On a generalized finite-capacity storage model

This paper considers a finite-capacity storage model defined on a Markov chain { X n ; n = 0, 1, ·· ·}, having state space J ⊆ {1, 2, ·· ·}. If X n = j , then there is a random ‘input' V n ( j ) (a negative input implying a demand) of ‘type' j , having a distribution function F j (·). We assume that { V n ( j )} is an i.i.d. sequence of random variables, taken to be independent of { X n } and of { V n ( k )}, for k ≠ j. Here, the random variables V n ( j ) represent instantaneous ‘inputs' of type j for our storage model. Within this framework, we establish certain limit distributions for the joint processes ( Z n , X n ) and ( Z n , Q n , L n ), where Z n (defined in (1.2)) is the level of storage at time n, Q n (defined in (1.3)) is the cumulative overflow at time n , and L n (defined in (1.4)) is the cumulative demand lost due to shortage of supply up to time n. In addition, an expression for the time-dependent distribution of ( Z n , X n ) is obtained.