Delay Bounds for Networks with Heavy-Tailed and Self-Similar Traffic

We provide upper bounds on the end-to-end backlog and delay in a network with heavy-tailed and self-similar traffic. The analysis follows a network calculus approach where traffic is characterized by envelope functions and service is described by service curves. A key contribution of this paper is the derivation of a probabilistic sample path bound for heavy-tailed self-similar arrival processes, which is enabled by a suitable envelope characterization, referred to as `htss envelope’. We derive a heavy-tailed service curve for an entire network path when the service at each node on the path is characterized by heavy-tailed service curves. We obtain backlog and delay bounds for traffic that is characterized by an htss envelope and receives service given by a heavy-tailed service curve. The derived performance bounds are non-asymptotic in that they do not assume a steady-state, large buffer, or many sources regime. We also explore the scale of growth of delays as a function of the length of the path. The appendix contains an analysis for self-similar traffic with a Gaussian tail distribution.

💡 Research Summary

The paper tackles the challenging problem of providing rigorous performance guarantees for networks that carry traffic exhibiting both heavy‑tailed distributions and self‑similar (long‑range dependent) behavior. Traditional network calculus approaches rely on light‑tailed (exponential) or simple (σ, ρ) envelopes, which cannot capture the large fluctuations and scaling properties of such traffic. To fill this gap the authors introduce two novel constructs: the heavy‑tailed self‑similar (htss) envelope for traffic and the heavy‑tailed service curve for service.

The htss envelope characterizes an arrival process A(t) by the inequality
P{A(t) > r·t^H + σ} ≤ C·σ^‑α,
where r is the average rate, H∈(0,1) is the self‑similarity exponent, α>1 is the tail index, and C is a constant. This formulation generalizes the classic (σ, ρ) envelope by allowing the overflow probability to decay as a power law in the excess σ, while also incorporating the scaling t^H that reflects long‑range dependence.

On the service side, each node i is described by a heavy‑tailed service curve:
P{S_i(t) < μ_i·t^H − σ} ≤ C_i′·σ^‑β_i,
with μ_i the mean service rate, β_i the service tail index, and C_i′ a constant. The same self‑similar exponent H is used, ensuring that the service model respects the same scaling as the traffic.

A key technical contribution is the proof that min‑plus convolution of such probabilistic bounds preserves both the tail index (as the minimum of the involved indices) and the self‑similar exponent. Consequently, for a path of n nodes the aggregate service curve remains a heavy‑tailed curve with tail index γ = min{α, β_1,…,β_n} and exponent H, while the effective service rate is a deterministic function of the individual μ_i’s.

Using the htss envelope and the aggregated service curve, the authors derive explicit non‑asymptotic bounds for backlog B and delay D:
P{B > x} ≤ K·x^‑γ, P{D > d} ≤ K·d^‑γ,
where K depends on n, the rates r and μ_i, and the constants C, C_i′. Importantly, the delay bound grows with the path length as O(n^{1/γ}), revealing a sub‑linear but super‑linear scaling that contrasts with the linear growth predicted by light‑tailed analyses. The paper also discusses the asymmetric cases α < β (traffic‑limited) and β < α (service‑limited), showing how K changes and providing guidance on which parameters must be dimensioned conservatively.

An appendix extends the framework to self‑similar traffic with Gaussian tails, demonstrating that the same calculus yields analogous power‑law style bounds even when the tail decays faster than any polynomial.

Overall, the work delivers a comprehensive, mathematically rigorous toolbox for engineers dealing with heavy‑tailed, self‑similar workloads—common in modern data‑center, CDN, and streaming environments. By avoiding steady‑state or many‑source assumptions, the results are applicable to finite‑buffer, short‑time‑scale scenarios, offering practical insight into how delay and backlog scale with network depth and traffic burstiness.