A stochastic geometry approach to wideband ad hoc networks with channel variations Steven Weber Drexel University, Dept. of ECE Philadelphia PA 19104 sweber@ece.drexel.edu Jeffrey G. Andrews The University of Texas at Austin, Dept. of ECE Austin TX 78712 jandrews@ece.utexas.edu Abstract— We present a methodology for determining the outage probability of wideband ad hoc networks with random wireless channels. Assuming that the nodes are Poisson dis- tributed and subject to a required SINR constraint, we develop a simple framework that gives upper and lower bounds on the outage probability. These bounds are important in that they can be manipulated to obtain bounds on the transmission capacity, i.e., the maximum permissible spatial density of transmissions ensuring an acceptably low outage probability. In this paper, we derive the outage probability of wireless ad hoc networks under path loss and shadowing, which are the dominant large-scale effects in wideband ad hoc networks. The analytical framework is rooted in stochastic geometry, employing marked point processes, void probabilities, Palm measure, and Campbell’s Theorem. I. I NTRODUCTION The long-term viability of decentralized wireless network- ing, compared to the more traditional centralized wireless networking, depends largely on the fundamental capabilities of a network of randomly distributed, mutually interfering, wireless nodes. This paper introduces a general framework, extending the framework in [1], for analyzing the effect of fading channels on ad hoc network outage probability as well as the transmission capacity. This framework can be used to compute both upper and lower bounds on the outage probability as well as the allowable intensity of transmitters in the network subject to a target outage probability (QoS constraint). A. Ad Hoc Network Capacity Ad hoc network capacity has been a highly active research area particularly since the seminal result of Gupta and Ku- mar [2], that found that the transport capacity of a large random wireless ad hoc network with n nodes scaled as (n log n) − 1 2 . Numerous interesting extensions of this result have since been derived, including for example [3], [4], and [5] which incorporated respectively mobility, bandwidth, and energy considerations. In these cases the scaling in n can be improved under certain conditions. More recently, the Gupta- Kumar result has been verified from an information theoretic perspective [6] using fairly simple probabilistic methods [7]. A stochastic geometric approach pioneered by Baccelli and others [8], [9] based on Poisson point processes and their accompanying rich body of theory has proven very useful in characterizing many key features of wireless networks. More recent work [10], [11] has used stochastic geometry to study connectivity and throughput scaling bounds for ad hoc networks. The Transmission Capacity is a capacity metric that gives the maximum spatial density of achievable transmissions in a wireless ad hoc network, subject to a quality of service constraint [1]. It has been recently used to show the impact of spread spectrum [1], interference cancellation [12], and scheduling [13]. All our prior work has assumed channels subject to only path loss effects. In this paper, we extend the framework to include shadowing effects. Path loss and shadowing are the the dominant large-scale effects in wide- band ad hoc networks. Our approach employs marked Poisson point processes, where the marks are random variables for each transmitter capturing shadowing effects for the channel connecting the transmitter to a reference receiver at the origin. B. Random Channels in Wireless Ad Hoc Networks A fundamental difference between ad hoc and centralized networks is the extent to which the node locations affect not only the quality of their own transmission, but that of all the other nodes. For this reason, most prior work on wireless ad hoc network capacity has focused on the geometric aspects of the network, and considered path loss channel models. Formally, it has typically been assumed that P ij = P j d −α ij , where P i j is the received power at node i from node j , d ij ≥ 1 is the transmission range between nodes i and j , α ≥ 2 is the path loss exponent, and P j is the far-field transmit power at node j (i.e. P ij = P j at d ij =1). While this simple model successfully captures the spatial interactions between the nodes, it has some important shortcomings. First, wireless channels are known to have random distributions that can change the received power by several orders of magnitude. Hence, a preferable channel model is P ij = P j d −α ij h ij , where h ij is a random variable. This random channel gain h ij can have various statistical distributions in different propaga- tion environments, including lognormal, exponential, Ricean, Nakagami, and compounds of these. A second motivation for including fading is that many capacity increasing techniques, such as transmit/receive diver- sity, spatial multiplexing, adaptive modulation, and multiuser diversity depend on variations in the channel (see [14], [15] 0-7803-9550-6/06/$20.00 © 2006 IEEE