Skip to main content Accessibility help

Sensor allocation problems on the real line

  • Evangelos Kranakis (a1) and Gennady Shaikhet (a2)


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.


Corresponding author

*Postal address: School of Computer Science, Carleton University, 1125 Colonel By Drive, Ottawa, ON, K1S 5B6, Canada. Email address:
** Postal address: School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, ON, H1S 5B6, Canada. Email address:


Hide All
[1] Atar, R.,Mandelbaum, A. and Reiman, M. I. (2004).Scheduling a multiclass queue with many exponential servers: asymptotic optimality in heavy traffic..Ann. Appl. Prob. 14,10841134.
[2] Burkhart, M.,von Rickenbach, P.,Wattenhofer, R. and Zollinger, A.. (2004).Does topology control reduce interference?. In Proceedings of the 5th ACM International Symposium on Mobile Ad Hoc Networking and Computing,ACM,New York, pp.919.
[3] Chen, H. and Yao, D., (YEAR).Fundamentals of Queueing Networks,Springer,New York.
[4] Czyzowicz, J. et al. (2009).On minimizing the maximum sensor movement for barrier coverage of a line segment. In Ad-Hoc, Mobile and Wireless Networks,Springer,Berlin, pp.194212.
[5] Czyzowicz, J. et al. (2010).On minimizing the sum of sensor movements for barrier coverage of a line segment. In Ad-Hoc, Mobile and Wireless Networks,Springer,Berlin, pp.2942.
[6] Huang, C.-F. Tseng, Y.-C. (2005).The coverage problem in a wireless sensor network.Mobile Networks Appl. 10,519528.
[7] Jacod, J. and Shiryaev, A. N. (2003).Limit Theorems for Stochastic Processes, 2nd edn.Springer,Berlin.
[8] Karatzas, I. and Shreve, S. E.(1991).Brownian Motion and Stochastic Calculus,Springer,New York.
[9] Kranakis, E. and Shaikhet, G.(2014).Displacing random sensors to avoid interference. In Computing and Combinatorics,Springer,Heidelberg, pp.501512.
[10] Kranakis, E. et al. (2013).Expected sum and maximum of displacement of random sensors for coverage of a domain. In Proceedings of the Twenty-Fifth Annual ACM Symposium on Parallelism in Algorithms and Architectures,ACM,New York, pp.7382.
[11] Kumar, S.,Lai, T. H. and Arora, A.. (2005).Barrier coverage with wireless sensors. In Proceedings of the 11th Annual International Conference on Mobile Computing and Networking,ACM,New York, pp.284298.
[12] Kuo, H.-H. (2006).Introduction to Stochastic Integration,Springer,New York.
[13] Meyn, S. and Tweedie, R. L.(2009).Markov Chains and Stochastic Stability, 2nd edn.Cambridge University Press.
[14] Moscibroda, T. and Wattenhofer, R.(2005).Minimizing interference in ad hoc and sensor networks. In Proceedings of the 2005 Joint Workshop on Foundations of Mobile Computing,ACM,New York, pp.2433.
[15] Prabhu, N. U. (1998).Stochastic Storage Processes: Queues, Insurance Risk, Dams, and Data Communication (Appl. Math. (New York) 15), 2nd edn,Springer,New York.
[16] Serfozo, R. (2009).Basics of Applied Stochastic Processes,Springer,Berlin.


MSC classification

Sensor allocation problems on the real line

  • Evangelos Kranakis (a1) and Gennady Shaikhet (a2)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.