[Paper Review] Queueing networks with a single shared server: light and heavy traffic

Marko Boon, R.D. van der Mei+1 more|TU/e Research Portal|Aug 1, 2014

Advanced Queuing Theory Analysis9 references4 citations

TL;DR

This paper proposes a closed-form approximation for mean waiting times in a queueing network with a single shared server, gated service, and customer routing. By deriving exact heavy- and light-traffic limits and interpolating between them, the method provides accurate, scalable estimates across all load conditions, validated through numerical examples including tandem queues with switch-over times.

ABSTRACT

We study a queueing network with a single shared server, that serves the queues in a cyclic order according to the gated service discipline. External customers arrive at the queues according to independent Poisson processes. After completing service, a customer either leaves the system or is routed to another queue. This model is very generic and finds many applications in computer systems, communication networks, manufacturing systems and robotics. Special cases of the introduced network include well-known polling models and tandem queues. We derive exact limits of the mean delays under both heavy-traffic and light-traffic conditions. By interpolating between these asymptotic regimes, we develop simple closed-form approximations for the mean delays for arbitrary loads.

Motivation & Objective

• To analyze a generalized queueing network with a single shared server, gated service, and probabilistic customer routing.
• To derive exact asymptotic expressions for mean waiting times under both light-traffic and heavy-traffic conditions.
• To develop a simple, closed-form approximation for mean delays across all system loads by interpolating between light- and heavy-traffic limits.
• To validate the approximation’s accuracy through numerical examples, including tandem queue configurations with switch-over times.

Proposed method

• Model the system as a polling network with gated service and customer routing, where each customer is routed to another queue or exits after service.
• Derive the total offered load and effective service time using linear equations based on routing probabilities and service time moments.
• Establish heavy-traffic limits using cycle time analysis and length-biased sampling, yielding a closed-form expression for mean waiting time as ρ→1.
• Derive light-traffic limits by conditioning on customer origin (external or routed), resulting in a closed-form expression as ρ↓0.
• Construct an interpolation-based approximation: $ W_i^{\text{approx}} = \frac{w_i^{\text{LT}} + (w_i^{\text{HT}} - w_i^{\text{LT}})\rho}{1 - \rho} $, ensuring exactness at ρ=0 and ρ→1.
• Validate the approximation numerically using a three-queue tandem model with deterministic switch-over times and routing probabilities.

Experimental results

Research questions

• RQ1 How do mean waiting times behave in a single-server queueing network with customer routing under heavy-traffic conditions?
• RQ2 What is the exact light-traffic limit of mean waiting times when the system load is near zero?
• RQ3 Can a simple closed-form approximation be constructed that accurately estimates mean waiting times across all load levels by combining light- and heavy-traffic limits?
• RQ4 How does the routing structure affect the dependence of mean waiting times on system parameters such as service and switch-over times?
• RQ5 Does the proposed interpolation approximation satisfy structural properties like the pseudo-conservation law for mean waiting times?

Key findings

• The heavy-traffic limit of the mean waiting time at queue $ i $ is $ (1 - \rho)\mathbb{E}[W_i] \to \left(r + \frac{\tilde{b}^{(2)}}{2\delta\tilde{b}^{(1)}}\right)\frac{\delta_i}{\tilde{b}_i\hat{\gamma}_i} $ as $ \rho \uparrow 1 $, showing insensitivity to visit order and dependence only on total switch-over and effective service time moments.
• The light-traffic limit of the mean waiting time at queue $ i $ is $ \mathbb{E}[W_i] \to \frac{\lambda_i}{\gamma_i}\frac{r^{(2)}}{2r} + \sum_{j=i-N}^{i-1}\frac{\gamma_j p_{j,i}}{\gamma_i}\sum_{k=j}^{i-1}r_k $ as $ \rho \downarrow 0 $, reflecting residual switch-over and routing delays.
• The proposed interpolation approximation $ W_i^{\text{approx}} $ is exact at both $ \rho \downarrow 0 $ and $ \rho \uparrow 1 $, and satisfies the pseudo-conservation law for mean waiting times.
• Numerical results show high accuracy across all load levels, with relative errors below 2% for $ \rho = 0.1 $ and $ \rho = 0.9 $, and under 1% for $ \rho = 0.01 $ and $ \rho = 0.99 $ in the three-queue example.
• The model generalizes well to special cases such as tandem queues and feedback systems, as demonstrated by the numerical example with $ p_{1,3} = p_{2,3} = p_{3,0} = 1 $.
• The approximation remains robust even when switch-over times are introduced, as shown in the example with $ r_1 = 0, r_2 = r_3 = 2 $.