# SMOR Seminar: Regular variation in a ﬁxed point problem for multiclass queues and branching processes

The busy period *R* in an *M*/*G*/1 queue and various quantities associated with branching processes satisfy fixed point equations of the form ; variants occur for example for the Google page rank algorithm. We present a simple derivation of the asymptotics of the tail of *R* under a regular variation condition somewhat more general than in the literature, and give a multivariate version applying to

multitype branching processes and multiclass queues.

Joint work with Sergey Foss.

Søren Asmussen, Professor of Applied Probability at Aarhus University since 2003, is a leading figure in applied probability. He has made major contributions to queueing theory, insurance risk, and simulation methodology in particular through his work on the theory of heavy tails, rare-event asymptotics and algorithms, Markov processes, and regenerative processes. Indeed, he has shaped modern applied probability as a whole through his tenure as Editor-in-Chief of the Applied Probability Journals from 2005 through to the end of 2015.

In addition over 150 journal publications, Prof. Asmussen is the author of four books, widely regarded as classics: *Branching Processes* (with H. Hering), *Ruin Probabilities* (2ed with Hansjoerg Albrecher), *Applied Probability and Queues*, and *Stochastic Simulation: Algorithms and Analysis* (with Peter W. Glynn).

His research excellence has been recognised through the *Marcel F. Neuts Applied Probability Award* in 1999; the *INFORMS Outstanding Simulation Publication Award* in 2002 and 2008; the *John von Neumann Theory Prize* in 2010; the *Conference in Honour of Søren Asmussen - New Frontiers in Applied Probability* in 2011 on occasion of his 65th birthday; the Gold Medal for *Great Contributions in Mathematics* awarded by the Sobolev Institute of Mathematics in 2011; and an Honorary Doctorate at Heriot--Watt University in 2013.