## Journal of Applied Probability

- J. Appl. Probab.
- Volume 53, Number 3 (2016), 667-687.

### Sensor allocation problems on the real line

Evangelos Kranakis and Gennady Shaikhet

#### 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.

#### Article information

**Source**

J. Appl. Probab., Volume 53, Number 3 (2016), 667-687.

**Dates**

First available in Project Euclid: 13 October 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.jap/1476370769

**Mathematical Reviews number (MathSciNet)**

MR3570087

**Zentralblatt MATH identifier**

1351.90080

**Subjects**

Primary: 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) [See also 90B30, 91D10, 91D35, 91E40]

Secondary: 68M20: Performance evaluation; queueing; scheduling [See also 60K25, 90Bxx]

**Keywords**

Sensor allocation queueing theory potential outflow reflected random walk Skorokhod map weak convergence

#### Citation

Kranakis, Evangelos; Shaikhet, Gennady. Sensor allocation problems on the real line. J. Appl. Probab. 53 (2016), no. 3, 667--687. https://projecteuclid.org/euclid.jap/1476370769