Some dynamical routing algorithms in large systems

Rather simple mathematical models of network are investigated. We consider a system with $N$ servers and a Poisson income flow of messages. The service time of a message is distributed exponentially. Two ways of network performance will be compared. In the first model at the arrival of each message two servers are selected randomly, and the message is directed to the least busy one. In the second model there are $N$ queues, the message is directed to the randomly selected queue, and a ``dispatcher'' selects a message to be served: two randomly chosen queues are compared and a message from the more busy one is served. The queue length distribution in the systems is investigated in the limit case as $N$ tends to infinity.