Dr. Nick Laskin nicklaskin.ca
Carleton University Breakthrough (2000–2002)

Alternative Fractional Stochastic Processes
& Internet Router Buffer Modeling

An alternative framework to the Kolmogorov-Mandelbrot fractional Brownian motion, extending the Norris buffer model for real-world network data accumulation.

1. The Carleton Breakthrough (2000–2002)

During his research tenure at Carleton University in 2000–2002, Dr. Nick Laskin developed a fundamental alternative for introducing and studying fractional stochastic processes.

Historically, fractional stochastic motion was defined using the classical Kolmogorov-Mandelbrot method for fractional Brownian motion (fBm). While Mandelbrot's fBm effectively captures long-range dependence, its Gaussian increment structure fails to model the extreme, non-Gaussian heavy tails observed in complex real-world data networks.

$$dY_H(t) = f(t, H) \, dB(t) + \int_0^t K_\alpha(t, s) \, dL_\alpha(s)$$

Laskin's formulation provides a flexible operator-based alternative that covers a much broader family of non-Markovian stochastic processes, including fractional Poisson and fractional Lévy processes.

2. Internet Traffic Statistics & Self-Similarity

High-speed internet packet arrivals do not follow simple Poisson distributions. Empirical measurements across enterprise routers demonstrate:

  • Self-Similarity: Traffic patterns look statistically identical across time scales ranging from milliseconds to hours.
  • Heavy-Tailed Bursts: Data arrivals exhibit extreme spike clusters that drastically exceed classical variance limits.
  • Long-Memory Dependence: Past packet arrivals strongly influence future queue buildup.

3. Laskin's Generalized Router Buffer Model

To assist network engineers in accurately sizing router RAM and avoiding catastrophic packet loss, Dr. Laskin created a generalized buffer model extending the classical Norris buffer model.

In the Norris model, buffer queue length $Q(t)$ is driven by Mandelbrot fBm. Laskin's generalized model incorporates fractional jump operators to yield an exact prediction for data accumulation $Q(t)$ and buffer overflow probability $P(Q > B)$:

$$P(Q(t) > B) \approx \exp\left( -C \cdot \frac{(B - (\lambda - C_{out})t)^{2 - \alpha}}{\sigma^\alpha t} \right)$$

This formulation allows network engineers to calculate the precise buffer depth $B$ required to guarantee quality-of-service (QoS) under heavy-tailed burst traffic.

4. Real-World Engineering Significance

Accurate RAM Sizing

Prevents router buffer bloat while eliminating under-provisioned memory crashes during DDoS or traffic spikes.

Non-Markovian Queueing

Replaces outdated M/M/1 queuing assumptions with exact fractional stochastic boundaries.

5. Interactive Router Memory & Queue Buffer Simulator

Compare packet queue accumulation $Q(t)$ across Poisson queues, Mandelbrot Norris fBm, and Laskin's Generalized Buffer Model under real-time parameter tuning.