Queues With the Dropping Function and Non-Poisson Arrivals
Andrzej Chydziński
Abstract
We deal with the single-server queueing system, in which an arriving job (packet, customer) is not allowed to the queue with the probability depending on the queue size. Such a rejected job is lost and never returns to the queue. The study is motivated, but not limited to, active queue management in Internet routers. The exponential service times and general interarrival times are assumed, what makes the model to be a generalization of classic G/M/1 and G/M/1/N queueing models. Firstly, a replacement for the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\rho < 1$ </tex-math></inline-formula> stability condition, which is too excessive in the considered system, is proven. Then, several popular performance characteristics are derived, including the distribution of the queue size, waiting time, workload and the time to reach a given level, as well as the loss ratio. Finally, numerical examples are presented, demonstrating the impact of the standard deviation of the interarrival time on the system performance, as well as the performance of the system for different parameterizations of the dropping function.