## Institute of Mathematical Statistics Collections

### Double Skorokhod Map and Reneging Real-Time Queues

#### Abstract

An explicit formula for the Skorokhod map Γ0,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,

Γ0,a(ψ)(t)=ψ(t)[(ψ(0)a)+infu[0,t]ψ(u)]sups[0,t][(ψ(s)a)infu[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.

#### Chapter information

Source
Stewart N. Ethier, Jin Feng and Richard H. Stockbridge, eds., Markov Processes and Related Topics: A Festschrift for Thomas G. Kurtz (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2008), 169-193

Dates
First available in Project Euclid: 28 January 2009

https://projecteuclid.org/euclid.imsc/1233152942

Digital Object Identifier
doi:10.1214/074921708000000372

Mathematical Reviews number (MathSciNet)
MR2574231

Zentralblatt MATH identifier
1170.60314

Rights

#### Citation

Kruk, Łukasz; Lehoczky, John; Ramanan, Kavita; Shreve, Steven. Double Skorokhod Map and Reneging Real-Time Queues. Markov Processes and Related Topics: A Festschrift for Thomas G. Kurtz, 169--193, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2008. doi:10.1214/074921708000000372. https://projecteuclid.org/euclid.imsc/1233152942

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