A Virtual Queue Approach for Online Estimation of Loss Probability Based on MVA Theory

Abstract: In network quality of service provisioning, premium services generally require to keep a very small loss probability, which is infeasible to measure directly. The proposed virtual queue scheme estimates the small packet loss probability of a real queueing system by measuring queue statistics in a set of separate virtual queues. A novel scaling property between the real queue and the virtual queues is deduced on the basis of the maximum variance asymptotic (MVA) theory. The new scheme retains the high accuracy and wide applicability of the MVA method for aggregated traffic while avoiding the high computational complexity in a direct application of the original MVA analysis in real time. This makes it suitable for online measurement applications such as network performance monitoring and measurement-based admission control.

• The paper introduces a virtual queue estimator that maps rare tail events to statistically feasible events using a novel MVA scaling property.
• The methodology leverages a variance reduction technique, achieving lower estimation variance for loss probabilities in the range 10⁻⁴ to 10⁻⁸ compared to direct measurement.
• The approach supports rapid, online loss probability estimation in high-speed, aggregated traffic environments, facilitating real-time network monitoring and admission control.

A Virtual Queue Method for Online Estimation of Loss Probability via Maximum Variance Asymptotics

Introduction

This paper introduces a virtual queue (VQ)-based methodology for real-time, online estimation of loss probability in networked queueing systems, particularly targeting rare loss events ($10^{-6} \sim 10^{-5}$) relevant for high QoS regimes. The approach synergistically leverages Maximum Variance Asymptotic (MVA) queueing analysis and an innovative probabilistic scaling property, thereby retaining the analytical rigor and generality of classical MVA theory while drastically reducing implementation complexity. This enables applicability in measurement-based admission control and performance monitoring for high-speed, aggregated-traffic environments.

Background: Maximum Variance Asymptotic (MVA) Theory

The MVA method estimates the queue length tail probability and buffer overflow probability for single-server, FIFO systems under the assumption of Gaussian aggregated input (justified via the central limit theorem for high fan-in). A key theoretical feature is the identification of the dominant time scale (DTS), i.e., the interval $\tau$ which minimizes the normalized buffer exceedance function $g(q, c, t)$, hence dominating overflow events. Specifically, buffer overflow probability $P\{Q > q\}$ can be approximated as an exponential function of $[g(q, c, \tau)]^2$.

While MVA exhibits high accuracy across a spectrum of traffic models—including LRD streams—it requires real-time measurement of traffic variance across a densely sampled set of time scales, with overall complexity $O(n)$ per estimation window. This is infeasible for real-time deployment on high-speed links with small packet service times and many relevant observation scales.

Methodology: Virtual Queue Estimator with Loss Probability Mapping

VQ Construction

The paper develops an estimator utilizing three VQs:

• VQ1 (bufferless, same service rate as real queue) directly measures $P_\text{L}(0)$.
• VQ2 (infinite buffer, same service rate) measures $P\{Q>0\}$.
• VQ3 (scaled buffer size $x'$ and service rate $c'$, same traffic) is used to indirectly estimate the tail probability $P\{Q>x\}$.

The key insight is the elimination of assumed knowledge about the buffer-size-to-loss-probability mapping, replaced with a derivation based on the scaling invariance property of MVA analysis.

Scaling Property

The authors show that by scaling the buffer threshold and service rate as:

• $x' = x / \alpha$
• $c' = c / \alpha + (1 - 1/\alpha) r$

(where $\alpha > 1$ and $r$ the mean traffic rate), the dominant time scale is preserved, and the rare tail event for the real system is mapped onto a more probable event in VQ3:

$P\{Q > x\} \approx [P\{Q'>x'\}]^{\alpha^2}$

Thus, measurement of a statistically feasible event in VQ3, combined with the above scaling, yields an accurate estimation of an extremely rare event in the real system.

Variance Reduction

A variance analysis is performed, quantifying the estimator's relative variance reduction $\eta$ compared to direct measurement, under binomial statistics for event counts. Notably, for $P\{Q>x\}$ in the range $10^{-4} \ldots 10^{-8}$, the variance reduction is substantial. The minimal variance is attained for an optimal choice of $\alpha$, corresponding to $P\{Q'>x'\} \approx 0.2032$. The estimator's stability and insensitivity to the precise value of $\alpha$ (for $\alpha \geq 2.0$) are advantageous for practical deployment.

Performance Evaluation

Extensive simulations are conducted on single-server FIFO systems with two types of multiplexed input: aggregated on-off (audio traffic) and Markov Modulated Poisson Process (MMPP, video traffic), using empirically grounded parameters. The methodology is tested across a wide range of buffer sizes and traffic loads.

Accuracy

With large measurement windows (25000 s), the VQ estimator accurately tracks the true loss probability, typically within one order of magnitude. This matches the intrinsic precision limits of the MVA methodology when applied to aggregated and LRD traffic, and outperforms alternative real-time feasible methods.

Transient Response

Experiments increasing the measurement window demonstrate rapid estimator convergence and robust transient behavior, with stable, low-variance results achieved with as little as 3 s of observation. In contrast, direct measurement of rare-event loss yields prohibitively long convergence times—packet loss events may not even occur within practical window sizes for $P_\text{L}(x) < 10^{-5}$. The VQ estimator thus supports reliable, low-latency on-line estimation even for tail events.

Implications and Future Work

The presented scheme constitutes a significant advance for online, real-time estimation of buffer overflow probabilities in packet-switched networks. Its computational efficiency ($O(1)$ per estimation window), combined with generality over traffic models and scalability to low-probability regions, allows practical integration into network monitoring and measurement-based control systems. The avoidance of ad hoc assumptions (e.g., buffer-size-to-loss mappings) increases robustness in diverse, dynamic contexts.

Theoretically, the introduction of scaling properties within the MVA framework broadens the applicability of asymptotic queueing analyses to operational settings, moving beyond their traditional offline, planning-focused role. The VQ paradigm as extended here suggests further research directions, including adaptation to multi-class queues, networks of queues, and exploration of additional scaling rules or mappings derived from other queueing-theoretic asymptotics.

Conclusion

The paper presents a VQ-based scheme for online estimation of small loss probabilities, grounded in a rigorous scaling property derived from MVA theory. It delivers high accuracy comparable to MVA without the computational barriers of traditional methods and demonstrates clear practical value in real-time, network-oriented applications. These results point toward a fruitful connection between advanced queueing analysis and real-time network measurement, with strong potential for further extension and integration.