Stochastics and Statistics Reliability function of a class of time-dependent systems with standby redundancy Amir Azaron a, * , Hideki Katagiri a , Masatoshi Sakawa a , Mohammad Modarres b a Department of Artificial Complex Systems Engineering, Graduate School of Engineering, Hiroshima University, Kagamiyama 1-4-1, Higashi-Hiroshima, Hiroshima 739-8527, Japan b Department of Industrial Engineering, Sharif University of Technology, Azadi Street, Tehran, Iran Received 13 January 2003; accepted 28 October 2003 Available online 3 March 2004 Abstract By applying shortest path analysis in stochastic networks, we introduce a new approach to obtain the reliability function of time-dependent systems with standby redundancy. We assume that not all elements of the system are set to function from the beginning. Upon the failure of each element of the active path in the reliability graph, the system switches to the next path. Then, the corresponding elements are activated and consequently the connection between the input and the output is established. It is also assumed each element exhibits a constant hazard rate and its lifetime is a random variable with exponential distribution. To evaluate the system reliability, we construct a directed stochastic network called E-network, in which each path corresponds with a minimal cut of the reliability graph. We also prove that the system failure function is equal to the density function of the shortest path of E-network. The shortest path distribution of this new constructed network is determined analytically using continuous-time Markov processes. Ó 2004 Elsevier B.V. All rights reserved. Keywords: Reliability; Markov processes; Graph theory 1. Introduction Many researchers in the second half of the last century have investigated system reliability evalu- ation and have developed a variety of methods in this regard (see Barlow and Proschan [2] or Grosh [9] for details). Yet, the existing analytical methods are usually constructed on the basis of some assumptions, which are quite restrictive and are not capable of analyzing all real systems. Thus, to calculate the systems reliability with special struc- tures, it is still necessary to design new methods. In this paper, a new approach is introduced to determine the reliability function of time-depen- dent systems with standby redundancy. The existing methods for time-dependent systems are * Corresponding author Tel.: +81-800-7905/5040; fax: +81- 800-6544. E-mail address: azaron@msl.sys.hiroshima-u.ac.jp (A. Aza- ron). 0377-2217/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2003.10.044 European Journal of Operational Research 164 (2005) 378–386 www.elsevier.com/locate/dsw