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Suppose in addition that B is reversible; this will happen if the matrix P is reversible. Then since both the residual lifetime and the current age have distribution Fjke , E B t jB 0 E B t jB 0 . Consequently, B 0 x is a local maximum, at t 0, of E B t jB 0 . Now suppose that there are n independent sources. Then as in Duf eld and Whitt [14], it follows by use of an appropriate functional law of large numbers that, as n 3 I under regularity conditions, the stochastic paths of the B process converge to this mean path. Thus we can identify, asymptotically as n 3 I, t 0 as a hitting time for the level x. Thus, we can use Eq. (17.42) to approximate the probability of this hitting time and t in Eq. (17.51) to approximate the associated recovery time. Example 17.8.1. Consider homogeneous two-level sources, that is, j P f1; 2g, 0 b1 < b2 , p1 p2 1 with mean lifetimes m1 , m2 , and P11 P22 1 P12 1 P21 0. With n sources and B 0 x we can calculate the pj in Eq. (17.46) directly from the relation x n 1 b1 p2 b2 for x P fnb1, p n b b1 b2 ; . . . ; b1 n 1 b2 ; nb2 g. Then r x n 1 b2 b1 =m1 p2 b1 b2 =m2 p nb2 x =m1 x nb1 =m2 : 17:52

CHAPTER SECTIONS: 1 / 2 / 3 / 4

As a concrete example, we let b1 1, b2 5, m1 3, m2 1, giving a mean bandwidth per source of m1 b1 m2 b2 = m1 m2 2. We also let n 50 and c 150. The parameters of the example were chosen as a caricature of video traf c on an OC3 link: take bi ; x; c in Mb=s, mi in seconds. We present in Table 17.1 some values of n1 ; n2 , the number of sources in each level for a given x, the approximate probability e I x of demand exceeding x (using I from Eq. (17.44)), and t. For comparison we give also the exact recovery time for the mean, calculated by using numerical transform inverstion methods [2], for particular

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Homogeneous Two-Level Sourcesa Initial Total Demand x 150 162 174 186 198 Steady-state Probability of x e I x 7.5 10 4 1.9 10 5 2.4 10 7 1.4 10 9 3.5 10 12 Linear Approx. 0 0.15 0.24 0.31 0.37 Recovery Time t Inversion Exponential Duration 0 0.16 0.29 0.41 0.50 Pareto Duration 0 0.19 0.41 0.64 0.91

Java has evolved more quickly than any other computer language in widespread use As the language itself has evolved, so, too, has the security model Here are some lessons we have learned as we have watched Java's approach to security change Type safety alone is not security 1, "Mobile Code and Security: Why Java Security Is Important," and 2, "The Base Java Security Model: The Original Applet Sandbox," explain why type safety is critical to Java's security model 5, "Attack Applets: Exploiting Holes in the Security Model," shows what happens when type safety is compromised Java security is much more than type safety, however Think of type safety as the foundation of a complete Java security solution The foundation is essential, but much needs to be built on top of the foundation to make a useful building Real security is more difficult than it sounds.

a Approximate hitting probabilities of aggregate demand x, together with recovery time t of mean from x, by linear approximation, and exact for (1) exponential duration and (2) Pareto duration of higher level; see Example 17.8.1.

models of level durations with the same means: (1) both level durations exponentially distributed; (2) lower level exponential, upper level duration Pareto with exponent 1.5, and hence cdf Gc x 1 2x 3=2 in order to give mean m2 1. The Pareto density g x a 1 x 1 a has a Laplace transform aes sa G a; s , where G a; z is the incomplete gamma function G a; z I

Recognizing Responsibilities Using Third-Party Security Review Summary Appendix A WWW Resources for Authentication, Authorization, and Access Control News and Information Appendix B Important Access Control and Security Terms Appendix C Critical Success Factors for Controls Design

t a 1 e t dt:

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