Covariance Structure

Useful characterizations of the aggregate and single-source bandwidth processes are their
(auto)covariance functions. The covariance function may help in evaluating the fitting. We now show that we can effectively compute the covariance function for our traffic source model.

Let $\{B(t):t \geq 0 \}$ and $\{B^i(t): t \geq 0\}$ be stationary versions of the aggregate and source-i bandwidth processes, respectively. Assuming that the single-source bandwidth processes are mutually independent, the covariance function of the aggregate bandwidth process is the sum of the single-source covariance functions, i.e.,  

$$R(t) \equiv \mbox{Cov}( B(0), B(t) ) = \sum^n_{i = 1} \mbox{Cov}(B^i(0), B^i(t))~.$$ (47)

Hence, it suffices to focus on a single source, and we do, henceforth dropping the superscript i.

In general,

 

R(t) = S(t) - m²,

where the steady-state mean m is as in (30) and (31) and

where g_je(x) = G^c_j(x)/m(G_j) is the density of G_je, and the second term captures the effect of the within-level variation process. In (56) b_j is the bandwidth in level j, p_j is the steady-state probability of level j, g_je(x) = G^c_j(x)/m(G_j) with G_j the level-j holding-time cdf and m(G_j) its mean, and P_jk(t|x) is the transition probability, whose matrix of Laplace transforms is given in Theorem 3.1 of [10]. We can thus calculate S(t) by numerically inverting its Laplace transform

To treat the second term in (58), we can assume an approximate functional form for the covariance of the within-level variation process W(t). For example, if  

$$Cov (W_j (0), W_j (t)) \approx \sigma_j^2 e^{-\eta_j t} ~, \quad t \geq 0 ~,$$ (49)

then

$$\sum_{j=1}^J p_j \int_0^\infty e^{-st} G_{je}^c (t) Cov (W_j (0), W_j (t)) dt$$

$$~~~~ \approx \sum_{j=1}^J p_j \sigma_j^2 \int_0^\infty e^{-(s + \eta_j )t} G_{je}^c (t) dt$$

 

$$~~~~ = \sum_{j=1}^J p_j \sigma_j^2 \displaystyle\frac{(1- \hat{g}_j (s+ \eta_j ))}{(s+ \eta_j) m(G_j )} ~.$$ (50)

Thus, with approximation (59), we have a closed-form expression for the second term of the transform $\hat{S}(s)$ in (58). For each required s in $\hat{S}(s)$, we need to perform one numerical integration in the first term of (58), after calculating the integrand as a function of x.

A major role is played by the asymptotic variance $\int_0^\infty R(t)dt$.For example, the heavy-traffic approximation for the workload process in a queue with arrival process $\int_0^t B(u) du,~ t \geq 0$,depends on the process $\{B(t):t \ge 0 \}$ only through its rate EB(0) and its asymptotic variance; see Iglehart and Whitt [17]. The input process is said to exhibit long-range dependence when this integral is infinite. The source traffic model shows that long-range dependence stems from level holding-time distributions with infinite variance.

Theorem 7132

If a level holding-time cdf G_j has infinite variance, then the source bandwidth process exhibits long-range dependence, i.e.,

$$\int_0^\infty R(t)dt = \infty~.$$

Proof. In (57) we have the component

$$\int_0^\infty g_{je}(x) P_{jj} (t\vert x) dx ~,$$

which in turn has the component

$$\int_0^\infty g_{je} (x) G_j^c (t\vert x) = G_{je}^c (t)~,$$

but

$$\int_0^\infty G_{je}^c (t) dt = \infty$$

if G_j has infinite variance. (As can be seen using integration by parts, the integral is the mean of G_je; see p. 150 of Feller [13]. In general, G_je has $k^{\rm th}$ moment m_k+1 (G_j )/(k+1)m₁ (G_j ), where m_k (G_j ) is the $k^{\rm th}$ moment of G_j.)

Note that if approximation (59) holds, then the level process contributes to long-range dependence, but the within-level variation process does not, because

$$\sum_{j=1}^J p_j \int_0^\infty G_{je}^c (t) Cov (W_j (0), W_n (t)) dt$$

$$~~~ \approx \sum_{j=1}^J p_j \sigma_j^2 \displaystyle\frac{(1- \hat{g}_j (\eta_j ))}{\eta_j m(G_j )} < \infty ~.$$