Probability Distributions in Periodic Random Environment and Their Applications

Periodic random environments and mechanisms of their effect on imbedded random variables are discussed. The variables under consideration represent either waiting time until some event occurs or the number of events within a given time interval. Their probability distributions have periodic residual lifetime functions and periodic failure rates. The form of the corresponding cumulative distribution functions is derived. Equivalent representations of these random variables as functions of other, suitably chosen independent random variables are established. Other probability properties such as almost lack of memory, invariance with respect to relevation transform, and preservation of service time distribution on nonreliable servers are additional characterizing features of this class of probability distributions. Nonstationary Poisson processes with periodic failure rates appear to be the closest extension of the homogeneous Poisson process to model the number of events imbedded into random environment of periodic nature. Some possible applications to reliability, queues, environmental studies, and other fields are briefly pointed out.