Discrete-time storage models with negative binomial inflow

The present paper studies a discrete-time storage process with discrete states. This model has inflow which is defined as independent random variables with a common negative binomial distribution and has a certain outflow discipline. Reversibility and quasi-reversibility for the process are investigated and the reversible measure is given. Thus, under a certain condition, it is shown that the process has time-reversibility with the stationary distribution constructed by the reversible measure. Also dynamic reversibility for the process is shown. As an application of the present results the paper considers an inventory model with a backlog for orders from substations. The relationship between the Lindley process and this model is discussed. Moreover, the paper deals with tandem storage models of an open or a closed network whose each node has the outflow discipline of the certain form. For each model, the invariant measure of the product form is obtained.