Distribution of the lifetime of consecutive k-within-m-out-of-n:F systems

The distribution of the lifetime (MTTF) of any consecutive k-within-m-out-of-n:F system, with independent exponentially distributed component lifetimes, is shown to be a convex combination of the distributions (MTTFs) of several convolutions of independent random variables, where each convolution represents a distinct path in the evolution of the system's history, and where in each convolution all but the last random variable is exponential. The last random variable in each convolution is either a zero lifetime or the lifetime of several disjoint consecutive k/sub i/ within m/sub i/-out-of-n:F systems in series with each k/sub i/>