Asymptotic analysis of the processor sharing multi-queue

Abstract

Queueing theory is a key tool for analyzing complex systems like cloud computing and networks. It helps understand how delays, congestion, and resource sharing behave under different regimes. This paper studies the asymptotic behavior of the fluid model solution associated with a network of processor sharing multi-queues. This model is particularly relevant to modern applications where multiple tasks share processing resources. The network consists of $J$ queues, each with a single server, an infinite waiting room and arbitrary interarrival and service time distributions. Under the processor-sharing discipline, all customers present in a queue are served simultaneously. In this system, customers may arrive at a queue either from outside the system or from the previous queue. Upon completing service at one queue, customers proceed to the next. Our results show that, as time approaches infinity, the fluid model solution converges in the critical regime and grows asymptotically linearly with time in the supercritical regime.