Markov Models Of Dual-Redundant Systems
Many practical systems require numerous sub-functions to be performed in order to support the overall function, and they are designed to be dual-redundant on each sub-function. Failure of the first component of any particular sub-function is annunciated, and repaired at a suitable rate to preclude complete failure. In addition, there is a possibility of a single failure from the "full-up" condition leading to overall functional failure. This type of system with n sub-functions is often represented by a Markov model as shown below.
If we designate the "full up" state with the number 0, the n degraded states with the numbers 1 through n, and the final failure state with the number n+1, then the equations of the system can be written as
For the steady-state condition we have dPj/dt = 0 for all j, so we can solve the central equations to give the values of P1 through Pn as a function of P0
where li,j signifies the rate of transition from state i to state j. Since the sum of all the state probabilities from P0 to Pn+1 is 1, we have
Also, the steady-state flow rate into State n+1 is
The exact failure rate for entering state n+1 is therefore
Naturally this rate is independent of ln+1,0, because the rate is, by definition, a measure of the propensity to enter a particular state for entities that are not presently in that state, which is clearly independent of the rate of leaving that state. Interestingly, if we define lj,j as infinite for each j, none of the state equations are affected, because the infinite "self-transition" flow Pjlj,j is both added to and subtracted from the jth equation, but this enables us to write the equation for the failure rate in the more unified form
This form emphasizes the fact that the overall rate for entering state n+1 is simply the weighted average of the individual transition rates lj,n+1 from each of the states 0, 1, 2, ..., n, with each rate weighted in proportion to the steady-state probability Pj of the respective state.