The Annals of Applied Probability

Continuity of a queueing integral representation in the M1 topology

Guodong Pang and Ward Whitt

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Abstract

We establish continuity of the integral representation y(t)=x(t)+0th(y(s)) ds, t≥0, mapping a function x into a function y when the underlying function space D is endowed with the Skorohod M1 topology. We apply this integral representation with the continuous mapping theorem to establish heavy-traffic stochastic-process limits for many-server queueing models when the limit process has jumps unmatched in the converging processes as can occur with bursty arrival processes or service interruptions. The proof of M1-continuity is based on a new characterization of the M1 convergence, in which the time portions of the parametric representations are absolutely continuous with respect to Lebesgue measure, and the derivatives are uniformly bounded and converge in L1.

Article information

Source
Ann. Appl. Probab., Volume 20, Number 1 (2010), 214-237.

Dates
First available in Project Euclid: 8 January 2010

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1262962322

Digital Object Identifier
doi:10.1214/09-AAP611

Mathematical Reviews number (MathSciNet)
MR2582647

Zentralblatt MATH identifier
1186.60098

Subjects
Primary: 60F17: Functional limit theorems; invariance principles 60K25: Queueing theory [See also 68M20, 90B22]
Secondary: 90B22: Queues and service [See also 60K25, 68M20]

Keywords
Many-server queues heavy-traffic limits Skorohod M_1 topology continuous mapping theorem bursty arrival processes

Citation

Pang, Guodong; Whitt, Ward. Continuity of a queueing integral representation in the M 1 topology. Ann. Appl. Probab. 20 (2010), no. 1, 214--237. doi:10.1214/09-AAP611. https://projecteuclid.org/euclid.aoap/1262962322


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References

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