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Space filling and depletion

Part of: Equilibrium statistical mechanics Special processes Markov processes

Published online by Cambridge University Press:  14 July 2016

Yuliy Baryshnikov ,
E. G. Coffman Jr  and
Predrag Jelenković
Yuliy Baryshnikov*
Affiliation:
Bell Laboratories
E. G. Coffman Jr*
Affiliation:
Columbia University
Predrag Jelenković*
Affiliation:
Columbia University
*
∗ Postal address: Bell Laboratories, 2C-261, 600 Mountain Avenue, Murray Hill, NJ 07974, USA
∗∗ Postal address: Department of Electrical Engineering, Columbia University, 1312 Seeley W. Mudd, Mail Code 4712, 500 West 120th Street, New York, NY 10027, USA
∗∗ Postal address: Department of Electrical Engineering, Columbia University, 1312 Seeley W. Mudd, Mail Code 4712, 500 West 120th Street, New York, NY 10027, USA
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Abstract

For a given k ≥ 1, subintervals of a given interval [0, X] arrive at random and are accepted (allocated) so long as they overlap fewer than k subintervals already accepted. Subintervals not accepted are cleared, while accepted subintervals remain allocated for random retention times before they are released and made available to subsequent arrivals. Thus, the system operates as a generalized many-server queue under a loss protocol. We study a discretized version of this model that appears in reference theories for a number of applications, including communication networks, surface adsorption-desorption processes, and reservation systems. Our primary interest is in steady-state estimates of the vacant space, i.e. the total length of available subintervals kX - ∑ℓ i , where the ℓ i are the lengths of the subintervals currently allocated. We obtain explicit results for k = 1 and for general k with all subinterval lengths equal to 2, the classical dimer case of chemical applications. Our focus is on the asymptotic regime of large retention times.

Keywords

Markov chainmany-server queuedimer packing

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 82B41: Random walks, random surfaces, lattice animals, etc.
Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 3 , September 2004 , pp. 691 - 702
Copyright
Copyright © Applied Probability Trust 2004 

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Footnotes

Supported by DIMACS.

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