How is reliability calculated for non-repairable systems?

For non-repairable systems, it is usually desirable to know what the probability of success (system reliability) is over a given period of time.  The basic steps involved in developing a system reliability model is first to define what is required for mission success and second to define the probability of being in each possible system state (good or failed).  The probability of successful system operation is the sum of the probabilities of being in a good state.  The following is an example of modeling dual redundant CD-ROM drives.  This example assumes perfect switching between drives.

System success definition: One of the two disk drives must work.

The possible states of the individual drives and the composite two drive "system" are shown below in Table 1 (G=Probability Drive is Good, F=Probability Drive Has Failed).

Table 1 - State Probabilities

The probability of successful system operation (one of the two disk drives operating) is equal to the probability of being in either states A, B, or C above.  Therefore:

                  P(Success)    = P(A)   + P(B)      + P(C)                         [ 1 ]

                                            = [G][G]  + [F][G]     + [G][F]

 Where:     [G] = Probability that the relevant item is “functional” and operating

                 [G] = 1 - [F]

                 [F] = Probability that the relevant item is not functional or operating

                 [F] = 1-[G]

 Thus far, no assumptions of failure distribution have been made.  Any probabilistic failure distribution can be substituted for "G" and "F" above; however, the most widely accepted failure distribution for electronics is the exponential (i.e., constant instantaneous failure rate (hazard function), l). The reliability function in the exponential case is: G=R(t)=e^-lt, where l is the failure rate and t is the period of time over which reliability is measured.  The probability of failure is F=1-R(t).  Thus to determine the probability of success assuming the exponential case, e^-lt and 1-e^-lt is substituted for G and F, respectively.  This substitution is shown in Table 2.

Table 2 - State Probabilities (Exponential Distribution)

The probability of the composite drive “system” being in a functionally working state (i.e., states A, B, or C) is:

          R(t) = P(A)      + P(B)             + P(C)

          R(t) = e^-2lt       + e^-lt - e^-2lt      + e^-lt - e^-2lt

          R(t) = 2e^-lt - e^-2lt                                                         [ 2 ]

If  we are interested  in the probability of success at the one year point and we know from historical field data that each drive has an MTBF of about 25,000 hours, then the probability of success is:

         R(t) = 2e^{-(.00004)(8760)} - e^{-2(00004)(8760)} = 0.91

        Where: t = (1 Yr.)(8760 Hr/Yr.) = 8760 Hours

         l = 1/MTBF = 1/25000 = .00004 Failures/Hour

This .91 probability of success is a 30% improvement over the probability of success for a single drive, which is : R(t)=e^-lt=R(8760)=e^{-(.00004)(8760)}=0.70.  However, it should be noted that cost increases for this added reliability in terms of hardware procurement costs and software routines to access the spare drive.