The calculation of the SIR distribution at the typical receiver (or, equivalently, the success probability of transmissions over the typical link) in Poisson bipolar and cellular networks with Rayleigh fading is relatively straightforward, but it only provides limited information on the success probabilities of the individual links. This paper introduces the notion of the meta distribution of the SIR, which is the distribution of the conditional success probability @math given the point process, and provides bounds, an exact analytical expression, and a simple approximation for it. The meta distribution provides fine-grained information on the SIR and answers questions such as "What fraction of users in a Poisson cellular network achieve 90 link reliability if the required SIR is 5 dB?". Interestingly, in the bipolar model, if the transmit probability @math is reduced while increasing the network density @math such that the density of concurrent transmitters @math stays constant as @math , @math degenerates to a constant, i.e., all links have exactly the same success probability in the limit, which is the one of the typical link. In contrast, in the cellular case, if the interfering base stations are active independently with probability @math , the variance of @math approaches a non-zero constant when @math is reduced to @math while keeping the mean success probability constant.
An Aloha-type access control mechanism for large mobile, multihop, wireless networks is defined and analyzed. This access scheme is designed for the multihop context, where it is important to find a compromise between the spatial density of communications and the range of each transmission. More precisely, the analysis aims at optimizing the product of the number of simultaneously successful transmissions per unit of space (spatial reuse) by the average range of each transmission. The optimization is obtained via an averaging over all Poisson configurations for the location of interfering mobiles, where an exact evaluation of signal over noise ratio is possible. The main mathematical tools stem from stochastic geometry and are spatial versions of the so-called additive and max shot noise processes. The resulting medium access control (MAC) protocol exhibits some interesting properties. First, it can be implemented in a decentralized way provided some local geographic information is available to the mobiles. In addition, its transport capacity is proportional to the square root of the density of mobiles which is the upper bound of Gupta and Kumar. Finally, this protocol is self-adapting to the node density and it does not require prior knowledge of this density.  The evaluation of optimum transmission ranges in a packet radio network in a fading and shadowing environment is considered. It is shown that the optimal probability of transmission of each user is independent of the system model and is p sub o spl sime 0.271. The optimum range should be chosen so that on the average there are spl chi (G b) sup 2 spl eta terminals closer to the transmitter than the receiver, where G is the spread spectrum processing gain, b is the outage signal-to-noise ratio threshold, spl eta is the power loss factor and spl chi depends on the system parameters and the propagation model. The performance index is given in terms of the optimal normalized expected progress per slot, given by spl thetav (G b) sup 1 spl eta where spl thetav is proportional to the square root of spl chi . A comparison with the results obtained by using deterministic propagation models shows, for typical values of fading and shadowing parameters, a reduction up to 40 of the performance index. >