Stochastic clearing systems

A stochastic clearing system is characterized by a non-decreasing stochastic input process $\text{{Y(t), t ≧ 0}}$, where Y(t) is the cumulative quantity entering the system in 0, t], and an output mechanism that intermittently and instantaneously clears the system, that is, removes all the quantity currently present. Examples may be found in the theory of queues, inventories, and other stochastic service and storage systems. In this paper we derive an explicit expression for the stationary (in some cases, limiting) distribution of the quantity in the system, under the assumption that the clearing instants are regeneration points and, in particular, first entrance times into sets of the form {y: y>q}. The expression is in terms of the sojourn measure W associated with $\text{{Y(t), t ≧ 0}: W{A} =}\text{E}${time spent in A by Y(t), 0 ≤ t < ∞}. The results are applied to compound input processes and processes with stationary independent increments. In particular, we show that, contrary to a wide-spread belief, the uniform stationary distribution characteristic of deterministic models does not usually carry over to genuinely stochastic models.