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Applied probability meets Bessel, Hermite, Kummer, Tricomi, Wiener & Hopf (and also Ornstein & Uhlenbeck)

Banff International Research Station for Mathematical Innovation and Discovery

We discuss a few applied probability models: a blood bank model, an insurance risk model and a queueing/inventory model. In each case, the process under study {X(t), t ≥ 0} has state space (−∞, ∞). In the blood bank case, X(t) indicates the amount of blood present at time t, if there is a positive amount present; otherwise, −X(t) denotes the total amount that is being demanded. In the insurance risk case, X(t) indicates the capital at time t, if there is a positive amount present; otherwise, −X(t) denotes the shortage. It is agreed that ruin does not immediately occur when X(t) becomes negative, but the process ends (bankruptcy) according to a bankruptcy rate function ω(−X(t)) when X(t) < 0. In the queueing/inventory model, X(t) indicates the amount of work present at time t, if there is a positive amount present; if no work is present, the tireless server keeps working, building up an inventory, and −X(t) then denotes the inventory level. The inventory is removed at a rate ω(−X(t)). Under Poisson assumptions for the various arrival processes (of blood donations and blood requirements; of claims; of service requirements) and ergodicity conditions we try to determine the steady-state distributions of the X(t) processes; in the insurance risk model, we determine the bankruptcy probability. This appears to involve various special functions, like those of Bessel, Hermite, Kummer and Tricomi.

Mathematics; Probability theory and stochastic processes; Statistics; Operation research

video/mp4

Author affiliation: Eindhoven University of Technology

2016-01-04

Attribution-NonCommercial-NoDerivs 2.5 Canada

http://hdl.handle.net/2429/56152

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/2.5/ca/