “Transient solution of the M/M/c queue with balking and reneging,” Computers & Mathematics with Applications, vol. 57, no. 8, pp. 1280-1285, 2009.

  • S. I. Ammar, M. M. Helan, and F. T. Al Amri, “The busy period of an M/M/1 queue with balking and reneging,” Applied Mathematical Modelling, vol. 37, no. 22, pp. 9223-9229, 2013.

  • R. Arumunganathan, and S. Jeyakumar, “Steady state analysis of a bulk queue with multiple vacations, set up times with N-policy and closedown times,” Applied Mathematical Modelling, vol. 29, pp. 972-986, 2005.

  • K. U. Chandrika, Transient analysis of a system with queue dependent servers, OPSEARCH, vol. 43, no. 2, pp. 178-189, 2006.

  • G. Chaudhury, and M. Paul, “A batch arrival queue with an additional service channel under N-policy,” Applied Mathematical Modelling, vol. 156, no. 1, pp. 115-130, 2004.

  • A. G. H. Díaz, and P. Moreno, “A discrete-time single-server queueing system with an N-policy, an early setup and a generalization of the Bernoulli feedback”, Mathematical and Computer Modelling: An International Journal, vol. 49, no. 5-6, pp. 977-990, 2009.

  • S. D. Woolford, and G. Douglas, “A preemptive priority queue with balking,” European Journal of Operational Research, vol. 164, no. 2, pp. 387-401, 2005.

  • D. Guha, V. Goswami, and A. D. Banik, “Algorithmic computation of steady-state probabilities in an almost observable GI/M/c queue with or without vacations under state dependent balking and reneging,” Applied Mathematical Modelling, vol. 40, no. 5-6, pp. 4199-4219, 2016.

  • Y. C. Hsieh, and K. H. Wang, “Reliability of a repairable system with spares and a removable repairman,” Microelectronics Reliability, vol. 35, no. 2, pp. 197-207, 1995.

  • S. Hur, J. Kim, and C. Kang, “An analysis of the M/G/1 system with N and T policy”, Applied Mathematical Modelling, vol. 27, no. 8, pp. 665-675, 2003.

  • M. Jain, “Optimal N-policy for single server Markovian queue with breakdown, repair and state dependent arrival rate,” International Journal of Management Systems, vol. 13, no. 3, pp. 245-260, 1997.

  • M. Jain, “M/M/m queue with discouragement and additional servers,” Journal of German Studies Review, vol. 36, no. 1-2, pp. 31-42, 1998.

  • M. Jain, “N- Policy for redundant repairable system with additional repairman,” OPSEARCH, vol. 40, no. 2, pp. 97-114, 2003.

  • M. Jain, Rakhee, and S. Maheshwari, “N policy for a machine repair system with spares and reneging,” Applied Mathematical Modelling, vol. 28, no. 6, pp. 513-531, 2004.

  • M. Jain, G. C. Sharma, and N. Singh, “Transient analysis of M/M/R machining system with mixed standbys, switching failures, balking, reneging and additional removable repairmen,” International Journal of Engineering, vol. 20, no. 2, pp. 169-182, 2007.

  • M. Jain, G. C. Sharma, and M. Singh, “M/M/R machine interference model with balking, reneging, spares and two modes of failure,” OPSEARCH, vol. 40, no. 1, pp. 24-41, 2003.

  • M. Jain, M. Singh, and K. P. S. Baghel, “M/M/C/K/N machine repair problem with balking, reneging, spares and additional repairman,” Journal of German Studies Review, vol. 26-27, pp. 49-60, 2000.

  • C. Kim, and A. Dudin, “Analysis of a queueing model with contingent additional server,” In Gaj P., Kwiecien A., Stera P. (eds.) Computer Networks, Communications in Computer and Information Science, vol. 608, 2016. Springer, Cham.

  • D. H. Lee, and W. S. Yang, “The N-policy of a discrete time Geo/G/1 queue with disasters and its application to wireless sensor networks,” Applied Mathematical Modelling, vol. 37, no. 23, pp. 9722-9731, 2013.

  • P. R. Parthasarathy, and R. Sudhesh, “Time-dependent analysis of a single server retrial queue with state dependent rates,” Operations Research Letters, vol. 35, no. 5, pp. 601-611, 2007.

