Large Queueing System where Messages are Transmitted via Several Routes

Let a system with $N$ servers be fed by a Poisson flow of rate $\lambda N$. Upon its arrival, a message is split into $n$ packets and each packet is sent to a randomly selected server independently of all other packets. The packet service time is distributed exponentially with mean 1. It is shown that if $\rho=\lambda_n<1$, then in the limit, as $N\to\infty$, the queue-length distribution at the servers tends to the queue-length distribution in an $M|M|1$ system with the input flow rate $\rho$. This permits one to conclude that if such a method of message transmission is used as the values of $\rho$ are small, the coding may speed up the delivery of messages. The case where a packet is formed by $M$ mini-packets and a mini-packet service time is distributed exponentially with mean $1/M$ is also briefly considered. As $M\to\infty$, the waiting-time distribution in such a system tends to the waiting-time distribution in the $M|D|1$ system.