We study the optimal transmission scheme that maximizes the local capacity in two-dimensional (2D) wireless networks. Local capacity is defined as the average information rate received by a node randomly located in the network. Using analysis based on analytical and numerical methods, we show that maximum local capacity can be obtained if simultaneous emitters are positioned in a grid pattern based on equilateral triangles. We also compare this maximum local capacity with the local capacity of slotted ALOHA scheme and our results show that slotted ALOHA can achieve at least half of the maximum local capacity in wireless networks.
Seminal work of Gupta & Kumar [1] and the following works, e.g., [2,3] quantify the capacity in wireless networks in the form of asymptotic scaling laws. However, these results may not be very useful for network protocol designers in comparing different medium access schemes that have different protocol overhead but follow the same scaling behavior. Our goal is to investigate the medium access scheme which optimizes the local capacity. Note that any such scheme may have no practical implementation but its evaluation is interesting in order to establish an upper bound on the local capacity in wireless networks. In our analysis, we will use first and second order differentiation of local capacity to prove that simultaneous emitters arranged in a grid pattern are locally optimal and, in 2D wireless networks, only square, hexagonal and triangular grid patterns are most optimal patterns. This article is organized as follows. In Section 2, we will discuss the model of our wireless network and define the local capacity. We will summarize some important related works in Section 3. The optimality of grid pattern based medium access schemes will be discussed in Section 4 and their local capacity will be analyzed in Section 5. Section 6 will discuss the local capacity of simple ALOHA based scheme. In Section 7, we will evaluate the local capacity of grid pattern schemes and slotted ALOHA and concluding remarks can be found in Section 8.
We consider a wireless network where nodes are uniformly distributed over an infinite plane centered at origin (0, 0). We assume that time is slotted and at any given slot, simultaneous emitters in the network are distributed like a set of points, S = {z 1 , z 2 , . . . , z n , . . .}, where z i is the location of emitter i. The distribution of set S depends on the medium access scheme employed by the nodes and we only assume that, in all slots, the set S has a homogeneous density equal to λ.
Let P i be the transmit power of node i and γ ij be the channel gain from node i to node j such that the received power at node j is P i γ ij . Therefore, transmission from node i to node j is successful if the following condition is satisfied
where N 0 is the background noise power and β is the minimum signal to interference ratio (SIR) required for successfully receiving the packet. We assume that all nodes use unit nominal transmit power and we only consider large-scale pathloss characteristics, i.e., γ ij = |z i -z j | -α , where α > 2 is the pathloss exponent and |.| is the Euclidean norm of the vector. We also assume that N 0 is negligible. Therefore, the SIR of emitter i at any point z in the plane is given by
We call the reception area of emitter i, the area of the plane, A i (λ, β, α), where this emitter is received with SIR at least equal to β. The area, A i (λ, β, α), also contains the point z i since here the SIR is infinite. The average size of A i (λ, β, α) is σ(λ, β, α) and it is independent of the location of z i .
We are interested in local capacity which is defined as the average information rate received by a node randomly located in the network. Consider a node at a random location z in the plane and let N(z, β, α) be the number of reception areas it belongs to. The expected value of
The average information rate received by the node, c(z, β, α), is equal to E(N(z, β, α)) multiplied by the nominal capacity. We assume unit nominal capacity and we have
In related works, focus has been on the medium access schemes like ALOHA, carrier sense multiple access (CSMA) or, in some instances, node coloring as well. Some of these works are as follows. [5] studied slotted ALOHA using a very simple geometric propagation model. Under a similar propagation model, [6] evaluated CSMA and compared it with slotted ALOHA in terms of throughput. [7] used simulations to analyze CSMA under a realistic SIR based interference model and compared it with ALOHA (both slotted and un-slotted). [8][9][10] introduced the concept of transmission capacity, defined as the maximum number of successful transmissions per unit area at a specified outage probability, and evaluated ALOHA, CSMA and code division multiple access (CDMA) protocols. [11] analyzed local (single-hop) throughput and capacity with slotted ALOHA, in networks with random and deterministic node placement, and TDMA, in 1D line-networks only. [12] determined the optimum transmission range under the assumption that interferers are distributed according to Poisson point process whereas [13] gave a detailed analysis on the optimal probability of transmission for ALOHA which optimizes the product of simultaneously successful transmissions per unit of space by the average range of each transmission. In contrast to these works, we investigate the most optimal medium access scheme which optimizes the local capacity in wireless networks. We will also compare this optimal scheme with slotted ALOHA. More detailed comparison with other schemes will be do
This content is AI-processed based on open access ArXiv data.