  • A. I. Pazgal, and S. Radas, “Comparison of customer balking and reneging behavior to queueing theory predictions: An experimental study,” Computers and Operations Research, vol. 35, no. 8, pp. 2537-2548, 2008.

  • J. F. Reynolds, “The stationary solution of a multi-server model with discouragement,” Operations Research, vol. 16, pp. 64-71, 1968.

  • J. A. Schwarz, G. Selinka, and R. Stolletz, “Performance analysis of time-dependent queueing systems: Survey and classification,” Omega, vol. 63, pp. 170-189, 2016.

  • O. P. Sharma, Markovian Queues, Ellis Horwood, London, 1990.

  • O. P. Sharma, and A. M. K. Tarabia, “A simple transient analysis of an M/M/1/N queue,” Sankhya: The Indian Journal of Statistics, vol. 62, no 2, pp. 273-281, 2000.

  • A. I. Shawky, “The single server machine interference model with balking, reneging and an additional server for longer queues,” Microelectronics Reliability, vol. 37, no. 1, pp. 355-357, 1997.

  • Shawky, A.I. (2000), “The machine interference model: M/M/C/K/N with balking, reneging and spares”, OPSEARCH, vol. 37, no. 1, pp. 25-35, 2000.

  • V. Sridaran, and P. R. Jayashree, “A note on the transient behavior and expected profit of a M/M/R machine repair problem with spares,” Stochastic Processes and their Applications, (eds. A. Vijayakumar and M. Sreenivasan), Narosa Pub. House, New Delhi, India, pp. 215-231, 1999.

  • W. Sun, S. Li, and E. Cheng-Guo, “Equilibrium and optimal balking strategies of customers in Markovian queues with multiple vacations and N-policy,” Applied Mathematical Modelling, vol. 40, no. 1, pp. 284-301, 2016.

  • N. Tian, and Z. G. Zhang, “A two-threshold vacation policy in queueing systems,” European Journal of Operational Research, vol. 168, no. 1, pp. 153-163, 2004.

  • J. Wang, X. Zhang, and P. Huang, “Strategic behavior and social optimization in a constant retrial queue with the N-policy,” European Journal of Operational Research, vol. 256, no. 3, pp. 841-849, 2017.

  • K. H. Wang, J. B. Ke, and J. C. Ke, “Profit analysis of the M/M/R machine repair problem with balking, reneging and standby switching failures,” Computers and Operations Research, vol. 34, no. 3, pp. 835-847, 2007a.

  • K. H. Wang, T. Y. Wang, and W. L. Pearn, “Optimal control of the N policy M/G/1 queueing system with server breakdowns and general startup times”, Applied Mathematical Modelling, vol. 31, no. 10, pp. 2199-2212, 2007b.

  • K. H. Wang, “Optimal operation of a Markovian queueing system with a removable and non-reliable server”, Microelectronics Reliability, vol. 35, no. 8, pp. 1131-1136, 1995.

  • M. Yadin, and P. Naor, “Queueing system with removable service station,” Opl. Research Quarterly, vol. 14, pp. 393-405, 1963.


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  • Sensitivity Analysis of Markovian Queue with Discouragement, Additional Servers and Threshold Policy

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Authors

Neetu Singh
World College of Technology and Management, Gurugram, Haryana, India

Abstract


In the present paper, an attempt has been made to study the optimal threshold policy for Markovian queueing model having additional servers along with permanent server. The incorporation of customer's balking and reneging behavior has been done. The customers arrive in Poisson fashion and their service times are exponentially distributed. The first server starts service when N (≥1) or more customers are accumulated and turns off when the system is empty. The (j+1)th (j=1, 2,...., r-1) server turns on when there are Nj+1 customers in the system and will be removed as soon as the number of customers drops to threshold level Nj. We use Laplace transform technique to derive transient probabilities and some other system characteristics such as the expected number of jobs in the system, throughput, and probability that jth (j=1,2,3,…,r) server being busy in rendering the service, etc.. The effects of system parameters on the performance characteristics have been examined by taking numerical illustrations.

Keywords


Threshold Policy, Markovian Queue, Balking, Reneging, Additional Servers, Transient Analysis, Throughput, Delay Time.

References