  This study investigates the problem of communication for a network composed
of two half-duplex parallel relays with additive white Gaussian noise. Two
protocols, i.e., \emph{Simultaneous} and \emph{Successive} relaying, associated
with two possible relay orderings are proposed. The simultaneous relaying
protocol is based on \emph{Dynamic Decode and Forward (DDF)} scheme. For the
successive relaying protocol: (i) a \emph{Non-Cooperative} scheme based on the
\emph{Dirty Paper Coding (DPC)}, and (ii) a \emph{Cooperative} scheme based on
the \emph{Block Markov Encoding (BME)} are considered. Furthermore, the
composite scheme of employing BME at one relay and DPC at another always
achieves a better rate when compared to the \emph{Cooperative} scheme. A
\emph{"Simultaneous-Successive Relaying based on Dirty paper coding scheme"
(SSRD)} is also proposed. The optimum ordering of the relays and hence the
capacity of the half-duplex Gaussian parallel relay channel in the low and high
signal-to-noise ratio (SNR) scenarios is derived. In the low SNR scenario, it
is revealed that under certain conditions for the channel coefficients, the
ratio of the achievable rate of the simultaneous relaying based on DDF to the
cut-set bound tends to be 1. On the other hand, as SNR goes to infinity, it is
proved that successive relaying, based on the DPC, asymptotically achieves the
capacity of the network.
