We present an asynchronous wakeup policy for wireless sensor networks that exploits the available path diversity for maximizing the expected network lifetime. We assume a random traffic generation model such that the rate is constant in time. Each node is assumed to have a set of forwarding neighbors, any of which may be used for forwarding its traffic to the sink. A node having data packet to send, transmits the packet to the first available node in its forwarding set. In order to maximize the network lifetime, we balance the power dissipation at the network nodes by adjusting the wakeup parameters at various nodes. Allowing different nodes to wakeup with different rates makes the scheme asymmetric. For ease of analysis, we restrict ourselves to static, open-loop policies. We show that the optimization problem is a Signomial Program (SP), that can be well approximated as a Geometric Program (GP). By extensive simulations, we compare the asymmetric policy thus obtained to the best possible symmetric policy obtained from the same optimization setup but ensuring additionally that the wakeup rates at all the nodes are the same (in which case the optimization problem is shown to be exactly a GP). The simulations show that allowing asymmetry can extend the network lifetime by effectively exploiting the available path diversity. Moreover, we also prove that, in case of symmetric policies, no piecewise static policy can beat the simple static policy that we use for comparison in our results. This shows that in the space of open-loop, asynchronous wakeup policies, employing the static, asymmetric policy presented in this paper is much more profitable than even the best piecewise static, symmetric policy.