Almost Sure Comparisons of Renewal Processes and Poisson Processes, with Application to Reliability Theory

Abstract
If the interarrival times of a renewal process {Si, i = 0, 1, 2…} have a failure rate function which is bounded away from 0 and ∞, then it is possible to construct (nonhomogeneous) Poisson processes {Ti0, i = 0, 1, 2, …} and {Ti1, i = 0, 1, 2, …} on the same probability space with {Si, i = 0, 1, 2, …} such that {T00, T10, T20, …} ⊂ {S0, S1, S2, …} ⊂ {T01, T11, T21, …} almost surely. This has applications to the reliability theory of maintained systems. If the components of a maintained coherent system have exponential lifetimes and NWU repair times, then an initially perfect system will have a NBU distribution of time until first system failure. Furthermore, transient availability is greater than steady-state availability.

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