[Paper Review] Fluid limits of many-server queues with state dependent service rates

Anup Biswas|arXiv (Cornell University)|Jun 25, 2012

Advanced Queuing Theory Analysis7 references3 citations

TL;DR

This paper establishes the existence and uniqueness of a fluid limit for a many-server queueing system with general service time distributions and state-dependent service rates, where server speeds adapt based on system load. Using measure-valued processes tracking residual service times, the authors derive a fluid model equation that characterizes the limiting behavior under suitable regularity conditions on the service rate function and arrival process.

ABSTRACT

We study a many-server queuing system with general service time distribution and state dependent service rates. The dynamics of the system are modeled using measure valued processes which keep track of the residual service times. Under suitable conditions, we prove the existence of a unique fluid limit.

Motivation & Objective

• To analyze the asymptotic behavior of many-server queues with general service time distributions and dynamic service rates that depend on system state.
• To extend fluid limit theory beyond constant-rate service models to capture realistic scenarios in call centers and service systems where service speed adjusts to congestion.
• To establish a rigorous fluid limit for a GI/G/n queue with state-dependent service rates using measure-valued processes tracking residual service times.
• To provide a deterministic fluid approximation that captures the macroscopic behavior of large-scale queueing systems under dynamic service rate policies.

Proposed method

• Models the system using a measure-valued process $\mathcal{Z}^n_t$ that tracks the residual service times of customers in service.
• Introduces a fluid-scaled process $\bar{X}^n_t = \frac{1}{n}X^n_t$ to analyze the system under diffusion scaling.
• Derives a fluid model equation (Equation 9) that characterizes the limit of the scaled measure-valued process $\frac{1}{n}\mathcal{Z}^n_t$.
• Uses a fixed-point argument and Gronwall's inequality to prove existence and uniqueness of solutions to the fluid model equation.
• Applies the Arzelà–Ascoli theorem to establish relative compactness of the sequence of fluid-scaled processes.
• Relies on techniques from [18] and [10], adapting them to handle dynamic service rates without requiring explicit compensators for departure processes.

Experimental results

Research questions

• RQ1 Does a fluid limit exist for a many-server queue with general service time distributions and state-dependent service rates?
• RQ2 Can the fluid limit be uniquely characterized by a deterministic integral equation when service rates depend on system load?
• RQ3 How does the dynamic adjustment of service rates affect the convergence and stability of the fluid limit?
• RQ4 What conditions on the service rate function ensure the existence and uniqueness of the fluid limit?
• RQ5 Can the measure-valued process tracking residual service times be used to derive a tractable fluid approximation in the presence of state-dependent service rates?

Key findings

• The fluid limit of the scaled measure-valued process $\frac{1}{n}\mathcal{Z}^n_t$ exists and is uniquely determined by a fluid model equation (Equation 9).
• The solution to the fluid model equation is shown to exist and be unique via a fixed-point argument and Gronwall’s inequality, under Lipschitz and boundedness conditions on the service rate and related functions.
• The sequence of fluid-scaled processes is relatively compact, ensuring convergence along subsequences.
• The fluid limit is characterized by an integral equation involving the cumulative service rate and residual time distributions, generalizing prior results with constant service rates.
• The approach avoids the need for explicit compensators of departure processes, which is a key challenge in systems with dynamic service rates.
• The results extend fluid limit theory to more realistic models of service systems where server speed adapts to congestion, such as in call centers with dynamic staffing or feedback-based rate adjustment.