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Livelock free routing schemes

We give a livelock free routing algorithm for any allowed network. Unlike some other solutions to this problem: 1) packets entering the network have an absolute upper bound on the time to reach their destination; 2) under light loads, packets are delivered to their destinations in nearly optimal time; 3) packets with desired paths far away from congested areas will have routing times far shorter than packets wanting to access congested areas; 4) if the network becomes congested and later clears, the network operates just as it would have when it was initially under a light load. The main ideas of this note appear in a different form in my 1994 patent 5,369,745. This note adds to those results and makes them more mathematical.