  High speed railways (HSRs) have been deployed widely all over the world in
recent years. Different from traditional cellular communication, its high
mobility makes it essential to implement power allocation along the time. In
the HSR case, the transmission rate depends greatly on the distance between the
base station (BS) and the train. As a result, the train receives a time varying
data rate service when passing by a BS. It is clear that the most efficient
power allocation will spend all the power when the train is nearest from the
BS, which will cause great unfairness along the time. On the other hand, the
channel inversion allocation achieves the best fairness in terms of constant
rate transmission. However, its power efficiency is much lower. Therefore, the
power efficiency and the fairness along time are two incompatible objects. For
the HSR cellular system considered in this paper, a trade-off between the two
is achieved by proposing a temporal proportional fair power allocation scheme.
Besides, near optimal closed form solution and one algorithm finding the
$\epsilon$-optimal allocation are presented.
