Some limit theorems for the general semi-Markov storage model

Abstract

In this paper we treat the general version of the semi-Markov storage model, introduced first by Senturia and Puri: transitions in the state of the system occur at a discrete sequence of time points, described by a two-state semi-Markov process. An input occurs at an instant of transition to state 1 and a demand for release occurs at an instant of transition to state 2. Assuming general distributions for all the variables involved, we show that the dam contents just after the nth input converges properly in distribution as n →∞ under conditions of stability; likewise that after the nth demand. We also show that the demand lost due to shortage of stock, accumulated over instants of demand as well as over time, obeys a strong law and a central limit theorem.

On the limit behavior of a multicompartment storage model with an underlying Markov chain

“…for h(-) an appropriate functional on C k + 1 [0,1]. That h(-) is continuous is clear from (4_6), so we have by the uniform convergence of (YO,n(-), ---, Yk,n(-)) to multivariate Brownian motion (see Billingsley (1968), Donsker (1951)…”

“…Initially, the random mechanism underlying the models was assumed to be independent, but in 1965 Lloyd and Odoom (1965) proposed a model in which a dependent structure was feasible by assuming an underlying Markov chain as part of the random mechanism. The stochastic model was later expanded to having an underlying semi-Markov process, and a specific one-compartment storage model with underlying semi-Markov process was considered by Puri (1978), Puri andSenturia (1972), (1975), Balagopal (1979), Puri and Woolford (1981), and Puri and Tollar (1985), and others.…”

The Renewal Equation for Markov Renewal Processes with Applications to Storage Models.

“…In a simpler form, the model was first proposed as a single compartment model by Senturia and Puri (1973), with subsequent research by Senturia and Puri (1974), Puri and Senturia (1975), Puri (1978), Balagopal (1979), Puri and Woolford (1981), and Puri and Tollar (1985).…”

On the limit behavior of certain quantities in a subcritical storage model

“…The present work deals with a storage model which allows inputs as well as releases to occur in random amounts and at random times according to an underlying semi-Markov process. While the reader may find other types of storage models elsewhere in the literature (see Moran [12], Prabhu [15], Lloyd [10], Ali Khan and Gani [1], for references) the present model is along the lines of Puri and Woolford [17], which itself is a generalization of a model considered previously by Senturia and Puri [19], [20] and Balagopal [3]. A special case of these models can be found in an earlier work (see Puri and Senturia [16]) which relates such models to a live situation arising in biology.…”

“…We shall adopt the terminology of saying that we are in the subcritical, critical or the supercritical case according to whether E 7TU is less than, equal to or greater than 0. In [3], [17], [19], and [20] various authors studied the limit behavior of quantities such as Ztr) and Lit), but only for the critical and 445 supercritical cases. The methods used by these authors did not lend themselves to the study of the joint limit behavior of (Ztr), L(t)) for the subcritical case.…”