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Sensor allocation problems on the real line

Part of: Markov processes Computer system organization

Published online by Cambridge University Press:  24 October 2016

Evangelos Kranakis  and
Gennady Shaikhet
Evangelos Kranakis*
Affiliation:
Carleton University
Gennady Shaikhet*
Affiliation:
Carleton University
*
*Postal address: School of Computer Science, Carleton University, 1125 Colonel By Drive, Ottawa, ON, K1S 5B6, Canada. Email address: [email protected]
** Postal address: School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, ON, H1S 5B6, Canada. Email address: [email protected]
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Abstract

A large number n of sensors (finite connected intervals) are placed randomly on the real line so that the distances between the consecutive midpoints are independent random variables with expectation inversely proportional to n. In this work we address two fundamental sensor allocation problems. The interference problem tries to reallocate the sensors from their initial positions to eliminate overlaps. The coverage problem, on the other hand, allows overlaps, but tries to eliminate uncovered spaces between the originally placed sensors. Both problems seek to minimize the total sensor movement while reaching their respective goals. Using tools from queueing theory, Skorokhod reflections, and weak convergence, we investigate asymptotic behaviour of optimal costs as n increases to ∞. The introduced methodology is then used to address a more complicated, modified coverage problem, in which the overlaps between any two sensors can not exceed a certain parameter.

Keywords

Sensor allocationqueueing theorypotential outflowreflected random walkSkorokhod mapweak convergence

MSC classification

Primary: 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
Secondary: 68M20: Performance evaluation; queueing; scheduling
Type
Research Papers
Information
Journal of Applied Probability , Volume 53 , Issue 3 , September 2016 , pp. 667 - 687
Copyright
Copyright © Applied Probability Trust 2016 

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