The Stationary Distribution and First Exit Probabilities of a Storage Process with General Release Rule

Source abstract

Consider a storage process X = {X(t), t ≥ 0} with compound Poisson input and a (state-dependent) release rule r(·) which is arbitrary except for the requirement that state zero be reachable in finite time from any positive starting state. We show that there exists a stationary distribution for X if and only if there is a limiting distribution independent of the initial state, in which case the stationary distribution is unique and coincides with the limiting distribution. A necessary and sufficient condition for the existence of a stationary distribution, as well as a general solution for the distribution when it exists, is given. We also give a general formula for U(x), the probability that level b is exceeded before level a is reached, starting from state x ∈ (a, b]. Both the stationary distribution and U(x) are expressed in terms of a certain positive kernel.