This paper addresses the following question, which is of interest in the design and deployment of a multiuser decentralized network. Given a total system bandwidth of W Hz and a fixed data rate constraint of R bps for each transmission, how many frequency slots N of size W N should the band be partitioned into to maximize the number of simultaneous transmissions in the network? In an interference-limited ad-hoc network, dividing the available spectrum results in two competing effects: on the positive side, it reduces the number of users on each band and therefore decreases the interference level which leads to an increased SINR, while on the negative side the SINR requirement for each transmission is increased because the same information rate must be achieved over a smaller bandwidth. Exploring this tradeoff between bandwidth and SINR and determining the optimum value of N in terms of the system parameters is the focus of the paper. Using stochastic geometry, we analytically derive the optimal SINR threshold (which directly corresponds to the optimal spectral efficiency) on this tradeoff curve and show that it is a function of only the path loss exponent. Furthermore, the optimal SINR point lies between the low-SINR (power-limited) and high-SINR (bandwidth-limited) regimes. In order to operate at this optimal point, the number of frequency bands (i.e., the reuse factor) should be increased until the threshold SINR, which is an increasing function of the reuse factor, is equal to the optimal value.
The transmission capacity framework introduced in @cite_3 is used to quantify the throughput of such a network, since this metric captures notions of spatial density, data rate, and outage probability, and is more amenable to analysis than the more popular transport capacity @cite_5 . Using tools from stochastic geometry @cite_6 , the distribution of interference from other concurrent transmissions at a reference receiving node The randomness in interference is only due to the random positions of the interfering nodes and fading. is characterized as a function of the spatial density of transmitters, the path-loss exponent, and possibly the fading distribution. The distribution of SINR at the receiving node can then be computed, and an outage occurs whenever the SINR falls below some threshold @math . The outage probability is clearly an increasing function of the density of transmissions, and the transmission capacity is defined to be the maximum density of successful transmissions such that the outage probability is no larger than some prescribed constant @math .
@cite_5 @cite_6 @cite_3
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In CSMA CA-based, multi-hop, multi-rate wireless networks, spatial reuse can be increased by tuning the carrier-sensing threshold (Tcs) to reduce the carrier sense range (dcs). While reducing dcs enables more concurrent transmissions, the transmission quality suffers from the increased accumulative interference contributed by concurrent transmissions outside dcs. As a result, the data rate at which the transmission can sustain may decrease. How to balance the interplay of spatial reuse and transmission quality (and hence the sustainable data rate) so as to achieve high network capacity is thus an important issue. In this paper, we investigate this issue by extending Cali's model and devising an analytical model that characterizes the transmission activities as governed by IEEE 802.11 DCF in a single-channel, multi-rate, multi-hop wireless network. The systems throughput is derived as a function of Tcs, SINR, beta, and other PHY MAC systems parameters. We incorporate the effect of varying the degree of spatial reuse by tuning the Tcs. Based on the physical radio propagation model, we theoretically estimate the potential accumulated interference contributed by concurrent transmissions and the corresponding SINR. For a given SINR value, we then determine an appropriate data rate at which a transmission can sustain. To the best of our knowledge, this is perhaps the first effort that considers tuning of PHY characteristics (transmit power and data rates) and MAC parameters (contention backoff timer) jointly in an unified framework in order to optimize the overall network throughput. Analytical results indicate that the systems throughput is not a monotonically increasing decreasing function of Tcs, but instead exhibits transitional points where several possible choices of Tcs can be made. In addition, the network capacity can be further improved by choosing the backoff timer values appropriately. This paper surveys and unifies a number of recent contributions that have collectively developed a metric for decentralized wireless network analysis known as transmission capacity. Although it is notoriously difficult to derive general end-to-end capacity results for multi-terminal or adhoc networks, the transmission capacity (TC) framework allows for quantification of achievable single-hop rates by focusing on a simplified physical MAC-layer model. By using stochastic geometry to quantify the multi-user interference in the network, the relationship between the optimal spatial density and success probability of transmissions in the network can be determined, and expressed-often fairly simply-in terms of the key network parameters. The basic model and analytical tools are first discussed and applied to a simple network with path loss only and we present tight upper and lower bounds on transmission capacity (via lower and upper bounds on outage probability). We then introduce random channels (fading shadowing) and give TC and outage approximations for an arbitrary channel distribution, as well as exact results for the special cases of Rayleigh and Nakagami fading. We then apply these results to show how TC can be used to better understand scheduling, power control, and the deployment of multiple antennas in a decentralized network. The paper closes by discussing shortcomings in the model as well as future research directions. Transmission capacity (TC) is a performance metric for wireless networks that measures the spatial intensity of successful transmissions per unit area, subject to a constraint on the permissible outage probability (where outage occurs when the signal to interference plus noise ratio (SINR) at a receiver is below a threshold). This volume gives a unified treatment of the TC framework that has been developed by the authors and their collaborators over the past decade. The mathematical framework underlying the analysis (reviewed in Section 2) is stochastic geometry: Poisson point processes model the locations of interferers, and (stable) shot noise processes represent the aggregate interference seen at a receiver. Section 3 presents TC results (exact, asymptotic, and bounds) on a simple model in order to illustrate a key strength of the framework: analytical tractability yields explicit performance dependence upon key model parameters. Section 4 presents enhancements to this basic model — channel fading, variable link distances (VLD), and multihop. Section 5 presents four network design case studies well-suited to TC: (i) spectrum management, (ii) interference cancellation, (iii) signal threshold transmission scheduling, and (iv) power control. Section 6 studies the TC when nodes have multiple antennas, which provides a contrast vs. classical results that ignore interference. We study a slotted version of the Aloha Medium Access (MAC) protocol in a Mobile Ad-hoc Network (MANET). Our model features transmitters randomly located in the Euclidean plane, according to a Poisson point process and a set of receivers representing the next-hop from every transmitter. We concentrate on the so-called outage scenario, where a successful transmission requires a Signal-to-Interference-and-Noise (SINR) larger than some threshold. We analyze the local delays in such a network, namely the number of times slots required for nodes to transmit a packet to their prescribed next-hop receivers. The analysis depends very much on the receiver scenario and on the variability of the fading. In most cases, each node has finite-mean geometric random delay and thus a positive next hop throughput. However, the spatial (or large population) averaging of these individual finite mean-delays leads to infinite values in several practical cases, including the Rayleigh fading and positive thermal noise case. In some cases it exhibits an interesting phase transition phenomenon where the spatial average is finite when certain model parameters (receiver distance, thermal noise, Aloha medium access probability) are below a threshold and infinite above. To the best of our knowledge, this phenomenon, which we propose to call the wireless contention phase transition, has not been discussed in the literature. We comment on the relationships between the above facts and the heavy tails found in the so-called "RESTART" algorithm. We argue that the spatial average of the mean local delays is infinite primarily because of the outage logic, where one transmits full packets at time slots when the receiver is covered at the required SINR and where one wastes all the other time slots. This results in the "RESTART" mechanism, which in turn explains why we have infinite spatial average. Adaptive coding offers another nice way of breaking the outage RESTART logic. We show examples where the average delays are finite in the adaptive coding case, whereas they are infinite in the outage case.