The finite dam II

Summary

A weir of capacity K is considered in which the water inflow is a process with stationary independent increments. Unless the weir is empty, there is a continuous release of water at unit rate; if K is finite the weir may become full in which case the excess water overflows instantaneously. A weir for which K is infinite will be referred to as infinite dam. For the latter the transient behaviour is well known if the input possesses a second moment (cf. e.g., Prabhu [7]) and serves as the starting point for the present paper. This result is first extended to yield the Laplace transform (L.T.) of the trivariate Laplace-Stieltjes transform (L.S.T.) of the content v(t) at time t, the input X(t) in (0, t) and the total time d(t) in the interval (0, t) during which the dam is dry. (Incidentally, the last two quantities, for relevant time intervals, will be carried throughout.) Then we use a relation between the latter and the L.S.T. of the expected number of downward level y crossings of the v(t) process established in Roes [9]. Since the dam processes considered are Markov processes, we have therewith the L.S.T. of the renewal function of the renewal process imbedded at level y. From this, one finds the L.S.T.'s of first entrance and taboo first entrance times (for their definition see introduction). Next we calculate the first skip times for the infinite dam from the first entrance times and the L.T. of the L.S.T. of v(t). It is then a routine matter to determine the taboo first skip times. From the (taboo) first entrance and skip times we derive the first entrance times for the finite dam, which in turn lead to the renewal functions of the renewal processes imbedded in the finite dam content process v*(t) and hence to the transient behaviour of the finite dam.

The advantage of the present approach over the one given in Roes [8] is that it is entirely probabilistic and avoids involved analytic arguments. As a result, the question of uniqueness of the solution does not arise, while more insight is obtained in the structure. The L.S.T. of several first entrance times and first skip times have been derived by Cohen [2] for compound Poisson input.