Optimal Sequential Wireless Relay Placement on a Random Lattice Path

Our work is motivated by impromptu (or “as-you-go”) deployment of wireless relay nodes along a path, a need that arises in many situations. In this paper, the path is modeled as starting at the origin (where there is the data sink, e.g., the control centre), and evolving randomly over a lattice in the positive quadrant. A person walks along the path deploying relay nodes as he goes. At each step, the path can, randomly, either continue in the same direction or take a turn, or come to an end, at which point a data source (e.g., a sensor) has to be placed, that will send packets to the data sink. A decision has to be made at each step whether or not to place a wireless relay node. Assuming that the packet generation rate by the source is very low, and simple link-bylink scheduling, we consider the problem of sequential relay placement so as to minimize the expectation of an end-to-end cost metric (a linear combination of the sum of convex hop costs and the number of relays placed). This impromptu relay placement problem is formulated as a total cost Markov decision process. First, we derive the optimal policy in terms of an optimal placement set and IThe research reported in this paper is an extension of the material published in the conference paper [1]. IIThis work was supported by (i) the Department of Science and Technology (India) via the Indo-Brazil Project WINSON, (ii) the Department of Electronics and Information Technology (India) and the National Science Foundation (USA) via an Indo-US collaborative project titled Wireless Sensor Networks for Protecting Wildlife and Humans, and (iii) the J.C. Bose National Fellowship (of the Govt. of India). Email addresses: [email protected] (Abhishek Sinha), [email protected] (Arpan Chattopadhyay), [email protected] (K.P. Naveen), [email protected] (Prasenjit Mondal), [email protected] (Marceau Coupechoux), [email protected] (Anurag Kumar) 1Corresponding author. This work was done while he was a Master of Engineering student in the ECE Department, Indian Institute of Science, Bangalore. 2This work was done while he was a Visiting Scientist in the ECE Department, Indian Institute of Science, Bangalore. Preprint submitted to Elsevier March 20, 2014 Figure 1: A wireless network being deployed as a person steps along a random lattice path. Inverted V: location of the deployment person; path drawn with a solid line: path already covered; circles: deployed relays; path drawn with a thick dashed line: a possible evolution of the remaining path. The source to be placed at the end is also shown as the black rectangle. show that this set is characterized by a boundary (with respect to the position of the last placed relay) beyond which it is optimal to place the next relay. Next, based on a simpler one-step-look-ahead characterization of the optimal policy, we propose an algorithm which is proved to converge to the optimal placement set in a finite number of steps and which is faster than value iteration. We show by simulations that the distance threshold based heuristic, usually assumed in the literature, is close to the optimal, provided that the threshold distance is carefully chosen.