## Asymptotic distributions and chaos for the supermarket model.(English)Zbl 1131.60005

Summary: In the supermarket model there are $$n$$ queues, each with a unit rate server. Customers arrive in a Poisson process at rate $$\lambda n$$, where $$0 < \lambda < 1$$. Each customer chooses $$d\geq 2$$ queues uniformly at random, and joins a shortest one.
It is known that the equilibrium distribution of a typical queue length converges to a certain explicit limiting distribution as $$n\to \infty$$. We quantify the rate of convergence by showing that the total variation distance between the equilibrium distribution and the limiting distribution is essentially of order $$n^{-1}$$; and we give a corresponding result for systems starting from quite general initial conditions (not in equilibrium). Further, we quantify the result that the systems exhibit chaotic behaviour: we show that the total variation distance between the joint law of a fixed set of queue lengths and the corresponding product law is essentially of order at most $$n^{-1}$$.

### MSC:

 60C05 Combinatorial probability 68R05 Combinatorics in computer science 90B22 Queues and service in operations research 60K25 Queueing theory (aspects of probability theory)
Full Text: