  We consider a half-duplex wireless relay network with hybrid-automatic
retransmission request (HARQ) and Rayleigh fading channels. In this paper, we
analyze the outage probability of the multi-relay delay-limited HARQ system
with opportunistic relaying scheme in decode-and-forward mode, in which the
\emph{best} relay is selected to transmit the source's regenerated signal. A
simple and distributed relay selection strategy is proposed for multi-relay
HARQ channels. Then, we utilize the non-orthogonal cooperative transmission
between the source and selected relay for retransmitting of the source data
toward the destination if needed, using space-time codes or beamforming
techniques. We analyze the performance of the system. We first derive the
cumulative density function (CDF) and probability density function (PDF) of the
selected relay HARQ channels. Then, the CDF and PDF are used to determine the
outage probability in the $l$-th round of HARQ. The outage probability is
required to compute the throughput-delay performance of this half-duplex
opportunistic relaying protocol. The packet delay constraint is represented by
$L$, the maximum number of HARQ rounds. An outage is declared if the packet is
unsuccessful after $L$ HARQ rounds. Furthermore, closed-form upper-bounds on
outage probability are derived and subsequently are used to investigate the
diversity order of the system. Based on the derived upper-bound expressions, it
is shown that the proposed schemes achieve the full spatial diversity order of
$N+1$, where $N$ is the number of potential relays. Our analytical results are
confirmed by simulation results.
