Probabilistic Assessment of Two-Unit Parallel System with Correlated Lifetime under Inspection Using Regenerative Point Technique

Document Type : Original Article


1 Mathematics, Yusuf Maitama Sule University, Kano, Nigeria

2 Department of General Requirements, Sur College of Applied Sciences, Oman.


This paper deals with the reliability analysis of a complex system consisting ofa two dissimilar unit’ in a parallel configuration with correlated lifetime distribution. The system stops functioning when both units stop working. Both units are inspected periodically as well as being examined before assigning to repair facility. Under consideration of the system have two states: Normal and failed. Regenerative point technique has been used for the mathematical formulation of the model. The system is analyzed using Laplace transforms to solve the mathematical equations. Reliability, Availability, MTSF, Busy Period of repairmen and Cost-effectiveness of the system has been computed. The computed results have been demonstrated by tables and graphs. The repair time of both the units follows the negative exponential distribution with different parameters in a joint probability density function. The inspection times are assumed to follow the general distribution. Some particular cases of the system have also been derived from seeing the practical importance of the model.


[1]     Gupta, R., and Goel, L.R. “Cost-benefit analysis of two-unit parallel standby redundant system with an administrative delay in repair.”International journal of system science, 1999, 21(7), pp. 1369-1379.
[2]     Gupta, R., R, Kishan, and P, Kumar. “A two non-identical unit parallel system with correlated lifetimes.”International journal of system science, 1999, 30(10), 1123-1129.
[3]     Kokolakis, G, andPapa Georgiou, E. A two-unit parallel system supported by (n-2) standbys with general and non-identical lifetimes” International Journal of Systems Science, 2004, Vol 35 (1), pp. 1-12.
[4]     Narender Singh,Dalip Singh,andMeenaxi. “Reliability and Profit Analysis of a Two-unit Non-identical Standby System in Snowstorm Weather Conditions” American Journal of Applied Mathematics and Statistics, 2018, Vol. 6(3), pp. 109-114.
[5]     Shivakar, Rakesh Gupta and Pawan Kumar. “A two-unitparallel system with repair machine failure and repair times,”Gujrat Statistical Review, 2003, 30, 19-24.
[6]     Murari, K., and Maruthachalam, C. “Two parallel unit system with periods of working and rest.”IEEE Transactions on Reliability, 1981, 30, 91.
[7]     Sharma, Deepankar. “Cost analysis availability and M.T.T.F of a complex system with the concept of waiting and human error” Bulletin of Pure and Applied Sciences, 2002, Vol 21E, pp. 553-559.
[8]     Rakesh Gupta, Ramkishan and P.K. Tyagi. “A two-unit system with correlated failure and repairs and random appearance of the repairman." International Journal of system science, 1996, 27(6), 561-566.
[9]     Pandey, D, Tyagi, S. K and Kumar, Vinesh. "Reliability analysis of a series and parallel network using triangular intuitionistic Fuzzy sets." Applications and Applied Mathematics, 2011, Vol 6(11), pp. 1845-1855.
[10]  Munday, V. J, Malik, S. C. "Stochastic analysis of computer system with Software Redundancy and priority to Hardware repair." Indian Journal of Computer Science and Engineering (IJCSE), 2016, Vol. 8(1) pp. 71-80.
[11]  Tiwari, Nidhi, Singh. S. B. "Analysis of two-unit system under marked process incorporating two repairmen with a vacation." Journal of Risk Analysis and Crisis, Response, 2015, Vol 5(4), pp.200-214.
[12]  Lado, A. Kareem, V. V. Singh, Kabiru, H. Ismail and Yusuf, Ibrahim. "Performance and cost assessment of the repairable complex system with two subsystems connected in series configuration." International Journal of Reliability and Applications, 2018, Vol.19(1)2 pp. 27-42.
[13]  Yusuf. Ibrahim. "Reliability modeling of a parallel system with a supporting device and two types of preventive maintenance". Int. J. Operation Research, 2016, Vol 25(3), pp. 269-287.
