On the generalized drift Skorokhod problem in one dimension

We show how to write the solution to the generalized drift Skorokhod problem in one-dimension in terms of the supremum of the solution of a tractable unrestricted integral equation (that is, an integral equation with no boundaries). As an application of our result, we equate the transient distribution of a reflected Ornstein–Uhlenbeck (OU) process to the first hitting time distribution of an OU process (that is not reflected). Then, we use this relationship to approximate the transient distribution of the GI/GI/1 + GI queue in conventional heavy traffic and the M/M/N/N queue in a many-server heavy traffic regime

A new Wiener-Hopf identity for a general class of reflected processes

We derive a new "Wiener-Hopf identity" for a class of preemptive-resume queueing systems, with batch arrivals and catastrophes that, whenever they occur, eliminate multiple customers present in the system. These processes are quite general, as they can be used to approximate Lévy processes, diffusion processes, and certain types of growth-collapse processes: thus, all of the processes mentioned above also satisfy this type of Wiener-Hopf identity. In the Lévy case, this identity simplifies to the well-known Wiener-Hopf factorization. We also show how the ideas can be used to derive transforms for some well-known state-dependent/inhomogeneous birth-death processes and diffusion processes