1984 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 55, NO. 10, OCTOBER 2007 Approximation for a Sum of ON-OFF Lognormal Processes With Wireless Applications C. Fischione, Member, IEEE, F. Graziosi, Member, IEEE, and F. Santucci, Senior Member, IEEE Abstract—In this paper, a lognormal approximation is proposed for the sum of lognormal processes weighted by binary processes. The analytical approach moves from the method early proposed by Wilkinson for approximating first-order statistics of a sum of lognormal components, and extends to incorporate second-order statistics and the presence of both time-correlated random binary weights and cross-correlated lognormal components in moments’ matching. Since the sum of weighted lognormal processes models the signal-to-interference-plus-noise ratio (SINR) of wireless systems, the method can be applied to evaluate in an effective and accurate way the outage occurrence rate and outage duration for different wireless systems of practical interest. In a frequency- reuse-based cellular system, the method is applied for various propagation scenarios, characterized by different shadowing correlation decay distances and correlations among shadowing components. A further case of relevant interest is related to power-controlled wideband wireless systems, where the random weights are binary random variables denoting the activity status of each interfering source. Finally, simulation results are used to confirm the validity of the analysis. Index Terms—Frequency-reuse-based radio cellular systems, level crossing analysis, lognormal distributions, power-controlled wideband wireless systems. I. INTRODUCTION W IRELESS communication systems are strongly af- fected by channel impairment and interference. In an interference-limited environment, the link between a mobile station (MS) and the base station (BS), to which it is connected, is said to be in outage in one of the two directions if the signal-to-interference-plus-noise ratio (SINR) at the receiver input goes below a minimum threshold. In this frame, the widely used performance measure is the outage probability, defined as the probability that, at a given time, the SINR is below the minimum quality threshold (see, e.g., [1] and [2], and references therein for definition and computation of outage probability in a variety of contexts of practical interest). However, as a consequence of the relevant correlation degree that fading phenomena usually exhibit, first order statistics of outage events may be incomplete for consistent performance Paper approved by C.-L. Wang, the Editor for Equalization of the IEEE Com- munications Society. Manuscript received July 9, 2005; revised August 28, 2006 and December 27, 2006. This work was supported in part by the HYCON Net- work of Excellence under Contract FP6-IST-511368. This paper was presented in part at the IEEE International Conference on Communications, 2004. C. Fischione was with the Royal Institute of Technology, School of Electrical Engineering, Stockholm SE-10044, Sweden. He is now with the Department of Electrical Engineering and Computer Sciences, University of California at Berkeley, Berkeley, CA 94720, USA (e-mail: fischion@eecs.berkeley.edu). F. Graziosi and F. Santucci are with the University of L’Aquila, Department of Electrical and Information Engineering and Center of Excellence Design Methodologies for Embedded Controllers, Wireless Interconnect and System- on-Chip (DEWS), L’Aquila I-67040, Italy (e-mail: graziosi@ing.univaq.it; santucci@ing.univaq.it). Digital Object Identifier 10.1109/TCOMM.2007.906432 prediction of a wireless interface. In fact, although the work of Gilbert and Elliot to efficiently model the memory prop- erty of fading channels dates back to decades ago, the subject has received major attention in recent years. The objective is to characterize the rate and duration of outage events, and to relate them to the performance of upper layers in the protocol stack (see, e.g., [3] and [4] for statement of the problem in ba- sic contexts and related analysis). In [5], a detailed analysis of second-order outage statistics in a selection combining diver- sity scheme is proposed for lognormal channels, and in [6], a general framework for incorporating second-order statistics of lognormal channels in performance evaluation of packet mo- bile radio systems is provided. While on one side, this line of research has the merit of including second-order statistics of the wireless channel for the user signal; on the other hand, it either neglects or oversimplifies interference statistics. In fact, the interference strength has been assumed to be either con- stant [4], or Gaussian distributed [3]. The interference has been neglected in [6] and [7], which is equivalent to retaining it as a constant. In all cases, no efforts have been paid to account for the autocovariance of the interference strength. Therefore, the next step is to incorporate interference statis- tics in second-order outage analysis. Characterization of the second-order statistics of the SINR is particularly useful in many situations of interest, since SINR is the link quality metric that is typically adopted for the design and tuning of modern wireless networking functionalities; relevant examples are rep- resented by handover algorithms, adaptive forward correction codes, modulation, spreading, etc. Recent publications have also proposed to include the outage statistics for adaptive power al- location problems with multiuser detection (see, e.g., [8] and references therein). However, it may be observed that the char- acterization of closed form expressions of the statistics of the SINR is a difficult task, due to the need of taking into account various interacting terms, such as multiaccess interference with data sources having different traffic pattern, bit rates, and quality of service specification, and channel fluctuations. A limited number of contributions can be found in recent liter- ature [2], and the analysis might be prohibitive in most context (e.g., composite channel models with correlated components and/or different statistical parameters for different signal com- ponents), so that simplifications in system model assumption are almost forced. Indeed, in relevant situations of practical inter- est, the expression of SINR includes the linear combination of correlated lognormal processes, with eventual (one-sided) auto- correlated random weights. This is the case of frequency-reuse narrow-band cellular systems with, e.g., lognormal channel model, where the lognormal shadowing components charac- terizing the links between the MS and each BS were assumed to 0090-6778/$25.00 © 2007 IEEE