IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 43, N. 3, AUG. 1994 1 Outage Probability in Multiple Access Packet Radio Networks in the Presence of Fading Michele Zorzi, Student Member, IEEE, and Silvano Pupolin, Senior Member, IEEE —The outage probability in a mobile communications envi- ronment, where the interferers are randomly distributed on the ground, taking into account the background noise, the co-channel interference and the random effects of propagation (log-normal shadowing and Rayleigh fading), is evaluated. Although not feasible in closed form, the computation can be performed numerically in a very fast and accurate manner. A com- parison to the results reported in the recent literature is presented, along with a sensitivity analysis and an example of application. — Outage probability, Fading, Shadowing, Packet Radio, Mo- bile Communications I. INTRODUCTION Self-interference is the major limiting factor in multiple-access radio systems. To combat the phenomenon, in order to increase the network throughput, several different modulation formats and random access strategies have been proposed in the liter- ature [1]. Most authors have considered only one of the two aspects outlined above, namely the interference or the random access protocol, while in the real world they are strongly related, and a joint optimization could improve substantially the network throughput. Hereafter, we consider the effects of a random at- tenuation on the signal-to-interference plus noise ratio generated in a mobile radio environment, looking for a simple result to be applied to the comparison of the performance of multiple access protocols. In the analysis, we modeled the interfering signal as the sum of many mutually independent random processes, each repre- senting the interfering signal generated by a specific radio. The statistics of each interferer depends on the power of the trans- mitter, the antenna gains, the distance from the receiver and the statistical characterization of the link attenuation. Typically, the following random effects should be included in the study [2; 3]: i) the random position of the interferers, ii) the random link attenuation due to the log-normal shadowing effect, and iii) the Rayleigh fading. Many authors have studied this problem with a different view; among them we recall Musa and Wasylkiwskyj [4], who performed the computation for a particular distribu- tion of the interferers, by ignoring also log-normal shadowing and Rayleigh fading; Sousa and Silvester [5], who took into account a deterministic propagation law; Benvenuto et al. [6], who employed intensive simulation; Sowerby and Williamson [7], who obtained some results only by costly multiple numerical integrations; Gosling [8], who considered Rayleigh fading only, ignoring log-normal shadowing; Prasad [9] and Linnartz [10], This work was supported by MURST, Italy. Part of this work has been presented at IEEE MILCOM’92, San Diego, CA, 11-14 October 1992. Michele Zorzi is with the Dipartimento di Elettronica e Informazione, Politecnico di Milano - ITALY. Silvano Pupolin is with the Dipartimento di Elettronica e Informatica, Universit` a di Padova - ITALY. who limited their analyses to fixed locations for the interferers, and to the absence of noise. In this paper, we take into account the superposition of several random features of both the channel characteristics and the pop- ulation of users. In the following, we evaluate in an extremely simple and rapid manner the outage probability, taking into ac- count different propagation laws, random propagation effects (fading and shadowing), background noise and possible spread spectrum modulations, and the obtained results are applied to the computation of a packet radio network throughput. Also, the interferers are distributed at random over some region, as is specified later. This accounts for the fact that many users, whose positions are not known a priori, may share the same communi- cations resources, and this affects the transmission performance of each of them. In this paper, we apply the outage probability concept, gen- erally used in cellular radio communications systems, to packet radio networks [11]. The motivation is related to the foresee- able transition from fixed-strategy resource assignment, used in cellular radio (e.g., TDMA and FDMA), to cellular packet com- munication networks [12]. Also, the contribution of this paper beyond the previous literature lies in the effort to take into ac- count an extreme randomness in the communications channel (as in [10]), along with the randomness of the locations of the users, coming up with a mathematically tractable analysis and a final expression which is easy to compute. More interestingly, on the track of [5], we tried to combine the considerations about the link performance and the outage probability techniques with a network approach. We believe that this effort to combine link level and network level is very important in pursuing an effective analysis of such systems. In Section II the model for propagation and interference is given, while in Section III the performance measure is introduced and discussed. Section IV provides a more precise formulation of the problem and gives the obtained formulas. Section V reports computational results in different situations, along with a comparison with the results obtained by Sousa and Silvester [5] in the deterministic situation, and a sensitivity analysis about several parameters. Finally, Section VI reports a simple example of application. II. SYSTEM MODEL In a mobile radio communication environment the path loss is well described by means of a random variable, whose average value is given by the deterministic propagation law , given by: 10 log 10 10 log 10 in dB 1