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excellent for sequences C and D, good for sequence A, and adequate for sequence B. The negative-binomial distribution is the discrete analog of the gamma distribution, and a discretized version of the latter can be used when it is more convenient to do so. Figure 12.2 shows the autocorrelation functions. The ordinate has a log scale, so geometric functions will appear as straight lines. The geometric property holds for at least 100 lags (2.5 seconds) for sequences B, C, and D, and for 50 lags for sequence A. For lags larger than 250, the geometric function underestimates the autocorrelation function. We examined sequences A, B, and C and concluded they possess longrange dependence. Since the autocorrelation functions shown in Fig. 12.2 are so large for small lags, it seem intuitive (to us, at least) that the short-range correlations should be the important ones to capture in a source model. We propose using the geometric function rk for the autocorrelation function. Since the negative-binomial and gamma distributions are speci ed by two parameters, these parameters can easily be esimated from the mean and the variance of the number of cells per frame by the method of moments. Only those two moments and the correlation coef cient (r) are needed to specify the key properties of VBR teleconference traf c. The correlation coef cient can be estimated from the geometric portion of the autocorrelation function by taking logarithms and doing a linear regression. 12.2.2 The DAR Model



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In this example, a web server gets its HTML files from an NFS server, and its forms rely on a user database. The database, file server, and network management applications all rely on a name service. Furthermore, the database requires help from an authentication service, which is also called by the web server. This example is oversimplified; we lump all file server access into a single bubble, when in reality most web sites are driven by multiple servers. We also left out routers, configuration tables, and other software-driven services that could affect reliability. These dependency diagrams should be directed, acyclic graphs. By directed, we mean that the relationship between the dependent and independent service is clear; an arrow points to the item that is dependent on something else. Acyclic refers to the property that there are no circle tours through the graph; if you start at any node, you can never get back to that same node simply by following the directed arrows. There are well-known algorithms for proving that graphs are without cycles; however, for the dozen or so dependencies you ll find, you can trace the possible paths out by hand. If you encounter a cycle, as shown in Figure 9.11, you have a deadly embrace between network services. If one goes down, it may crash another, leading to a deadlock condition that can t be broken through normal, automated methods. The example shows a pair of file servers that cross-mount each other; when one crashes, the other is taken out as well. Look at the number of graph nodes that have no incident arrows; these are the root or independent services. Typically, it s naming, DNS and LDAP, and occasionally an authentication server. There are many design points to be drawn from these types of graphs:





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Our rst investigations of these sequences with Tabatabai [16] and Heeke [18] focused on multiplexing issues. First, we established that the time series were stationary. This was done by examining plots of smoothed versions of the time series and boxplots of many partitions of the time series. Next, we showed a Markov chain provided a good description of the time series. This was done via simulations as described in Section 12.2.2.1. This means that the marginal distributions of the time series can be viewed as the steady-state distributions of the Markov chain. A Markov chain that has a geometric autocorrelation function and whose steady-state distribution can be speci ed is the DAR(1) process introduced by Jacobs and Lewis [20]. The only member of the DAR k family that is used here is the DAR(1), so the (1) will be deleted. The transition matrix is given by P rI 1 r Q; 12:1

Make sure the root services are highly replicated and available, because failures will trickle through to appear in the applications that are at the end of the dependency graph.

Evaluation of g*

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We work under the assumption that the scaling sequence fv*; t 1; 2; . . .g satis es t Condition (ii) of Proposition 9.6.2. Fix y > 0. Setting z 0 in Eq. (9.78) we obtain b y c rin K g y : Next, going back to the expression (9.49) we note that vt yx L y sup lim inf inf x>y h tx y>0 t3I vt sup lim inf inf x c rin y x>y h tx 0<y<1 t3I vt y lim inf inf x c rin t3I x>y h tx so that vt x c rin a y Kg y lim inf inf t3I x>y h tx 9:79

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