Distributed Computing Column 38: Models for algorithm design in wireless networks

The signal-to-interference & noise ratio (SINR) model is one of the most commonly studied physical (or fading channel) models for wireless networks. We survey some recent studies aiming at achieving a better understanding of the SINR model and its structural properties and developing efficient design algorithms and communication protocols for it. Physical models and SINR Traditional (wired, point-to-point) communication networks can be described satisfactorily using a graph representation. A station s is able to transmit a message to another station s′ if and only if there is a wire connecting the two stations. This condition is independent of the locations, connections and activities of the two stations or of other nearby stations1. Accurately representing a wireless network is considerably harder, since it is nontrivial to decide whether a transmission by a station s is successfully received by another station s′; this may depend on the positioning and activities of s and s′, and on other nearby stations, whose activities might interfere with the transmission and prevent its reception This means that a transmission from s may reach s′ in some settings but fail to reach it under other settings. Moreover, the question of successful reception is more complex, since connections can be of varying quality and capacity. In fact, there are many other relevant factors, such as the presence of physical obstacles, the directions of the antennae at s and s′, the weather, and more. Obtaining an accurate solution taking all of those factors into account involves solving the corresponding Maxwell equations. Since this is usually far too complicated, the common practice is to resort to approaches based on approximation models. For instance, one way to predict the behavior of wireless systems in an urban environment is using a ray tracing model, based on the assumption that radio waves behave according to geometric optics where walls are modeled as reflective mirrors. The uncertainly involved in the dynamics of a radio channel can be modeled using a Markov process. For more information we refer the interested reader to Chapters 2 and 3 of [9]. For the purposes of the current discussion, we follow the approach of ignoring those complicating factors, and assuming a relatively clean abstract setting where the only players are the transmitting and listening ∗Partially supported by a gift from Cisco research center and by the Israel Science Foundation, grant 894/09. excepting certain types of wired local area networks. ACM SIGACT News 74 June 2010 Vol. 41, No. 2 stations, and the antennae are omni-directional. In this setting, the rules governing the reception quality of wireless transmissions can be described schematically by physical or fading channel models. Among those, one of the most commonly studied is the signal-to-interference & noise ratio (SINR) model. In this model, the energy of a signal fades with the distance to the power of the path-loss parameter α. When a station s transmits, the message is successfully received by a listening station s′ if and only if the strength of the signal received by s′, divided by the strength of the interferences from other simultaneous transmissions (plus the background noise N ), exceeds some threshold β. Hence for a collection S = {s1, . , sn} of simultaneously transmitting stations in the plane, it is possible to identify with each station si a reception region H(si) around it, consisting of the points where the transmission of si is received correctly. More precisely, denote by dist(p, q) the Euclidean distance between p and q, and assume that each station si transmits with power Ei. At an arbitrary point p, the transmission of station si is correctly received if Ei · dist(p, si) N + ∑ j 6=iEj · dist(p, sj)−α ≥ β . This formula represents a rather general model concerning the allowed transmission power, referred to as the power control model, in which each station can control the power with which it transmits. A simpler (and weaker) model is the uniform wireless network model, which assumes that all transmissions use the same amount of energy, i.e., Ei = 1 for every i.