International Journal of Nonlinear Analysis and Applications

Analysis of Markovian queueing system with server failures, N-policy and second optional service

Document Type : Research Paper

Authors

1 Department of Statistics, Acharya Nagarjuna University, Guntur and Data Curator (SBDM), ICRISAT, Hyderabad, India

2 GRIET, Hyderabad, India

3 Theme Leader and Principal Scientist (SBDM), ICRISAT, Hyderabad, India

4 Acharya Nagarjuna University, Guntur, India

Abstract

The current work details the behaviour of a finite Markovian queueing system with a vacation in which the server may face problems of breakdowns while in service. The repair process does start immediately after a breakdown which immediately resumes the service. During this period any new customer is allowed to join the system. Whenever the server finds nobody, the server goes on vacation and resumes service after N customers are accumulated. Meanwhile, it triggers pre-service called start-up. Further, we considered two types of repair facilities for the broken-down server with an optional probability. The server first provides essential service to all customers and the second optional service will be provided with a probability of “p”. The customer may renege in the first phase of service. We adopted Runge-Kutta Method to find Transient state probabilities and computed various performance indices like the expected length of the system, the mean waiting time etc. We then performed the sensitivity analysis to explore the effect of different parameters.

Keywords

[1] 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, Comput. Math. Appl. 57 (2009) 1280–1285.
[2] C.J. Ancker Jr and A.V. Gafarian, Some queueing problems with balking and reneging, II (1963) 88–100.
[3] C.J. Ancker Jr and A.V. Gafarian, Some queueing problem with balking and reneging II, Oper. Res. II (1963) 928–937.
[4] B.T. Doshi, A note on stochastic decomposition in GI/M/1 queue with vacations or startup time, J. Appl. Probab. 22(2) (1985) 419–428.
[5] D. Gross and C.M. Harris, Fundamentals of Queueing Theory, 2nd Edition, John Wiley and Sons, New York, 1985.
[6] S.M. Gupta, Machine interference problem with warm spares, server vacations and exhaustive service, Performance Eval. 29(3) (1997) 195–211.
[7] F.A. Haight, Queueing with balking, Biometrika 44 (1957) 360–369.
[8] Y.C. Hsies and K.H. Wang, Reliability of a repairable system with spares and removable repairmen, Microelectron. Reliab. 35 (1995) 197–208.
[9] M. Jain, M/M/R/ machine repair problem with spares and additional servers, Indian J. Pure Appl. Math. 29(5) (1998) 517–524.
[10] M. Jain, N policy redundant reparable system with additional repairman, OPSEARCH, 40(2) (2003) 97–114.
[11] J.C. Ke, The analysis of a general input queue with N policy and exponential vacations, Queueing Syst. 45(2) (2003) 135–160.
[12] J.D. Kim, D.W. Choi and K.C. Chae, Analysis of queue length distribution of the M/G/1 with working vacation, (M/G/1/WV), Proc. Int. Conf. Statist. Related, Honolulu, Hawaii, USA, (2008).
[13] L. Liu and V. Kulkarni, Explicit solutions for the steady state distributions in M/PH/1 queues with workload-dependent balking, Queueing Systems, 52 (2006) 251–260.
[14] L. Liu and V. Kulkarni, Busy period analysis for M/PH/1 queues with workload-dependent balking, Queueing Syst. 59(1) (2008) 37–51.
[15] V.N. Rama Devi, A.A. Rao and K. Chandan, Analysis of a M/M/1 queueing system with two-phase, N-policy, server failure and second optional batch service with customers impatient behavior, J. Phys. Conf. Ser.1344 (2019) 012015.
[16] V.N. Rama Devi, Y. Saritha and K. Chandan, M/EK/1 queueing system with vacation, two types of repair facilities and server timeout, Int. J. Adv. Sci. Technol. 29(8s) (2020) 846–855.
[17] V.N. Rama Devi, Y. Saritha and K. Chandan, M/G/1 queue with vacation, two cases of repair facilities and server timeout, Test Engin. Manag. 82 (2020) 16358–16363.
[18] S.S. Rao, Queueing models with balking, reneging, and interruptions, Oper. Res. 13(4) (1965) 596–608.
[19] N. Tan and Z.G. Zhang, The discrete-time GI/Geo/1 queue with multiple vacations, Queueing Syst. 40(3) (2002) 283–294.
[20] V. Vasanta Kumar and K. Chandan, Cost analysis of a two-phase MX/EK/1 queueing system with N-policy, OPSEARSCH 45(2) (2008) 155–174.
[21] N.R.D. Vedala, Y. Saritha, A.A. Rao and G. Sridhar, Study of MX/M/1 queueing system with vacation, two kinds of repair facilities and server timeout, Adv. Sci. Technol. Eng. Syst. J. 4(6) (2019) 339–342.
[22] K.-H. Wang and Y.-C. Chang, Cost analysis of a finite M/M/R queueing system with balking, reneging, and server breakdowns, Math. Methods Oper. Res. 56(2) (2002) 169–180.
[23] W. Whitt, Improving service by informing customers about anticipated delays, Manag. Sci. 45 (1999) 192–207.
[24] J. Wu, J. Wang, Z. Liu, A discrete-time Geo/G/1 retrial queue with preferred and impatient customers, Appl. Math. Modell. 37(4) (2013) 2552–2561.
International Journal of Nonlinear Analysis and Applications
Volume 13, Issue 1
March 2022
Pages 3073-3083
  • Receive Date: 09 September 2021
  • Revise Date: 14 October 2021
  • Accept Date: 19 December 2021