General standby systems with component lifetimes following independent and nonidentical phase-type (PH) distributions are presented in a state-space model using state transition block matrices. The model is constructed by identifying a block matrix representing each system state and a block matrix that causes a transition from one system state to another. This general model is applicable to hot, warm, or cold standby and any combination of them in K-out-of-N general standby structures. The resulting model becomes a PH representation of the system lifetime distribution and is thus useful for exact dynamic system reliability analysis. The advantage is that many functional system reliability measures, such as the reliability, hazard, and mean residual life functions, can be obtained by simple matrix algebra. These functions are shown to be useful for determining optimal component ordering. Comparisons with other methods from previous publications are presented.

Original languageEnglish
Article number107615
JournalComputers and Industrial Engineering
Publication statusPublished - Nov 2021


  • Exact analysis
  • K-out-of-N general standby structure
  • Mean residual life function
  • Phase-type distribution
  • System hazard function
  • System reliability function


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