Dynamic Routing in the Mean-Field Approximation
Résumé
A queuing system is considered, with stations $1$, $2$, where station $i$ contains a collection $S_i$ of infinite-buffer FCFS single servers. The number of servers in each $S_i$ is $N$, and we assume that $N$ is large (formally, $N\to\infty$). The system is fed by a Poisson flow of rate $2\l N$, with i.i.d. exponential service times of mean one. Upon arrival, a customer chooses two servers according to the following rule. He performs two independent trials, each time choosing station $1$ with probability $p_1$ and $2$ with probability $p_2=1-p_1$ and then choosing a server at random from the corresponding collection $S_i$. He then selects the server with a shortest queue from the two, breaking ties at random if necessary. We assume that $p_1\geq p_2\geq 0$; the main result is that (a) the inequalities $\lambda <1$, $2\lambda p_1^2 <1$, are necessary and sufficient for the existence (and uniqueness) of a (stable) stationary regime, and (b) under these inequalities, a stationary queue-size distribution has a super-exponentially decaying tail. These results are in a striking contrast with a `linear' model where a customer simply chooses a station with probabilities $p_1$ and $p_2$ and then selects a server at random from the corresponding $S_i$. Here, the necessary and sufficient condition is $2\lambda p_1<1$, and the queue-size distribution is geometric.