[Paper Review] A Tight Bound on the Maximum Interference of Random Sensors in the Highway Model

Evangelos Kranakis, Danny Kriz̧anc+3 more|arXiv (Cornell University)|Jul 13, 2010

Mobile Ad Hoc Networks10 references8 citations

TL;DR

This paper analyzes interference in a one-dimensional wireless sensor network (the 'highway model') where n sensors are placed uniformly at random in (0,1). It proves that assigning each sensor a transmission range equal to the maximum distance to its two adjacent neighbors results in an expected maximum interference of Θ(√(ln n)), establishing a tight bound that holds with high probability. This resolves a key open question about interference scaling in random sensor deployments under a natural connectivity-preserving range assignment.

ABSTRACT

Consider $n$ sensors whose positions are represented by $n$ uniform, independent and identically distributed random variables assuming values in the open unit interval $(0,1)$. A natural way to guarantee connectivity in the resulting sensor network is to assign to each sensor as its range, the maximum of the two possible distances to its two neighbors. The interference at a given sensor is defined as the number of sensors that have this sensor within their range. In this paper we prove that the expected maximum interference of the sensors is $Θ(\sqrt{\ln n})$.

Motivation & Objective

• To analyze the interference experienced by sensors in a one-dimensional wireless network with random, uniformly distributed node placements.
• To determine the expected maximum interference when each sensor is assigned a range equal to the maximum distance to its two nearest neighbors, ensuring network connectivity.
• To establish a tight asymptotic bound on the expected maximum interference under this natural range assignment scheme.
• To contrast this random model with worst-case configurations where interference can be as high as Ω(n), highlighting the benefit of random placement.

Proposed method

• Models n sensors as i.i.d. uniform random variables in the open unit interval (0,1).
• Assigns each sensor a transmission range equal to the maximum of its distances to its left and right neighbors, ensuring connectivity.
• Defines interference at a sensor as the number of other sensors whose transmission ranges include it.
• Uses a stochastic process to bound short-range left-interference by analyzing sequences of inter-node distances and their cumulative sums.
• Applies concentration inequalities and exponential tail bounds to show that the probability of high interference drops rapidly with n.
• Symmetrically bounds right-interference and combines both to derive a high-probability upper bound on total interference.

Experimental results

Research questions

• RQ1 What is the expected maximum interference in a random sensor network on a line when each sensor uses the maximum distance to its two neighbors as its transmission range?
• RQ2 How does the interference scale with the number of sensors n in the random highway model compared to worst-case configurations?
• RQ3 Can a tight asymptotic bound be established for the expected maximum interference under this natural range assignment?
• RQ4 What is the probability that any sensor experiences interference exceeding √(c log n) for a given constant c > 0?

Key findings

• The expected maximum interference in the random highway model is Θ(√(ln n)), providing a tight asymptotic bound.
• With high probability, the maximum interference is at most √(c log n) for any constant c > 0, and this bound holds with probability at least 1 - n^(-Ω(c)).
• The analysis shows that the probability of interference exceeding √(c log n) decays exponentially with c, confirming the tightness of the bound.
• The bound holds despite the worst-case interference being Ω(n) in adversarial configurations, demonstrating a significant improvement under random placement.
• The result is robust to both left- and right-interference, with symmetric bounds derived for each direction.
• The proof technique relies on a recursive stochastic process that models interference propagation and uses exponential tail bounds to control rare high-interference events.