## Section: New Results

### Analyzing the choice of the least loaded queue between two neighboring queues

A model of N queues, with a local choice policy, is studied. Each one-server queue has a Poissonian arrival of customers. When a customer arrives
at a queue, he joins the least loaded queue between this queue and the next one, ties solved
at random. Service times have exponential distribution. The system is stable if the arrival-to-service
rate ratio, also called load, is less than one. When the load tends to zero, we derive the first
terms of the expansion in this parameter for the stationary probabilities that a queue has few customers. Then we provide explicit asymptotics, as the load tends to zero, for the
stationary probabilities of the queue length. We used the analyticity of the stationary probabilities as a function of the load. It shows the
behavior difference between this local choice policy and the 2-choice policy (*supermarket model*).