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Now let us consider a link with nite buffer B and bandwidth C. Suppose that the link carries a uid traf c process R t . Recall that tn ; tn 1 is the nth interval between transitions of the state of R t . The modeling assumption on R t is determined by the distribution of Dtn , the length of the interval tn ; tn 1 . We are curious about the transient loss behavior of R t at the link under two con icting assumptions in traf c modeling: R t is a Markov process. Consequently, Dtn is exponentially distributed. R t is an LRD process, which implies that Dtn obeys some (asymptotically) heavy-tailed distribution. To compare the transient loss behavior of Markov and LRD traf c, we consider the following two cases. 13.3.1 Loss Behavior in Single States



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To compare the loss behavior of multistate Markov and LRD uids in single states, we use the traf c loss probability and the expected traf c loss ratio de ned in the interval tn ; tn 1 where n ! 1, as the transient loss measures. Both the above loss measures are conditioned on the amount of traf c left in the system at tn . Suppose R n r. for convenience of exposition, we simply let tn 0 and tn 1 S, where S is a random variable with distribution FS s PfS sg that may depend on r. The amount of traf c in the system at time t is represented by w t , where t P 0; S . Denote by w 0 the initial amount of traf c left in the system at the beginning of interval 0; S . Accordingly, the loss probability and the expected loss ratio are denoted, respectively, by PflossjR n r, w 0 wg and E ljR n r, w 0 w , where l is the fraction of traf c lost in 0; S . The following two lemmas show how to compute the loss measures. Lemma 13.3.1 8 r C; < 0; PflossjR n r; w 0 wg 1; r > C; w B; : PfS > t w g; r > C; w < B; 13:1





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where t w B w : r C 13:2

temperature and humidity levels in your computing environments Failure of an air-conditioning unit can result in the total failure of the data center The effects of extended exposure to extreme heat or humidity can be subtle or, in some cases, can be quite noticeable The clearances between disk heads and their associated disk platters have become remarkably small Heat can cause expansion in disk head arms, causing the disk head to make contact with the platter, destroying your disk and its data The glue that holds the disk heads in place can soften or melt under extreme conditions, causing it to move out of line In more extreme cases, solders could melt or the fire suppression system could go off, giving all the computers a nice, cool bath or sucking all the oxygen out of the room and causing a potential personnel problem.

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Proof. Clearly, if r C, then PflossjR n r, w 0 wg 0, and r > C together with w B implies that PflossjR n r, w 0 wg 1. On the other hand, if r > C and w < B, then the random event that traf c loss due to buffer over ow occurs in the interval is equivalent to existing t P 0; S such that w t B for t P t; S and hence PflossjR n r, w 0 wg PfS > tg. It is easy to see that t B w : r C j

Table 7.1 (continued)

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COMMENT 13.3.2. The physical meaning of t w is the instant in 0; S after which traf c loss due to buffer over ow begins immediately in the interval. Since t w depends on bandwidth and buffer allocated to the underlying traf c, it can be viewed as a control parameter. Lemma 13.3.3 8 > 0; < u0 E ljR n r; w 0 w u0 t w > P S> du; : u0 u 0 where t w is given by Eq. (13.2) and u0 1 C : r 13:4 r C; 13:3

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