Popularity of wireless access technologies manifests itself in the form of ever increasing penetration of wireless local area networks. For example, it is estimated that household penetration of WiFi networks is 61% in the USA, 73% in the UK, and 80% in South Korea as of 2011[1].
This trend has a number of profound implications, particularly for densely populated urban areas. First, as more WiFi networks that are in close physical proximity share a common spectrum, the amount of spectrum per network drops due to interference. This effect cannot be controlled in unlicensed bands, but its extent may be bounded by estimating the maximum number of neighboring networks that is possible in view of the limited WiFi transmission range.
Second, and more subtly, dense deployment of WiFi networks leads to large systems of weakly interacting networks. That is, while each network contends with its immediate neighbors to access the spectrum, it also interacts obliviously with further away networks through intermediaries that form chains of neighbors. In that case, whether or not the performance of individual networks depend on the size of the alluded system is of critical importance: Because while this latter effect cannot be controlled either, in contrast to the former effect, the system size cannot easily be estimated and tends to be very large in metropolitan areas. Hence, if performance of individual networks indeed degrades persistently with the aggregate system size, then the resulting operating regime would essentially be practically unacceptable under dense deployment of such networks.
The objective of this article is to investigate key parameters that delineate practically relevant regimes of dense spectrum usage. Our focus is on delaysensitive applications and random spectrum access with carrier sensing (CSMA). Specifically, we seek succinct conditions that predict excessive dependence between channel access delay and system size. Our ultimate interest is in understanding the relationship among throughputs, access delays, and system size; and thereby in identifying throughput levels that entail admissible access delay regardless of the system size.
Using a Markov model, it was recently shown that randomized CSMA is throughputoptimal. That is, if a collection of pernetwork throughputs in a given system topology can be attained by some transmission scheduling algorithm, then it can also be attained by a randomized CSMA algorithm that operates with appropriate parameters[2, 3]. Such feasible throughputs are coined the throughput region for that system topology. It was also observed that in certain parts of the throughput region CSMA displays shortterm unfairness[4]: Namely, theoretically computed throughputs emerge as time averages only if such averages are computed over longtime intervals. Over shorttime intervals, however, one constellation of networks in the system tends to enjoy virtually unobstructed channel access whereas the remaining networks starve. Hence, in the short term, channel access is unfair among constituent networks of the system. Although different constellations take effect in the long term interval, this operational regime leads to high temporal variation in access delay. When it exists, this variation becomes more pronounced with increasing system size[5].
This phenomenon is related to the mixing time of the underlying system dynamics, and in turn to the concept of phase transitions. In statistical physics, phase transition refers to the existence of multiple equilibrium distributions in a graphical model of infinite size. In a finite, prelimit graphical model, a phase transition typically manifests itself in the form of a unique equilibrium distribution that has multimodal nature. That is, most of the probability measure is concentrated around several quasistable states. Transitions between such states become rare as the system size increases, leading to multiple distinct equilibrium distributions in the limit.
Alternatively, such shortterm behavior is suggested by existence of longrange dependence in a graphical system model. In more specific terms, if states of distant nodes in the graph are strongly correlated (either negatively or positively), then such correlation is indicative of the constellations that take effect over shorttime horizons.
Our main contributions are as follows:

We claim that the shortterm fairness among the interacting wireless transmitters is affected by the degree of the conflict graph of these transmitters if the conflict graph is a random regular graph where each vertex has the same number of neighbors. A denser deployment results in an increase in the number of contending neighbors of a network and our results suggest that the practically useful portion of the throughput region reduces as the number of neighboring networks increases.

We demonstrate the implications of our study on a practical citywide WiFi deployment scenario. Our results indicate that shortterm fairness has to be sacrificed to improve coverage in such a system. To improve coverage, the density of the deployment has to be increased which causes the nodal degree of the system to increase. This in turn reduces shortterm fairness.

We discuss if there is a reduction in the performance of interacting networks as the system size increases. Our results suggest that there is a weak dependence on the system size if the density of deployment is kept unchanged and the deployment has a random regular conflict graph. On the other hand, the performance of networks with a grid conflict graph may severely degrade with system size if all networks operate at high throughputs.

We highlight the results from the statistical physics and theoretical computer science literatures on the longrange dependence in physical systems and identify a relationship between CSMA systems and physical systems. Despite the discrepancies between the physical models and the practical networking scenarios, we point out similarities between the shortterm fair capacity region and the phase transition thresholds of the physical models.
The remainder of this article is organized as follows: Section 2. surveys related work and Section 3. describes the system model. We explain the shortterm fairness metrics that we use in Section 4. A mathematical analysis of the shortterm fairness of the tree topology is given in Section 5. Section 6. presents a simulationbased analysis of the tree, grid, and random topologies. Section 7. illustrates the tradeoff between shortterm fair capacity and coverage for a practical WiFi deployment scenario. Several observations on the relationship between the phase transitions of the hardcore model and the CSMA network are presented in Section 8. A summary and discussion of results are given in Section 9.