We consider a fully connected queueing network in which customers have one direct and many alternative routes through the network, and where customer routeing is dynamic. We obtain an asymptotically optimal routeing policy, taking the limit as the number of queues of the network increases. We observe that good policies route customers directly, unless there is a danger of servers becoming idle, in which case customers should be routed alternatively so as to avoid such idleness, and this leads to policies that perform well in moderate-sized networks.