[14]  Yusuf. I, Yusuf. B, Babagana. M, Sani. B, Lawan. A. and Muhammad. “Some reliability characteristics of a linear consecutive 2-out-of-4 system connected to supporting device for operation”. International Journal of Engineering & Technology, 2018, Vol 7(1), pp.135-139.
[15]  Permila and Malik. S. C. “Analysis of a 2-out-of-2: G system with single cold standby unit with priority to repair and arrival time of the server”. Int. J. Agricult. Stat. Sci, 2016, Vol 11 (1) pp. 27-36.
[16]  V. V. Singh, S. B. Singh, Mangey Ram & C. K. Goel. “Availability, MTTF and cost analysis of a system having two units in a series configuration with the controller." InternationalJournal of System Assurance and Management, 2013, Vol. 4(4), pp. 341-352.
[17]  V. V. Singh and Chiwa Musa Dalah. “Study of reliability measures of a two-unit standby system under the concept of switch failure using copula distribution." American Journal ofComputational and Applied Mathematics, 2014, Vol. 4(4), pp. 118-129.
[18]  V.V. Singh, Hamisu I. Ayagi. “Stochastic analysis of a complex system under preemptive resume repair policy using Gumbel- Hougaard family copula, International Journal of Mathematics in Operation Research, 2018, Vol 12(2), pp. 273- 291.
[19]  V. V. Singh, Hamishu, H. I. Ayagi, “Study of Reliability measures of a system consisting of two subsystems in the series configuration using copula.”Palestine Journal of Mathematics,2017 Vol 6. Special Issue II pp 102-111, 2017.
[20]  Ghasemi, S. H. and Nowak, A. S, Target reliability for bridges with consideration of unlimited limit state, Engineering Structures, 2017, Vol. 152, pp. 226-237.
[21]  Ghasemi, S. H. and Nowak, A. S., Reliability index for non – normal distribution of limit state functions, Structural Engineering and Mechanics, 2017, Vol. 62 (3), pp, 365-372.
[22]  Ghasemi. S. H, Kalantari, H., Abdulahi, S. S and Nowak, A. S. “Fatigue reliability analysis of medial tibial stress syndrome, Material Science and Engineering, 2019, Vol. 99, pp 387-393.
[23]  Ghasemi, S. H. and Nowak, A. S, Reliability analysis of circular tunnels with consideration of strength limit state, Geomechanics Engineering, 2018, Vol 15(3), pp. 879-888.
[24]  Ghasemi, S. H. and Nowak, A. S, Mean maximum value of non- normal distributions for different time periods, International Journal of Reliability and safety, 2016, Vol. 10(2), pp. 99-109.
[25]  Ghasemi, S. H. and Nowak, A. S.,  "Statistical Parameters of In-A-Lane Multiple Truck Presence and a New Procedure to Analyze the Lifetime of Bridges," Journal of Structural Engineering International, Association for Bridge and Structural Engineering IABSE, 2016, Vol. 26, No. 2, pp. 150-159.
[26]  Yanaka, M., Ghasemi, S. H. and Nowak, A. S., Reliability-based and life cycle cost-oriented design recommendations for prestress concrete bridge Girders” Structural concrete, 2016, Vol 18(1), pp 836-847.
[27]  Yusuf. I., and Fagge N.J.  "Evaluation of some reliability characteristics of a single unit system requiring two types of supporting devices for operation." Application and Applied Mathematics, 2017, Vol 12(2), pp.803-812.
[28]  Gahlot, Monika, V. V. Singh H. Ismail Ayagi and C.K. Goel, “Performance assessment of repairable system in the series configuration under different types of failure and repair policies using Copula Linguistics," International Journal of Reliability and Safety, 2018, Vol.12 (4), pp.367-374. 
[29]  Kumar, Praween and Gupta, R. Reliability analysis of a single unit M|G|1 system model with helping unit. Journal of Combinatorics, Information & System Sciences, J. of Comb. Info. & System Sciences, 2007, Vol. 32, No. 1-4, pp. 209-219.
[30]  Barak M.S., Neeraj, and Kumari S.  “Profit analysis of a two-unit cold standby system operating under different weather conditions subject to inspection”. Application and Applied Mathematics, 2018, Vol 13(1), pp.67-83.
[31]  Kumar, Praween and Sirohi, A., Profit analysis of a two-unit cold standby system with the delayed repair of the partially failed unit and better utilization of units. International Journal of Computer Applications, 2015, 117 (1), 41-46.
Volume 2, Issue 1
June 2019
Pages 5-14
  • Receive Date: 22 March 2019
  • Revise Date: 05 May 2019
  • Accept Date: 06 May 2019
  • First Publish Date: 01 June 2019