Here is a simple heuristic (approximation algorithm) for routing a message from node A to node B in a reasonable amount of time. The algorithm assumes that every node in the network has at least two outgoing links.
Assume A has n outgoing links, $n \geq 2$, with delays to the nodes on the other end of $a_1, a_2, \ldots, a_n$. Select the link with the shortest delay and send the message on that line. If the message arrives at node $\mathrm{B}$, you are done. Otherwise, repeat this process until the message does arrive at its intended destination, but do not send the message directly back to the node from where it just came. That is, if the message was sent from A to $C$, then $C$ should not return the message to A. In this case, choose the second-lowest delay value for the outgoing line.
Will this heuristic always deliver the message from node A to node B? If not, explain why.