A Capacity Upper Bound for Large Wireless Networks with Generally Distributed Nodes Guoqiang Mao, Zihuai Lin Wei Zhang School of Electrical and Information Engineering School of Electrical Engineering and Telecommunications The University of Sydney The University of New South Wales Email: {guoqiang.mao, zihuai.lin}@sydney.edu.au Email: w.zhang@unsw.edu.au Abstract—Since the seminal work of Gupta and Kumar, extensive research has been done on studying the capacity of large wireless networks under various scenarios. Most of the existing work focuses on studying the capacity of networks with uniformly or Poissonly distributed nodes. While uniform and Poisson distribution form an important class of spatial distributions, their capability in capturing the spatial distribution of users in various scenarios and application settings is limited. Therefore it is critical to investigate to what extent, the aforementioned results on capacity of networks with uniformly or Poissonly distributed nodes depend on the underlying node distribution being uniform or Poisson. In this paper, we study the capacity of networks under a general node distribution. A capacity upper bound on networks with generally distributed nodes is obtained, which is valid for both finite networks and asymptotically infinite networks. By imposing some mild conditions on the transmission range, we further simplify the result and show that the asymptotic capacity upper bound can be expressed as a product of four factors, which represents respectively the impact of node distribution, link capacity, number of source destination pairs and the transmission range. The upper bound is shown to be tight in the sense that for the special case of networks with uniformly distributed nodes, the bound is in the same order as known results in the literature. Index Terms—Capacity, node distribution, wireless networks I. I NTRODUCTION Since the seminal work of Gupta and Kumar [1], extensive research has been done on studying the capacity of large wire- less networks under various scenarios [1]–[9]. More specifi- cally, in [1], Gupta and Kumar considered an ad-hoc network with a total of n nodes uniformly and i.i.d. on an area of unit size. Each node in the network is capable of transmitting at W bits/s and using a fixed and identical transmission range. It was shown that when each node randomly and independently chooses another node in the network as its destination, the transport capacity and the achievable per-node throughput are Θ n  W  n log n  and Θ n  W √ n log n  respectively 1 . When the This research is supported by ARC Discovery projects DP110100538 and DP120102030. 1 The following notations are used throughout the paper. For two positive functions f (x) and h (x): • f (x)= ox (h (x)) iff (if and only if) limx→∞ f (x) h(x) =0; • f (x)= ωx (h (x)) iff h (x)= ox (f (x)); • f (x)=Θx (h (x)) iff there exist a sufficiently large x 0 and two positive constants c 1 and c 2 such that for any x>x 0 , c 1 h (x) ≥ f (x) ≥ c 2 h (x); • f (x) ∼x h (x) iff limx→∞ f (x) h(x) =1; nodes are optimally and deterministically placed to maximize throughput, the transport capacity and the achievable per-node throughput become Θ n (W √ n) and Θ n  W √ n  respectively. In [2], Franceschetti et al. considered essentially the same random network as that in [1] except that nodes are now allowed to use two different transmission ranges. The link capacity is determined by the associated SINR through the Shannon–Hartley theorem. By having each source-destination pair transmitting via the so-called “highway system”, formed by nodes using the smaller transmission range, it was shown in [2] that the transport capacity and the per-node throughput can also reach Θ n ( √ n) and Θ n  1 √ n  respectively even when nodes are randomly deployed. The existence of such highways was analytically proved using the percolation theory [10]. The key to achieving a higher capacity in the network considered in [2] is that nodes are restricted to use the smaller transmission range as often as possible and the larger transmission range can only be used by source (destination) nodes to access their respective nearest highway nodes. In this way, the number of concurrent transmissions that can be accommodated in the network area is maximized, hence the improvement in capacity. In [4] Grossglauser and Tse showed that in mobile ad hoc networks, by leveraging on the nodes’ mobility, a per- node throughput of Θ n (1) can be achieved at the expense of unbounded delay. Their work [4] has sparked huge interest in studying the capacity-delay tradeoffs in mobile networks assuming various mobility models and the obtained results often vary greatly with the different mobility models being considered, see [3], [5], [11]–[14] and references therein for examples. In [7], Chen et al. studied the capacity of wireless networks under a different traffic distribution. More specifi- cally, they considered a network with a set of n randomly deployed nodes transmitting to single sink or multiple sinks where the sinks can be either regularly-deployed or randomly- deployed. Under the above settings, it was shown that with single sink, the transport capacity is given by Θ n (W ); with k sinks, the transport capacity is increased to Θ n (kW ) when k = O n (n log n) or Θ n (n log nW ) when k =Ω n (n log n). There is also a significant amount of work studying the impact of infrastructure nodes [6] and multiple-access protocols [9] • An event ξx depending on x is said to occur asymptotically almost surely (a.a.s.) if its probability tends to one as x →∞. Globecom 2013 - Ad Hoc and Sensor Networking Symposium 978-1-4799-1353-4/13/$31.00 ©2013 IEEE 347