Ageing Orders of Series-Parallel and Parallel-Series Systems with Independent Subsystems Consisting of Dependent Components


1 Department of Mathematics and Statistics, McMaster University, Hamilton, CANADA

2 Department of Statistics, University of Zabol, Sistan and Baluchestan, IRAN

3 Department of Mathematics, University of Zabol, Sistan and Baluchestan, IRAN



In this paper, we consider series-parallel and parallel-series systems with independent subsystems consisting of dependent homogeneous components whose joint lifetimes are modeled by an Archimedean copula. Then, by considering two such systems with different numbers of components within each subsystem, we establish hazard rate and reversed hazard rate orderings between the two system lifetimes, and also discuss how these systems age relative to each other in terms of hazard rate and reversed hazard rate functions.


  1. Billionnet, A. (2008), Redundancy allocation for series-parallel systems using integer linear programming. IEEE Transactions on Reliability, 57, 507-516. [DOI:10.1109/TR.2008.927807]
  2. Coit, D. W., and Smith, A. E. (1996), Reliability optimization of series-parallel systems using a genetic algorithm. IEEE Transactions on Reliability, 45, 254-260. [DOI:10.1109/24.510811]
  3. Ding, W., and Zhang, Y. (2018), Relative ageing of series and parallel systems: Effects of dependence and heterogeneity among components. Operations Research Letters, 46, 219-224. [DOI:10.1016/j.orl.2018.01.005]
  4. El-Neweihi, E., Proschan, F., and Sethuraman, J. (1986), Optimal allocation of components in parallel-series and series-parallel systems. Journal of Applied Probability, 23, 770-777. [DOI:10.2307/3214014]
  5. Fang, L., Balakrishnan, N., and Jin, Q. (2020), Optimal grouping of heterogeneous components in series-parallel and parallel-series systems under Archimedean copula dependence. Journal of Computational and Applied Mathematics, 377, 112916. [DOI:10.1016/]
  6. Kalashnikov, V. V., and Rachev, S. T. (1986), Characterization of queueing models and their stability. Theory and Mathematical Statistics, 2, 37-53.
  7. Kotz, S., Balakrishnan, N., and Johnson, N. L. (2000), Continuous Multivariate Distributions. New York: John Wiley & Sons, 1(2)
  8. Levitin, G., and Amari, S. V. (2009), Optimal load distribution in series-parallel systems. Reliability Engineering & System Safety, 94, 254-260. [DOI:10.1016/j.ress.2008.03.001]
  9. Ling, X., Wei, Y., and Li, P. (2018), On optimal heterogeneous components grouping in series-parallel and parallel-series systems. Probability in the Engineering and Informational Sciences, 33, 564-578. [DOI:10.1017/S0269964818000499]
  10. Muller, A., and Stoyan, D. (2002), Comparison Methods for Stochastic Models and Risks. Hoboken, New Jersey: John Wiley & Sons.
  11. Nelsen, R. B. (2006), An Introduction to Copulas. New York: Springer.
  12. Ramirez-Marquez, J. E., Coit, D. W., and Konak, A. (2004). Redundancy allocation for series-parallel systems using a max-min approach. IIE Transactions, 36, 891-898. [DOI:10.1080/07408170490473097]
  13. Rezaei, M., Gholizadeh, B., and Izadkhah, S. (2015). On relative reversed hazard rate order. Communications
  14. in Statistics-Theory and Methods, 44, 300-308.
  15. Sarhan, A .M., Al-Ruzaiza, A. S., Alwasel, I. A., and El-Gohary, A. I. (2004). Reliability equivalence of a series-parallel system. Applied Mathematics and Computation, 154, 257-277. [DOI:10.1016/S0096-3003(03)00709-4]
  16. Shaked, M., and Shanthikumar, J. G. (2007) Stochastic Orders. New York: Springer.
  17. Sun, M. X., Li, Y. F., and Zio, E. (2019). On the optimal redundancy allocation for multi-state series-parallel systems under epistemic uncertainty. Reliability Engineering & System Safety, 192, 106019. [DOI:10.1016/j.ress.2017.11.025]
Volume 20, Issue 1
June 2021
Pages 83-100
  • Receive Date: 23 July 2022
  • Revise Date: 26 February 2024
  • Accept Date: 23 July 2022