# Mittagsseminar (in cooperation with J. Lengler, A. Steger, and D. Steurer)

__Mittagsseminar Talk Information__ | |

**Date and Time**: Tuesday, May 17, 2005, 12:15 pm

**Duration**: This information is not available in the database

**Location**: This information is not available in the database

**Speaker**: Malwina Luczak (London School of Economics and Political Science)

## On the maximum queue length and asymptotic distributions in the supermarket model

There are n queues, each with a single server. Customers arrive
in a Poisson process at rate λn, where 0 < λ < 1.
Upon arrival each customer selects d ≥ 2 servers uniformly at
random, and joins the queue at a least-loaded server among those
chosen. Service times are independent exponentially distributed
random variables with mean 1. We show that the system is rapidly
mixing, and then investigate the maximum length of a queue in the
equilibrium distribution. We prove that with probability tending
to 1 as n → ∞ the maximum queue length takes at
most two values, which are ln ln n / ln d +O(1).

It is known that the equilibrium distribution of a typical queue
length converges to a certain explicit limiting distribution as n
→ ∞. We quantify this 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 propagation of chaos: 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}.

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