[Paper Review] Fluid limit of a heavily loaded EDF queue with impatient customers

Laurent Decreusefond, Pascal Moyal|ArXiv.org|Dec 30, 2005

Advanced Queuing Theory Analysis8 references16 citations

TL;DR

This paper establishes the fluid limit of a heavily loaded Earliest-Deadline-First (EDF) queue with impatient customers by modeling the system via a measure-valued process tracking residual time credits. The fluid limit is shown to satisfy an integrated transport equation, enabling approximation of the normalized number of waiting and lost customers through deterministic fluid-level processes.

ABSTRACT

In this paper we present the fluid limit of an heavily loaded Earliest Deadline First queue with impatient customers, represented by a measure-valued process keeping track of residual time-credits of lost and waiting customers. This fluid limit is the solution of an integrated transport equation. We then use this fluid limit to derive fluid approximations of the processes counting the number of waiting and already lost customers.

Motivation & Objective

• To analyze the fluid limit of a heavily loaded EDF queue where customers have deadlines and may leave if not served in time.
• To model the system using a measure-valued process tracking residual time credits of waiting and lost customers.
• To derive deterministic fluid approximations for the number of waiting and lost customers in the heavy-traffic regime.
• To establish a rigorous connection between the stochastic queueing process and its fluid-scale limit via a transport equation.

Proposed method

• Model the system as a measure-valued Markov process where each customer is represented by a unit mass at their residual time credit.
• Derive the infinitesimal generator of the process, incorporating arrival, departure, and continuous time-credit decay at unit rate.
• Use distributional calculus to rigorously define the spatial derivative term arising from the continuous decrease of time credits.
• Establish the fluid limit as the solution to an integrated transport equation derived from the generator.
• Normalize the process and prove weak convergence to a deterministic fluid limit in the functional space of measures.
• Apply the fluid limit to approximate the normalized processes counting waiting and lost customers.

Experimental results

Research questions

• RQ1 What is the fluid-scale behavior of an EDF queue with impatient customers under heavy traffic?
• RQ2 How can the residual time credit process be modeled as a measure-valued Markov process with a continuous decay mechanism?
• RQ3 What is the limiting deterministic equation that describes the mean behavior of the system at fluid scale?
• RQ4 How can the fluid limit be used to approximate the number of waiting and lost customers?
• RQ5 What is the relationship between the fluid limit and known results for M/GI/∞ and M/M/1 queues with impatience?

Key findings

• The fluid limit of the EDF queue with impatient customers is the solution to an integrated transport equation involving a spatial derivative term due to continuous time-credit decay.
• The normalized number of waiting customers converges fluidly to a process given by $\left(1 + (\lambda - \mu)t\right)^+ \mathbf{1}_{\left\{t \leq \frac{\rho d - \mu^{-1}}{\rho - 1}\right\}} + \lambda d \, \mathbf{1}_{\left\{t \geq \frac{\rho d - \mu^{-1}}{\rho - 1}\right\}}$.
• The normalized number of lost customers converges fluidly to $\left(1 + \lambda(t - d) - \mu t\right)^+ \mathbf{1}_{\left\{t \geq \frac{\rho d - \mu^{-1}}{\rho - 1}\right\}}$.
• The fluid limit recovers known results for the M/GI/∞ system when the service rate is set to zero, confirming consistency with prior work.
• The fluid approximation provides a tractable, deterministic alternative to numerical simulations for loss probability assessment in EDF systems.
• The method extends to general service time distributions and allows for asymptotic analysis of performance metrics under heavy traffic.