Fluid limits of G/G/1+G queues under the non-preemptive earliest-deadline-first discipline

Published 12 May 2013 in math.PR | (1305.2587v2)

Abstract: A single-server queuing model is considered with customers that have deadlines. If a customer's deadline elapses before service is offered, the customer abandons the system (customers do not abandon while being served). When the server becomes available, it offers service to the customer having earliest deadline among those that are in the queue. We obtain a fluid limit of the queue length and abandonment processes and for the occupation measure of deadlines, in the form of measure-valued processes. We characterize the limit by means of a Skorohod problem in a time-varying domain, which has an explicit solution. The fluid limits also describe a certain process called the frontier, that is well known to play a key role in systems operating under this scheduling policy.

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Summary

The paper introduces a fluid limit model that reduces queue dynamics to a one-dimensional Skorokhod reflection problem for both queue length and abandonment rates.
It employs a measure-valued process to explicitly characterize the empirical distribution of remaining customer lead times and the evolving frontier process.
The study distinguishes between subcritical and supercritical regimes, offering actionable insights for analyzing performance in deadline-driven service systems.

Fluid Limits of G/G/1+G Queues Under Non-Preemptive Earliest-Deadline-First

Introduction and Problem Statement

This paper rigorously investigates the fluid limit behavior of single-server G/G/1 queues incorporating customer deadlines and reneging under the non-preemptive Earliest-Deadline-First to beginning-of-service (EDF-b) discipline. Customers arrive with i.i.d. lead times and abandon the system if their deadline elapses prior to starting service. Upon server availability, the queued customer with the shortest remaining lead time is selected, and preemption is disallowed.

Analysis centers on the law-of-large-numbers regime, with system and arrival/service rates accelerated by a scaling parameter $N$ , and studies the limiting behavior as $N \to \infty$ . The focus is on characterizing the fluid limits of:

The (scaled) queue length process.
The scaled abandonment process.
The evolution of the empirical distribution of remaining customer lead times (the measure-valued queue-length process).
The associated "frontier" process, delineating the maximal lead time ever associated with a customer at the queue’s head.

Technical Framework and Main Results

A measure-valued approach is adopted, extending and generalizing the analytic apparatus developed for processor-sharing, FIFO, and shortest-processing-time disciplines to the setting of G/G/1+G under non-preemptive EDF-b.

The core departure from prior work (notably on preemptive EDF-e [Kruk et al.^1, Doytchinov et al.^2], and M/M/1+G EDF-b [Decreusefond & Moyal^3]) is the incorporation of both general interarrival, service, and lead time distributions and a non-preemptive policy, with technical innovations required to accommodate the lack of Markovian structure.

Skorohod Problem and Explicit Fluid Limit

The principal technical achievement is the reduction of the limiting queue length and abandonment processes to the solution of a one-dimensional Skorohod reflection problem on a time-inhomogeneous, deterministic barrier. Specifically, if $Q^N(t)$ is the queue length and $R^N(t)$ the number of abandonments by time $t$ , the processes

$\left(\frac{1}{N} Q^N, \frac{1}{N} R^N\right) \rightarrow (\phi, \eta)$

where $(\phi,\eta)$ solve

$\begin{aligned} \eta(t) &= \sup_{s \le t} \left[\psi(s) - H(0,s)\right]^+ \ \phi(t) &= \psi(t) - \eta(t) \end{aligned}$

with $\psi(t) = Q(0) + (\lambda - \mu)t$ , and $H$ is a deterministic function capturing the potential queue size if there were no service, as defined by the inputs and lead time distribution:

$H(x, t) = G_0(x + t) + \int_0^t G(x + t - s) ds$

where $G_0$ is the initial distribution and $G$ the lead time distribution.

This result provides an explicit and constructive characterization of the LLN limits for both the queue and abandonment rates, offering computational tractability for the evaluation of performance under a broad range of parameters.

Measure-Valued Process and Frontier Behavior

The limit of the measure-valued queue-length process is also characterized, governed by the limit frontier process $F(t)$ , which is the maximal lead time ever seen at the head of the queue, evolved as

$F(t) = \chi(\phi(t), t)$

with $\chi$ the (generalized) inverse of $H$ . The limiting empirical lead time measure is given in closed form in terms of this frontier:

$\mu_t(B) = \xi_0(B \cap [F(t), \infty) + t) + \int_0^t \nu(B \cap [F(t), \infty) + t - s) ds$

where $\xi_0$ and $\nu$ denote initial and lead time distributions, respectively. The paper underscores that, in the limit, the mass of the queue is concentrated above the current frontier, with negligible measure in $[C(t), F(t))$ (where $C(t)$ is the instantaneous head-of-queue lead time).

Subcritical and Supercritical Regimes

The analysis distinguishes between the critical/supercritical ( $\lambda \geq \mu$ ) and subcritical ( $\lambda < \mu$ ) regimes, establishing in the former that the fluid-scaled system remains persistently busy and reneging is governed exclusively by the Skorokhod reflection at the time-varying barrier, while in the latter, the queue eventually vanishes and remains empty.

Methodological Advances

A salient novelty is that the Skorokhod mapping and explicit representation of limits are derived directly from the prelimit dynamics of the queue length, abandonment, and potential queue-length processes, sidestepping the technical machinery of reflected measure-valued process analysis required in prior heavy-traffic or preemptive studies.

Furthermore, the approach departs from existing methods by offering a streamlined proof that the tight coupling between the queue-length process and the barrier (potential queue process) holds in the limit, even in the absence of preemption, and under general distributional assumptions with nontrivial support.

Numerical Instantiations

The paper provides concrete instances (e.g., exponential lead times, general arrival/service) where the explicit Skorokhod-based solutions yield closed-form expressions for the limiting processes, demonstrating the practicality and flexibility of the main results.

Implications and Future Directions

From a theoretical perspective, the results deliver a precise LLN characterization of non-Markovian deadline-driven queues under non-preemptive EDF, illuminating the deterministic structure underlying queue and abandonment dynamics in the fluid regime.

Practically, these results provide system designers with explicit tools for predicting long-term aggregate performance of real-time systems with general arrival, service, and patience distributions, including the rate and circumstances under which deadline-missing (reneging) occurs.

The generalized approach to the frontier process is anticipated to be extensible to multi-server and many-server scenarios, as suggested in the paper, particularly in the critically loaded and overloaded regimes with exponential service/lead times—a direction directly connected to recent developments in mean-field and many-server queueing theory.

Conclusion

This work establishes, under minimal distributional assumptions, the fluid limits for key performance metrics in G/G/1+G queues governed by the non-preemptive EDF-b discipline, using a deterministic barrier Skorokhod reflection framework. The explicit formulae for fluid limits of queue lengths, abandonments, and empirical lead time measures deepen the theoretical understanding of deadline-oriented queuing and provide concrete methodologies for analyzing large-scale time-sensitive service systems.

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