Abstract
We consider a queue to which only a finite pool of n customers can arrive, at times depending on their service requirement. A customer with stochastic service requirement S arrives to the queue after an exponentially distributed time with mean S-αfor some [Formula: see text]; therefore, larger service requirements trigger customers to join earlier. This finite-pool queue interpolates between two previously studied cases: α = 0 gives the so-called [Formula: see text] queue and α = 1 is closely related to the exploration process for inhomogeneous random graphs. We consider the asymptotic regime in which the pool size n grows to infinity and establish that the scaled queue-length process converges to a diffusion process with a negative quadratic drift. We leverage this asymptotic result to characterize the head start that is needed to create a long period of activity. We also describe how this first busy period of the queue gives rise to a critically connected random forest.
Highlights
This paper introduces the Δα(i)/G/1 queue that models a situation in which only a finite pool of n customers will join the queue
We considered a generalization of the Δ(i)/G/1 queue, which we coined the Δα(i)/G/1 queue, a model for the dynamics of a queueing system in which only a finite number of customers can join
Under a suitable heavy-traffic assumption, the diffusion-scaled queue-length process embedded at service completions converges to a stochastic process W(·)
Summary
This paper introduces the Δα(i)/G/1 queue that models a situation in which only a finite pool of n customers will join the queue. The busy periods of this queue will be completely characterized by the initial number of customers i and the random variables (A(k))k≥1. The total number of vertices in the forest is N This random forest is exemplary for a deep relation between queues and random graphs, perhaps best explained by interpreting the embedded Δα(i)/G/1 queue as an exploration process, a generalization of a branching process that can account for dependent random variables (A(k))k≥1. For a given random graph, the exploration process declares vertices active, neutral, or inactive. More vertices are already explored (inactive) or discovered (active), fewer vertices are neutral This phenomenon is known as the depletion-of-points effect and plays an important role in the scaling limit of the random graph.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
