  This note aims to verify a Laplace transform pair, previously published
without proof, concerning the expected stock-out that may occur in a
production-inventory systems when demand is Poisson, and the time horizon is
finite. Stock-out, or "backlog", is understood as the non-negative difference
between cumulative demand, a stochastic variable, and production over a finite
time interval. The hoped for outcome is greater clarity on how the Laplace
transform pair of interest might be arrived at, and a possible improvement in
terms of accuracy on the original result.
