[Paper Review] Waiting time asymptotics in the single server queue with service in random order

Onno Boxma, Sergey Foss+2 more|Data Archiving and Networked Services (DANS)|Jul 18, 2012

Advanced Queuing Theory Analysis3 citations

TL;DR

This paper analyzes the waiting time tail asymptotics in a single-server queue with service in random order (ROS), focusing on heavy-tailed service time distributions. It establishes that for Poisson arrivals and regularly varying service times with index $-\nu$, the waiting time distribution is also regularly varying with index $1-\nu$, and explicitly derives the pre-factor in the asymptotic expression, providing a precise heavy-traffic limit for the $M/G/1$ queue with infinite variance service times.

ABSTRACT

We consider the single server queue with service in random order. For a large class of heavy-tailed service time distributions, we determine the asymptotic behavior of the waiting time distribution. For the special case of Poisson arrivals and regularly varying service time distribution with index -ν, it is shown that the waiting time distribution is also regularly varying, with index 1 -ν, and the pre-factor is determined explicitly. Another contribution of the paper is the heavy-traffic analysis of the waiting time distribution in the M/G/1 case. We consider not only the case of finite service time variance, but also the case of regularly varying service time distribution with infinite variance.

Motivation & Objective

• To determine the asymptotic behavior of waiting time distributions in single-server queues with service in random order (ROS) under heavy-tailed service time distributions.
• To extend heavy-traffic limit theorems to the $M/G/1$ queue with infinite variance service time distributions.
• To establish exact tail asymptotics for the waiting time when service times are regularly varying with index $-\nu$.
• To provide a probabilistic interpretation of the dominant contributions to large waiting times in the ROS queue.
• To demonstrate that the tail behavior of the waiting time distribution inherits the regular variation property of the service time distribution under ROS discipline.

Proposed method

• Uses a decomposition of the waiting time exceeding a large value $x$ into four probabilistically interpretable terms, each corresponding to distinct rare event scenarios.
• Applies the theory of subexponential and long-tailed distributions to identify the dominant contribution to the tail, where one large service time or residual service dominates.
• Employs a powerful lemma by Bingham and Doney on Laplace-Stieltjes transforms of regularly varying distributions to derive the asymptotic waiting time tail from Le Gall's expression for the waiting time LST.
• Uses a probabilistic argument based on the 'Typical Event Theorem' to interpret the asymptotic behavior in terms of rare events involving large service times or large numbers of customers.
• Applies a reduction lemma (Lemma D.1) showing that for tail asymptotics, the arrival process can be replaced by a deterministic one with the same rate, simplifying analysis to a $D/G/1$-like system.
• Performs a detailed asymptotic analysis of a sum over customer indices, using uniform convergence and integral approximation to derive the final tail expression.

Experimental results

Research questions

• RQ1 What is the exact asymptotic behavior of the waiting time distribution in the $M/G/1$ queue with service in random order when service times are regularly varying with infinite variance?
• RQ2 How does the tail of the waiting time distribution relate to the tail of the service time distribution in the ROS queue?
• RQ3 What are the dominant probabilistic mechanisms leading to extremely long waiting times in the ROS queue under heavy-tailed service times?
• RQ4 Can the arrival process be replaced by a deterministic one without affecting the tail asymptotics of the waiting time distribution?
• RQ5 What is the precise pre-factor in the asymptotic expression for the waiting time tail when service times are regularly varying with index $-\nu$?

Key findings

• For Poisson arrivals and regularly varying service time distribution with index $-\nu$, the waiting time distribution is regularly varying with index $1-\nu$.
• The pre-factor in the asymptotic expression for the waiting time tail is explicitly determined as $\frac{\rho}{1-\rho}h(\rho)$, where $h(\rho)$ is a function of the traffic load $\rho$.
• The asymptotic behavior of the waiting time tail is dominated by the event that a residual service time exceeds $x$, contributing $\rho \cdot \mathsf{P}(B^{fw} > x)$ to the tail.
• The paper establishes a heavy-traffic limit theorem for the $M/G/1$ queue with service in random order, even when the service time variance is infinite.
• The tail asymptotics are derived using two independent methods: one via Laplace-Stieltjes transform analysis and another via direct probabilistic decomposition of the waiting time tail.
• The analysis confirms that the waiting time tail inherits the regular variation property of the service time distribution, with a precise scaling factor dependent on the traffic intensity $\rho$.