[Paper Review] Asymptotics of Queue Length Distributions in Priority Retrial Queues

Joris Walraevens, Dieter Claeys+1 more|arXiv (Cornell University)|Jan 22, 2018

Advanced Queuing Theory Analysis25 references3 citations

TL;DR

This paper derives asymptotic tail behavior of the orbit length distribution in a two-class priority retrial M/G/1 queue using singularity analysis of the probability generating function. It identifies two distinct regimes—'priority regime' with exponential decay modulated by a power law factor, and 'retrial regime' where the retrial rate alters the sub-exponential decay, with similar asymptotics to non-priority retrial queues.

ABSTRACT

We calculate asymptotics of the distribution of the number of customers in orbit in a two-class priority retrial $M/G/1$-type queueing model. In this model, priority customers wait in line while non-priority customers join an orbit and retry later. Although the generating function and moments of the number of customers in orbit has been analyzed before, asymptotics of the distribution have not been thoroughly investigated. We use singularity analysis of the probability generating function to do just that. Our results show that different regimes exist for these asymptotics in case of light-tailed service times: in what we call the `priority regime', the tail asymptotics have the same decay ($\sim cn^{-3/2}R^{-n}$) as in the priority non-retrial queue and the retrial rate only influences the constant $c$. In the `retrial regime', the retrial rate also influences the sub-exponential factor of the asymptotics. In this regime, asymptotics are very similar to asymptotics in retrial queues without (priority) waiting line. Finally, we also analyze the case that the service time distribution is power law (with or without exponential cut-off) using the same technique.

Motivation & Objective

• To investigate the asymptotic behavior of the number of customers in orbit in a two-class priority retrial M/G/1 queue, a model where priority customers queue while non-priority customers retry after blocking.
• To extend existing knowledge on queue length asymptotics, which has largely focused on conventional queues or single-class retrial systems, to the more complex priority retrial setting.
• To determine how the retrial rate and service time distribution influence the heavy-tailed or light-tailed decay of the orbit length distribution.
• To provide a refined understanding of system performance under extreme congestion, particularly in applications like cellular networks and call centers.

Proposed method

• The study employs singularity analysis of the probability generating function (PGF) of the orbit length to extract asymptotic tail behavior.
• It decomposes the PGF into components representing the priority queue and the retrial orbit, analyzing their singularities near the dominant singularity at z=1.
• The analysis distinguishes between light-tailed and power-law service time distributions, applying different asymptotic techniques accordingly.
• For light-tailed service times, the method identifies two regimes: 'priority regime' and 'retrial regime', based on the influence of the retrial rate on the sub-exponential decay factor.
• For power-law service times, the method uses the smooth implicit-function schema to characterize the asymptotic behavior of the PGF and derive power-law tail decay.
• The derivation relies on the implicit function theorem and the existence of a solution to the system G(r,s)=s and G_w(r,s)=1 to locate the dominant singularity R_h.

Experimental results

Research questions

• RQ1 How does the retrial rate affect the tail decay of the orbit length distribution in a priority retrial queue?
• RQ2 Are there distinct asymptotic regimes in the orbit length distribution depending on the service time distribution and retrial mechanism?
• RQ3 How do the asymptotics of the orbit length compare to those of standard priority queues and non-retrial retrial queues?
• RQ4 What is the impact of power-law service time distributions on the tail behavior of the orbit length?
• RQ5 Can singularity analysis of the PGF accurately capture the sub-exponential decay factors in the orbit length distribution?

Key findings

• In the 'priority regime' for light-tailed service times, the orbit length tail decays as ∼cn⁻³/²R⁻ⁿ, with the retrial rate only affecting the constant c, not the decay rate.
• In the 'retrial regime', the retrial rate influences the sub-exponential factor, leading to asymptotics similar to those in non-priority retrial queues.
• For power-law service times with exponent α_B*, the orbit length distribution decays as ∼n^(α_B*−1), with the exponent determined by the service time tail and system parameters.
• The asymptotic behavior of the orbit length distribution q(n) is governed by the dominant singularity R_h of the PGF, which is determined by the solution to G(r,s)=s and G_w(r,s)=1.
• The derived asymptotics for r(n) and p₂(n) are identical and scale as ∼n^α_B*, indicating that the retrial and priority components share the same heavy-tailed behavior under power-law service times.
• The analysis confirms that the smooth implicit-function schema applies to the generating function h(z) when the service time distribution is of type 1, ensuring the existence of a dominant square-root singularity R_h.