Mk'roelectrun. Reliab., VoL 20, pp. 875-880. 0026-2714/80/1201 --0875502.00/0 Pergamon Pre~ Ltd, 1980. Printed in Great Britain MULTI,STATE HOMOGENEOUS MARKov MODELS IN RELIABILITYANALYSIS ANDREA BOBBIO, AMeDEO PREMOLI and CLiniC SARACCO Gruppo Sistemi e Circuiti, Istituto Elettrotecnico Nazionale Galileo Ferrm'is, I-- 10135Torino, Italy (Received for publication 12 December 1979) A~tract--In the standard Markov technique applied to reliability analysis, components are characterized by two gates: an up state and a down state. The present paper explores the possibility of studying system reliability, by modelling each component with a multi-state ~ u s o M a r k o v med¢l (MHMM). It is shown that this approach is of value both in approximating non-exponential probability distributions and in helping to build up suitable models for phys/cal processes. Examples are presented which illustrate how thcmulti-state technique fits many practical situations. Finally some open problems on this topieare suggested. 1. INTRODUCTION The standard theory of homogeneous Markov processes applied to system reliability analysis assumes that each component is characterized by two states: the up state and the down state. In this approach, the transition rates are time-independent, that i~ failure and repair times are extmnefitially distributed. However, .many physical systems do not follow this simple pattern. In this case, one of the methods proposed in the literaturecousists m representing the transition of each component from its working to its failed coadition (or vice versa) by a multi-state homo- 8eaeous Markov model (MHMM). This technique is very attractive since it enables.to approximate, as closely as desired, non-Markov processes by homogeneous Markov processes, but it has received, in the authors' opinion, little attention. In particular, in the literature on this topic distinction is made between multi-state model [!-6] (as, for instance, in the case of components with more than one failure mode) and stage device [7-9] (when the transition from the up to the down state is non- exponential). This point of view is revised in this paper, where it is shown that both techniques are particular cases of a general philosophy, consisting in modelh'ng components (or systems) by more than two states. The present paper deals with homogeneous models (in which the transition rates are time-independent) since it is the most interesting case for practical numerical applications, however some particular examples of multi-state non-homogeneous models have already appeared in the literature [4, 10]. Section 2 is devoted to outline the general criteria for the application of the method to system reliability analysis. In Section 3 the physical meaning of the multi- state models is explored, and it is shown how the use of this approach may help to formulate a more adequate model for the component. Finally, Section 4 indicates some open problems concerning non series- parallel MHMMs, and the determination of the model parameters. 7- APPLICATION OF THE MULTI-STA'U~ TECHNIQUE TOSYSTEM RELIABILITY ANALYS/S Some examples of application of the multi-state technique to the analysis of system.~eliability have been reported [11, 12]. In this sedtion.~ problem is tackled in a more syst~.matic way and general criteria are presented. Let a system be composed of N componeats and let the ith component be r~esented by a MHMM of M~ states (i = 1, 2 ..... N). The number of mutually excltisive states Of the whole system Ms m given by IV" -Ms = H M,. (1) i=l This formula obviously reduces to the weft-known formula Ms= 2 ~, if each component is dc~l'ibed by only two state~ The set ~Ahe Ms states may be divided into two mutually exclusive subsets M~r and Ms: Ms = My u My, (2) which represent the up and down states of the.system, respectively.This decomposition depends both onthe configuration of the system and on the configuration of the MHMM of each component. Under the above assumptions, the process is unequivocally characterized by a time-independent Ms x Ms transition matrix, and the the0ry of homogeneous Markov processes can be di/ectly applied. In particular, the reliability R(t) of the system (or its availability A(t) in the presence of repair) is given by R(t) = ie~u PAt), (3) whemPj(t) is the probability Of being in thejth state at time t. Fm't hermore, Tollowing this pattern, an interest- ing procedure for the computation of the mean time between fiiilures in repa~able systems has been presented [13]. For the s~e Of clarity, |et-'us illustrate the previous concepts on a two-component system. 875