Fig. 17.2 Recovery curves for two-level sources: linear approximation and numerical transform inversion with exponential and Pareto durations in the higher level; see Example 17.8.1.

various offered loads, the approximate probability e I x t of a demand at least x t as a function of t, where t c x t =r x t ; that is, x r is the demand from which the recovery time to the level c is t, using the linear approximation. Figure 17.3 shows that the two criteria together impose more constraints on what sets of

Duration (seconds)

Fig. 17.3 Design criteria: estimated probability of overdemand of at least duration t, for various offered loads; see Section 17.8.

sources are acceptable. Expressed differently, for the same probability of occurrence, rare congestion events can have very different recovery times.

Useful characterizations of the aggregate and single-source bandwidth processes are their (auto)covariance functions. The covariance function may help in evaluating the tting. We now show that we can effectively compute the covariance function for our traf c source model. Let fB t : t ! 0g and fBi t : t ! 0g 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; that is, R t  Cov B 0 ; B t

Cov Bi 0 ; Bi t :

17:54

Hence, it suf ces 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 Eq. (17.31) and (17.32) and S t  EB 0 B t I

17:55

bk Pjk tjx 17:56

c pj Gje t Cov Wj 0 ; Wj t ;

where gje x Gjc x =m Gj is the density of Gje , and the second term captures the effect of the within-level variation process. In Eq. (17.56) bj is the bandwidth in level j, pj is the steady-state probability of level j, gje x Gjc x =m Gj with Gj the level-j holding-time cdf and m Gj its mean, and Pjk tjx is the transition probability, whose matrix of Laplace transforms is given in Theorem 17.4.1. We can thus calculate S t by numerically inverting its Laplace transform ^ S s  I

J J P bj pj I c P ^ S t dt Gj x bk Pjk sjx dx j 1 m Gj 0 k 1 I J P c e st Gje t Cov Wj 0 ; Wj t dt: pj

j 1 0

17:57

To treat the second term on the right in Eq. (17.57), we can assume an approximate functional form for the covariance of the within-level variation process W t . For example, if Cov Wj 0 ; Wj t s2 e Zj t ; j then

t ! 0;

17:58

17:59

Thus, with approximation (17.58), we have a closed-form expression for the second ^ ^ term of the transform S s in Eq. (17.57). For each required s in S s , we need to perform one numerical integration in the rst term of Eq. (17.57), after calculating the integrand as a function of x. I A major role is played by the asymptotic variance 0 R t dt. For example, the heavy-traf c approximation for the workload process in a queue with arrival process t 0 B u du, t ! 0, depends on the process fB t : t ! 0g only through its rate EB 0 and its asymptotic variance; see Iglehart and Whitt [22]. The input process is said to exhibit long-range dependence when this integral is in nite. The source traf c model shows that long-range dependence stems from level-holding-time distributions with in nite variance. Theorem 17.9.1. If a level-holding-time cdf Gj has in nite variance, then the source bandwidth process exhibits long-range dependence, that is, I

R t dt I: