Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

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.

Upcoming talks     |     All previous talks     |     Talks by speaker     |     Upcoming talks in iCal format (beta version!)

Previous talks by year:   2023  2022  2021  2020  2019  2018  2017  2016  2015  2014  2013  2012  2011  2010  2009  2008  2007  2006  2005  2004  2003  2002  2001  2000  1999  1998  1997  1996  

Information for students and suggested topics for student talks

Automatic MiSe System Software Version 1.4803M   |   admin login