We consider the multihop broadcasting problem for @math nodes placed uniformly at random in a disk and investigate the number of hops required to transmit a signal from the central node to all other nodes under three communication models: Unit-Disk-Graph (UDG), Signal-to-Noise-Ratio (SNR), and the wave superposition model of multiple input single output (MISO). In the MISO model, informed nodes cooperate to produce a stronger superposed signal. We do not consider the problem of transmitting a full message nor do we consider interference. In each round, the informed senders try to deliver to other nodes the required signal strength such that the received signal can be distinguished from the noise. We assume sufficiently high node density @math . In the unit-disk graph model, broadcasting needs @math rounds. In the other models, we use an Expanding Disk Broadcasting Algorithm, where in a round only triggered nodes within a certain distance from the initiator node contribute to the broadcasting operation. This algorithm achieves a broadcast in only @math rounds in the SNR-model. Adapted to the MISO model, it broadcasts within @math rounds. All bounds are asymptotically tight and hold with high probability, i.e. @math .
When the energy of each sender is constrained, in @cite_17 it is shown that the SNR-model does not give much improvement compared to the UDG-model. So, @cite_17 incorporates the unit disk model with the SINR (Signal to Interference and Noise) model. The focus are TDMA scheduling schemes to enhance the network capacity. The path-loss exponent in the SINR model is chosen with @math and interference has only local effects for unsynchronized transmitters. In this context, the SNR model is used for each sender separately. So, the problem of broadcast mainly reduces to range assignment and scheduling problem, for which the number of rounds approaches the diameter @cite_2 .
@cite_2 @cite_17
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Modeling communications in wireless networks is a challenging task, since it requires a simple mathematical object on which efficient algorithms can be designed but which must also reflect the complex physical constraints inherent in wireless networks, such as interferences, the lack of global knowledge, and purely local computations. As a tractable mathematical object, the unit disk graph (UDG) is a popular model that has enabled the development of efficient algorithms for crucial networking problems. In a ρ-UDG, two nodes are connected if and only if their distance is at most ρ, for some ρ ≪ 0. However, such a connectivity requirement is basically not compatible with the reality of wireless networks due to the environment of the nodes as well as the constraints of radio transmission. For this purpose, the signal interference plus noise ratio model (SINR) is the more commonly used model. The SINR model focuses on radio interferences created over the network depending on the distance to transmitters. Nevertheless, due to its complexity, this latter model has been the subject of very few theoretical investigations and lacks of good algorithmic features. In this paper, we demonstrate how careful scheduling of the nodes enables the two models to be combined to give the benefits of both the algorithmic features of the UDG and the physical validity of the SINR. Precisely, we show that it is possible to emulate a 1 √n ln n-UDG that satisfies the constraints of the SINR over any set of n wireless nodes distributed uniformly in a unit square, with only a O(ln3 n) time and power stretch factor. The main strength of our contribution lies in the fact that the scheduling is set in a fully distributed way and considers non-uniform power ranges, and it can therefore fit the sensor network setting. Moreover, our scheduling is optimal up to a polylogarithmic factor in terms of throughput capacity according to the lower bound of Gupta and Kumar. This paper addresses the following question, which is of interest in the design 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 in order to maximize the number of simultaneous links in the network? Dividing the available spectrum results in two competing effects. On the positive side, a larger N allows for more parallel, non- interfering communications to take place in the same area. On the negative side, a larger N increases the SINR requirement for each link because the same information rate must be achieved over less bandwidth. Exploring this tradeoff and determining the optimum value of N in terms of the system parameters is the focus of the paper. Using stochastic geometry, the optimal SINR threshold - which directly corresponds to the optimal spectral efficiency - is derived for both the low SNR (power-limited) and high SNR (interference-limited) regimes. This leads to the optimum choice of the number of frequency bands N in terms of the path loss exponent, power and noise spectral density, desired rate, and total bandwidth. We consider the problem of resource allocation in downlink OFDMA systems for multi service and unknown environment. Due to users' mobility and intercell interference, the base station cannot predict neither the Signal to Noise Ratio (SNR) of each user in future time slots nor their probability distribution functions. In addition, the traffic is bursty in general with unknown arrival. The probability distribution functions of the SNR, channel state and traffic arrival density are then unknown. Achieving a multi service Quality of Service (QoS) while optimizing the performance of the system (e.g. total throughput) is a hard and interesting task since it depends on the unknown future traffic and SNR values. In this paper we solve this problem by modeling the multiuser queuing system as a discrete time linear dynamic system. We develop a robust H∞ controller to regulate the queues of different users. The queues and Packet Drop Rates (PDR) are controlled by proposing a minimum data rate according to the demanded service type of each user. The data rate vector proposed by the controller is then fed as a constraint to an instantaneous resource allocation framework. This instantaneous problem is formulated as a convex optimization problem for instantaneous subcarrier and power allocation decisions. Simulation results show small delays and better fairness among users. Cognitive radio networks (CRNs) have the potential to utilize spectrum efficiently and are positioned to be the core technology for the next-generation multihop wireless networks. An important problem for such networks is its capacity. We study this problem for CRNs in the SINR (signal-to-interference-and-noise-ratio) model, which is considered to be a better characterization of interference (but also more difficult to analyze) than disk graph model. The main difficulties of this problem are two-fold. First, SINR is a nonconvex function of transmission powers; an optimization problem in the SINR model is usually a nonconvex program and NP-hard in general. Second, in the SINR model, scheduling feasibility and the maximum allowed flow rate on each link are determined by SINR at the physical layer. To maximize capacity, it is essential to follow a cross-layer approach, but joint optimization at physical (power control), link (scheduling), and network (flow routing) layers with the SINR function is inherently difficult. In this paper, we give a mathematical characterization of the joint relationship among these layers. We devise a solution procedure that provides a (1- ) optimal solution to this complex problem, where is the required accuracy. Our theoretical result offers a performance benchmark for any other algorithms developed for practical implementation. Using numerical results, we demonstrate the efficacy of the solution procedure and offer quantitative understanding on the interaction of power control, scheduling, and flow routing in a CRN.