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This note aims to verify a Laplace transform pair, previously publishedwithout proof, concerning the expected stock-out that may occur in aproduction-inventory systems when demand is Poisson, and the time horizon isfinite. Stock-out, or "backlog", is understood as the non-negative differencebetween cumulative demand, a stochastic variable, and production over a finitetime interval. The hoped for outcome is greater clarity on how the Laplacetransform pair of interest might be arrived at, and a possible improvement interms of accuracy on the original result.

Verifying the inverse Laplace transform of total expected stock-outs when demand is Poisson