A Unified Treatment of Queueing and Storage Problems — Elementary Probability Distributions

Abstract

The level of a storage system (a reservoir, an inventory system, etc) depends on its input and its output (or release). In a simple storage model we have input from one source and only one form of output; we may have complex systems in which we have inputs of various types from various sources and the outputs are also of various forms going into various systems. We shall consider the simple system: if Z_t be the level at time t (which is for the time being taken to be discrete) before an input X_t goes into it and Y_t the release at the end of the time period (t,t+l), then we get

$${Z_{{t + 1}}} = {[{Z_{t}} + {X_{t}} - {Y_{t}}]^{ + }}$$

((1.1))

suggesting that Z_t cannot be negative. If the storage system has a capacity k, we write

$${Z_{{t + 1}}} = {[{Z_{t}} + {X_{t}} - {Y_{t}}]^{ + }}$$

((1.2))

.