Double Skorokhod Map and Reneging Real-Time Queues

Abstract

An explicit formula for the Skorokhod map Γ_0,a on [0,a] for a>0 is provided and related to similar formulas in the literature. Specically, it is shown that on the space $\mathcal{D}$[0,∞) of right-continuous functions with left limits taking values in ℝ,

Γ_0,a(ψ)(t)=ψ(t)−[(ψ(0)−a)⁺∧inf_u∈[0,t]ψ(u)]∨sup_s∈[0,t][(ψ(s)−a)∧inf_u∈[s,t]ψ(u)]

is the unique function taking values in [0,a] that is obtained from by minimal “pushing” at the endpoints 0 and a. An application of this result to real-time queues with reneging is outlined.