Laplace-transforms and fast-repair approximations for multiple availability and its generalizations

Stochastic models, describing multiple availability, are analyzed for a system with periods of operation and repair that form an alternating renewal process with exponential times to failure and repair. For the simplest case multiple availability is defined as the probability that the system is available in the interval [0, t) at each moment of demand. Instants of demand form a homogeneous Poisson process. This setting is generalized to considering a possibility of one or more points of unavailability in [0, t) as well as time redundancy. The corresponding integral equations are derived and solved (wherever possible) via the Laplace transform. A fast repair approach is also applied to each case under consideration and simple approximate relations for multiple availability are obtained. The fast repair approximation makes it possible to derive approximate solutions for problems that cannot be solved by the first approach. The accuracies of the fast repair approximations are analyzed. Generalizations to arbitrary failure and repair distributions are also discussed.