Interacting particles, series Jackson networks, and non-crossing probabilities

Stochastics Seminar
Thursday, October 22, 2009 - 3:00pm for 1 hour (actually 50 minutes)
Skiles 269
Ton Dieker – (ISyE, Georgia Tech)
Yuri Bakhtin
In this talk, we study an interacting particle system arising in the context of series Jackson queueing networks. Using effectively nothing more than the Cauchy-Binet identity, which is a standard tool in random-matrix theory, we show that its transition probabilities can be written as a signed sum of non-crossing probabilities. Thus, questions on time-dependent queueing behavior are translated to questions on non-crossing probabilities. To illustrate the use of this connection, we prove that the relaxation time (i.e., the reciprocal of the ’spectral gap’) of a positive recurrent system equals the relaxation time of a single M/M/1 queue with the same arrival and service rates as the network’s bottleneck station. This resolves a 1985 conjecture from Blanc on series Jackson networks. Joint work with Jon Warren, University of Warwick.