In this paper, we present delay allocation strategies that minimize the total transmit power in a multihop wireless network. The focus is on guaranteeing an end-to-end average delay bound for a single variable-bit-rate flow on a multihop fading channel. We first compute an analytical approximation for the transmit power that is required to send a variable-bit-rate source over a finite-state fading channel. We then use this approximation to derive a low-complexity and near-optimal delay allocation method for multihop networks when the fading processes on the multiple hops are independent. Properties of the optimal delay allocation are also studied; in the special case of a Gaussian network, the optimal delay allocation strategy is completely characterized. The tradeoff between single-hop transmission and multihop transmission is studied under an end-to-end delay constraint.