Analysis of a non-work conserving Generalized Processor Sharing queue

We consider in this paper a non work-conserving Generalized Processor Sharing (GPS) system composed of two queues with Poisson arrivals and exponential service times. Using general results due to Fayolle \emph{et al}, we first establish the stability condition for this system. We then determine the functional equation satisfied by the generating function of the numbers of jobs in both queues and the associated Riemann-Hilbert problem. We prove the existence and the uniqueness of the solution. This allows us to completely characterize the system, in particular to compute the empty queue probability. We finally derive the tail asymptotics of the number of jobs in one queue.