Heavy-traffic limit for a feed-forward fluid model with heterogeneous heavy-tailed On/Off sources

Abstract

We consider a multi-station fluid model with arrivals generated by a large number of non-homogeneous heavy-tailed On/Off sources. If the model is feed-forward in the sense that fluid cannot flow from one station to other with lighter tail distributions, we prove that under heavy-traffic, the scaled workload converges in distribution to a reflected fractional Brownian motion process with a multi-dimensional Hurst parameter. As an application, we analyze the impact of having independent streams with variable parameters in high-speed telecommunication networks, on the asymptotic behavior of the maximum queue length.

Appendix

The invariance principle of Williams and its corollary, Theorem 4.1, and Corollary 4.3 [14], respectively, are stated and proved for a reflected Brownian motion, but these results, whose proofs rely heavily on the oscillation inequality (Theorem 5.1 [14]), do not depend on fact on the law of the process, as can be checked by following the steps of the proof. For convenience of the reader, we present the particular (simplified) version of Corollary 4.3 [14] we use in our proof:

Version of Corollary 4.3 [14]

Assume that matrix R satisfies (HR). For each positive integer n, let R ⁿ be a d×d matrix and W ⁿ,X ⁿ,Y ⁿ be processes with paths in defined on some probability space such that

(i)′:

W ⁿ(0)=0 and $W^{n}(t)\in\mathbb{R}^{d}_{+}$ for all t≥0, a.s.,

(ii)′:

W ⁿ=X ⁿ+R ⁿ Y ⁿ a.s.,

(iii)′:

X ⁿ converges in distribution as n→+∞ to some process X whose paths live in ,

(iv)′:

Y ⁿ has non-decreasing paths, and for each j=1,…,d, a.s., $Y^{n}_{j}(0)=0$ and $\int_{0}^{+\infty} 1_{\{W^{n}_{j}(s)>0\}}\,dY^{n}_{j}(s)=0$,

(v)′:

R ⁿ converges to R as n→+∞.

Then there exists ${\operatorname {\textsf {D}\!-\!\lim }_{n\to+\infty} (W^{n},X^{n},Y^{n})=(W,X,Y)}$, where the limit (W,X,Y) satisfies conditions $\mathrm{(i)}$, $\mathrm{(ii)}$ and $\mathrm{(iv)}$ of Definition 2 (with W(0)=X(0)=Y(0)=0), that is, W=X+RY is a Skorokhod decomposition.

Brief justification

First, note that we use Theorem 4.1 [14] in a particular situation in which the probability measure ν on $\mathbb{R}^{d}_{+}$ gives probability 1 to the point $0\in\mathbb{R}^{d}_{+}$, α ⁿ=γ ⁿ=δ ⁿ=0 for all n≥1, and hypothesis (iii) of Theorem 4.1 is replaced by (iii)′. The proof of this theorem does not uses any specific property of the Brownian motion process. Indeed, the tightness of sequence {X ⁿ} _n is a consequence of (iii)′ and {(W ⁿ,X ⁿ,Y ⁿ)} _n inherits tightness from it (the necessary and sufficient conditions for tightness given in Corollary 3.7.4 of Ethier and Kurtz [7] are verified). Moreover, by Theorem 3.10.2 [7], any (weak) limit point of {(W ⁿ,X ⁿ,Y ⁿ)} _n has continuous paths. Let (W,X,Y) be a (weak) limit point, that is, there is a subsequence $\{(W^{n_{k}},X^{n_{k}},Y^{n_{k}})\}_{k}$ such that $\operatorname {\textsf {D}\!-\!\lim }_{k\to +\infty} (W^{n_{k}},X^{n_{k}},Y^{n_{k}})=(W,X,Y)$. Then the Skorokhod representation theorem (Theorem 3.1.8 [7]) is used to replace the above sequence of processes by one that is term-by-term equivalent in distribution to the original one and which a.s. converges uniformly on compact intervals. With this simplification, it is easily seen that the limit triplet (W,X,Y) inherits properties (i), (ii) and (iv) of Definition 2 from properties $\mathrm{(i)}'$, $\mathrm{(ii)}'$, and $\mathrm{(iv)}'$, except for

$$\int _{0}^{+\infty} 1_{\{W_j(s)>0\}} dY_j(s)=0, $$

whose proof is technically more complicated and can be seen on pages 21 and 22 of [14]. Now, instead of using Theorem 3.1 [14] to ensure that all (weak) limit points of {(W ⁿ,X ⁿ,Y ⁿ)} _n have the same law, as is done in the proof of Theorem 4.1 [14], we use that by assumption (HR) on matrix R, the law of the pair (W,Y) is unique (see Remark 2), which gives the desired result. Combining this uniqueness with tightness, it follows that the whole sequence {(W ⁿ,X ⁿ,Y ⁿ)} _n converges in distribution to a triplet (W,X,Y) which satisfies conditions (i), (ii), and (iv) of Definition 2 (with W(0)=X(0)=Y(0)=0). □