Probability Seminar
Thursday, September 15, 2016, 4:15pm, UNC, 125 Hanes Hall
Haya Kaspi
An Infinite-Dimensional Skorohod Map and Continuous Parameter Priorities

Abstract:

(Joint work with Rami Atar, Anup Biswas and Kavita Ramanan) The Skorokhod map on the half-line has proved to be a useful tool for studying processes with non-negativity constraints. In this lecture I will introduce a measure-valued analog of this map that transforms each element of a certain class of càdlàg paths that take values in the space of signed measures on the positive half line to a càdlàg path that takes values in the space of non-negative measures on that space. This is done in a way that for each point x > 0, and a signed measure valued process (m(t)) the path t to m(t)[0, x] is transformed via the classical Skorokhod map on the half-line, and the regulating functions for different x > 0 are coupled. We show that the map provides a convenient tool for studying queueing systems in which tasks are prioritized according to a continuous parameter. Three such well known models are the EDF-earliest-deadline-first, the SJF-shortest-job-first and the SRPT-shortest-remaining-processing-time scheduling policies. Concentrating in this talk on the EDF, I will show how the map provides a framework within which one forms fluid model equations, proves uniqueness of solutions to these equations and establishes convergence of scaled state processes to the fluid model. In particular, for this model, our approach leads to new convergence results in time-inhomogeneous settings, which appear to fall outside the scope of existing approaches.

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