Singular perturbation for a two-class processor-sharing queue with impatience

• Published in: Stochastic models Vol. 39; no. 3; pp. 502 - 536
• Main Authors: Nasri, R., Simatos, F., Simonian, A.
• Format: Journal Article
• Language: English
• Published: Philadelphia Taylor & Francis 03-07-2023; Taylor & Francis Ltd
• Subjects: Asymptotic methods; Asymptotic series; Customers; Decay rate; Engineering Sciences; Microprocessors; Other; Processor-Sharing queue with impatience; Queues; singular perturbation; Singular perturbation methods; two time-scales Markov processes; Processor-sharing; Singular perturbation; Heavy traffic
• ISSN: 1532-6349; 1532-4214
• DOI: 10.1080/15326349.2022.2133142

A two-class Processor-Sharing queue with one impatient class is studied. Local exponential decay rates for its stationary distribution
are established in the heavy traffic regime where the arrival rate of impatient customers grows proportionally to a large factor A. This regime is characterized by two time-scales, so that no general Large Deviations result is applicable. In the framework of singular perturbation methods, we instead assume that an asymptotic expansion of the solution of associated Kolmogorov equations exists for large A and derive it in the form
with explicit functions g and H.
This result is then applied to the model of mobile networks proposed in [Olivier and Simonian] and accounting for the spatial movement of users. We give further evidence of a unusual growth behavior in heavy traffic in that the stationary mean queue length
and
of each customer-class increases proportionally to
with system load
tending to 1, instead of the usual
growth behavior.