Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-27T07:34:51.317Z Has data issue: false hasContentIssue false
  • English
  • Français

Space filling and depletion

Part of: Special processes Equilibrium statistical mechanics 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
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Supported by DIMACS.

References

Bertozzi, A., and McKenna, J. (1993). Multidimensional residues, generating functions and their applications to queueing networks. SIAM Rev. 35, 239–268.10.1137/1035045Google Scholar
Coffman, E. G. Jr., Jelenković, P., and Poonen, B. (1999). Reservation probabilities. Adv. Perf. Anal. 2, 129–158.Google Scholar
Coffman, E. G. Jr., Flatto, L., Jelenković, P., and Poonen, B. (1998). Packing random intervals on-line. Algorithmica 22, 448–476.10.1007/PL00009233Google Scholar
Evans, J. W. (1993). Random and cooperative sequential adsorption. Rev. Mod. Phys. 65, 1281–1329.10.1103/RevModPhys.65.1281Google Scholar
Ferrari, P., and Garcia, N. L. (1998). One-dimensional loss networks and conditioned M/G/∞ queues. J. Appl. Prob. 35, 963–975.10.1239/jap/1032438391Google Scholar
Flory, P. J. (1939). Intramolecular reaction between neighboring substituents of vinyl polymers. J. Amer. Chem. Soc. 61, 1518–1521.10.1021/ja01875a053Google Scholar
Kelly, F. P. (1987). One-dimensional circuit-switched networks. Ann. Prob. 15, 1166–1179.10.1214/aop/1176992089Google Scholar
Mackenzie, J. K. (1962). Sequential filling of a line by intervals placed at random and its application to linear adsorption. J. Chem. Phys. 37, 723–728.10.1063/1.1733154Google Scholar
Page, E. S. (1959). The distribution of vacancies on a line. J. R. Statist. Soc. B 21, 364–374.Google Scholar
Pemantle, R., and Wilson, M. C. (2002). Asymptotics of multivariate sequences. I. Smooth points of the singular variety. Combin. Theory Ser. A 97, 129–161.10.1006/jcta.2001.3201Google Scholar
Privman, V., and Nielaba, P. (1992). Diffusional relaxation in dimer deposition. Europhys. Lett. 18, 673.10.1209/0295-5075/18/8/002Google Scholar
Rényi, A. (1958). On a one-dimensional problem concerning random space filling. Magyar Tud. Akak. Mat. Kutató Int. Közl. 3, 109–127 (in Hungarian).Google Scholar
Talbot, J., Tarjus, G., and Viot, P. (2000). Adsorption-desorption model and its application to vibrated granular materials. Phys. Rev. E 61, 5429–5438.10.1103/PhysRevE.61.5429Google Scholar
Talbot, J., Tarjus, G., Van Tassel, P. R., and Viot, P. (2000). From car parking to protein adsorption: an overview of sequential adsorption processes. Colloids Surfaces A 165, 287–324.10.1016/S0927-7757(99)00409-4Google Scholar
Zachary, S., and Ziedins, I. (1999). Loss networks and Markov random fields. J. Appl. Prob. 36, 403–414.10.1239/jap/1032374461Google Scholar
Ziedins, I. (1987). Quasi-stationary distributions and one-dimensional circuit-switched networks. J. Appl. Prob. 24, 965–977.10.2307/3214219Google Scholar