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Fig. 13.4 Transient loss versus steady-state loss for C 0:8 and B 1, 2, 3, 4. The horizontal axis represents w scaled by B. (a) The loss probability of the Markov process. (b) The expected loss ratio of the Markov process. (c) The loss probability of LRD process I. (d) The expected loss ratio of LRD process I. (e) The loss probability of LRD process II. (f) The expected loss ratio of LRD process II.



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From the numerical results, we see that, in the steady state, the loss behavior of each of the above two-state traf c processes is indeed signi cantly different from its loss behavior in the transient state. Since steady-state loss measures are weighted by the marginal distributions of the underlying traf c process, the random variance in transient loss measures is completely averaged out in the steady state. As a result, the steady-state loss measures are simply some constants. Unlike the loss measures in the steady state, the transient loss measures are functions of the initial value w that summarizes the history of the system. For all the

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A Java JNA wrapper for Tesseract OCR API. Tess4J is released and distributed under the Apache License, v2.0 and is also available from Maven Central ...

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traf c processes considered, we observe that as the initial value varies, the transient loss measures will become greater than the corresponding steady-state loss measures That is, steady-state loss measures can underestimate actual loss in the transient state As the buffer size B or the service rate C increases, we can still observe such underestimation Although increasing C can decrease the probability for w to take larger values, due to the randomness of on and off periods, it is still possible for w to take values greater than some critical value beyond which the steady-state loss measures will underestimate transient loss We even observe that if C (or B) is large enough, then for all w P 0; B , the steady-state loss measures are strictly less than the corresponding transient loss measures; for example, see the case of C 0:9 in Fig 133(d) 13.

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5 THE IMPACT OF LONG-RANGE DEPENDENCE IN NETWORK TRAFFIC If bandwidth allocated to a traf c source equals the peak bit rate of the source, then whether there exists long-range dependence in the underlying traf c has no impact at all on traf c modeling and engineering So in the following discussions, we assume that the bandwidth allocated to an LRD traf c source is less than its peak rate First of all, perhaps the most important issue is whether long-range dependence is relevant to traf c engineering [6, 9, 10, 16, 17] It has been shown that in the steady state, when the buffer size is small, long-range dependence may not have signi cant impact [10, 17] However, the above conclusion is drawn in the steady state, so it may not be necessarily valid beyond the region of steady state.

Our analysis and numerical results have shown that, in the transient state, there exists signi cant difference in traf c loss behavior between traditional Markov models and LRD models Analysis results presented in Section 133 describe a scenario regarding transient loss performance impact of long-range dependence in multistate uid traf c In nontrivial cases, after the uid enters a state, it will pour more traf c into the buffer than what the bandwidth can transmit However, according to Lemma 1331, traf c loss will not take place until the time spent by the uid in the state exceeds the value of t w , which is the instant after which loss of traf c begins From Theorems 1336 and 133.

7, we see that there exists some critical value of t w and y determined only by the property of the underlying LRD traf c, beyond which Markov models can underestimate loss performance degradation for LRD traf c Since large values of t w and y imply stringent loss performance requirements, the above results in fact say that, in the transient state, long-range dependence is more likely to have signi cant impact on applications with stringent loss constraints Although a small buffer may alleviate the impact of long-range dependence in the steady state [9, 10], it does not help in the transient state if the loss constraint is stringent Let us take the conditional loss probability as an example Since t w B w = r C depending on buffer and bandwidth allocated to the uid, it can be viewed as a control variable From Lemma 133.

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