Worst closed-loop controlled bulk distributions of stochastic arrival processes for queue performance

This paper presents basic queueing analysis contributing to teletraffc theory, with commonly accessible mathematical tools. This paper studies queueing systems with bulk arrivals. It is assumed that the number of arrivals and the expected number of arrivals in each bulk are bounded by some constraints B and λ, respectively. Subject to these constraints, convexity argument is used to show that the bulk-size probability distribution that results in the worst mean queue performance is an extremal distribution with support {1, B} and mean equal to λ. Furthermore, from the viewpoint of security against denial-of-service attacks, this distribution remains the worst even if an adversary were allowed to choose the bulk-size distribution at each arrival instant as a function of past queue lengths; that is, the adversary can produce as bad queueing performance with an open-loop strategy as with any closed-loop strategy. These results are proven for an arbitrary arrival process with bulk arrivals and a general service model.