Average Outage and Non-Outage Duration of Selective Decode-and-Forward Relaying Nikola Zlatanov † , Robert Schober † , Zoran Hadzi-Velkov †† , and George Karagiannidis ††† † The University of British Columbia, E-mail: {zlatanov, rschober}@ece.ubc.ca †† Ss. Cyril and Methodius University, Skopje, E-mail: zoranhv@feit.ukim.edu.mk ††† Aristotle University Thessaloniki, E-mail: geokarag@auth.gr Abstract— In this paper, assuming transmission with fixed rate and fixed power, we derive the average duration of capacity outage and non-outage events of selective decode-and-forward relaying with repetition coding over slow Rayleigh fading channels. Furthermore, we develop high signal-to-noise ratio (SNR) approx- imations for both durations which provide significant insight into the impact of various system and channel parameters. For high SNR, on a double logarithmic scale, both the average outage duration (AOD) and the average non-outage duration (ANOD) become straight lines when plotted as functions of the SNR. However, while the slope of the ANOD improves with increasing diversity order, the slope of the AOD is -1/2 independent of the diversity order. I. I NTRODUCTION Cooperative diversity is an efficient technique to exploit the inherent spatial diversity of wireless networks [1]-[4]. In particular, opportunistic relaying, where only one out of the set of available relays is selected, extracts the full diversity of the channel and minimizes the loss in spectral efficiency caused by the repetition at the relays [1], [3]. For fixed rate transmission, the random nature of the propagation environment may lead to capacity outage events if the transmission rate exceeds the capacity of the end-to-end channel. Under these conditions, the performance of cooperative diversity systems is generally characterized by the outage probability. However, in systems with mobile nodes, the channel gains are slowly time varying and the outage events in neighboring coding blocks become correlated. This phenomenon is not captured by the outage probability itself but can be characterized by the average outage duration (AOD) and the average non-outage duration (ANOD). The AOD and ANOD are useful measures for the design of systems which exploit the temporal changes of the channel such as systems using automatic repeat request (ARQ) and multi-user scheduling. In this paper, we derived the AOD and ANOD of a simple se- lective repetition coding (SRC) based decode-and-forward (DF) relaying protocol introduced in [1]. Besides the accurate results, we develop high signal-to-noise ratio (SNR) approximations which provide significant insight into the impact of the various system and channel parameters. Related work includes [5] and [6]. Compared to this paper, in [5], the direct link between the source and the destination was ignored and a different relay selection policy was considered. In [6], cooperative protocols with only one relay were studied. Furthermore, to the best of our knowledge, the ANOD and its asymptotic behavior have not been considered in any other work before. Organization : In Section II, the system model and some def- initions are presented. The AOD and ANOD for the considered SRC DF protocol are derived in Section III, and illustrated based on a numerical example in Section IV. Conclusions are drawn in Section V. II. PRELIMINARIES In this section, we present the adopted system and channel models as well as the definitions of the AOD and ANOD. A. System Model We consider a cooperative network consisting of a source S,a destination D, and M relays denoted as R i , i ∈{1, 2,...,M }, operating in a Rayleigh fading enviroment. The channel gain of the direct link (S → D) is denoted by X(t), the channel gain of the link between the source and relay R i (S → R i ) is denoted by Y i (t), and the channel gain of the link between relay R i and the destination (R i → D) is denoted by Z i (t). The average channel gain powers are defined as E[X 2 ]  Ω X , E[Y 2 i ]  Ω Y , and E[Z 2 i ]  Ω Z , where E[·] denotes expectation 1 . All nodes are assumed to transmit with equal power P T , and Γ 0  P T /N 0 denotes the transmit SNR, where N 0 is the power spectral density of the underlying additive white Gaussian noise. We assume two-dimensional isotropic scattering around all nodes. Furthermore, S, R i , i ∈{1, 2,...,M }, and D are assumed to be mobile, thus introducing maximum Doppler shifts f mS , f mR , and f mD , respectively. Under these conditions, the channel gains X(t), Y i (t), and Z i (t) are time-correlated Rayleigh random processes and their auto-covariance functions and Doppler spectra are given in [7]. The time derivative of the channel gain α(t), α ∈{X,Y,Z }, is denoted by ˙ α(t). This derivative is independent of the channel gain itself and is a zero-mean Gaussian random variable (RV) with variance [8] σ ˙ α  πf mα  Ω α , (1) where f mX   f 2 mS + f 2 mD , f mY   f 2 mS + f 2 mR , and f mZ   f 2 mR + f 2 mD . B. Relay Protocol The considered relaying protocol was introduced in [1]. The transmission is performed in two phases. In the first phase, the source broadcasts a codeword to an a priori selected best relay and to the destination. The relay decodes the codeword and forwards it to the destination in the second phase using repetition coding. The destination coherently combines the signals received in both phases and attempts to decode the transmitted message. The single best relay for this SRC DF protocol is chosen as follows. Assume that for a relay to decode successfully its source-relay channel, Y i , has to exceed some threshold Y 0 . We collect the indices i of all relays R i , i ∈{1, 2,...,M }, with Y i >Y 0 in set D(t). The selected best relay, R j , has the maximum relay-destination channel gain among all relays with indices in D(t), i.e., Z j (t) = max {Z i (t)} i∈D(t) . We note 1 Throughout this paper we omit reference to time t whenever possible without giving rise to ambiguity. 978-1-4577-0742-1/11/$26.00 ©2011 IEEE 94