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Modeling and Analysis of an Expiration-Based.. - Hou, Pan, Li, Tang.. (2002) (Correct)
....within the hierarchical caching system. A. Average TTL Behavior We start our analysis with the most general form of a caching system. Suppose we are at a cache server of level 6 is the height of the tree. Denote the (remaining) TTL at the cache server as is a renewal process [8], with the renewal point starting at time # when just decreases to ( and , see Fig. 2) It should be clear that when . 0 , then for all 1 2 3 . This is because that under our basic model, an object maintained at a ....
S.M. Ross, Introduction to Probability Models, Fourth Edition, (Chapter 7), Academic Press, Inc., 1989.
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....the solution to (1.1) is given by ( V ) where N( is the cumulative distribution function (cdf) for a standard normal random variable. The inverse cdf is easy to either compute using mathematics software libraries or to look up in a table for the normal cdf (see for example [37, 52]) 1.2 Generalizing the method from the example The method from the example generalizes [43, 33] Suppose that the vector of relative changes R in the market risk factors is a multivariate normal random variable with the mean and the covariance matrix C. Similar to the example, assume that the ....
....preferred over models for prices. We consider a model in which the returns, sampled at equally spaced points in time, form a sequence fR i g i=1 of independent and identically distributed random variables. This means that stock price are discrete time Markov chains with an in nite state space [52]. Choosing di erent distributions gives di erent models in this family. By examining time series of stock prices, we get an indication about how the distribution should be chosen for a realistic model. Figure 2.1 shows the daily closing prices over 4 years for two Canadian stocks traded on the ....
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....= 2) Then the probing rate will use to generate a new sleeping period t s according to the probability density function f(t s ) e t s . 2.2.1 Explanation We now explain why the above algorithm keeps the aggregate around the desired d . From the probability theory [8], the exponentially distributed intervals between successive wakeups observe a Poisson process of wakeup events. Probings from di erent sleeping neighbors still construct a Poisson process, but with a parameter , the sum of all sleeping nodes rates i : i ; 3) We also tried a ....
....the average interval T s , we can derive the aggregate rate . This is exactly what (1) does. To obtain an accurate estimate that is close to the actual , the constant k in (1) has to be large enough. Because the intervals are i.i.d. random variables, we apply the central limit theorem [8] to estimate how large k should be for a reasonably good measurement. It turns out that when k 16, with over 99 con dence the measured average has only 1 error compared with the real value. We select k = 32 based on experimental studies. This also accounts for the short random time each ....
S. Ross. Introduction to Probability Models, 6th Edition. Acedemic Press, 1997.
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....arbitrary time. Following Takagi ( 15] pp. 77 79, 109) we have M [X(ii) n) I. m n 1 1 where L i. was defined in section 5.2. J Since we assume that the system is stable (ergodic) it follows that t ,oo t C = EC w.p. 1. On the other hand, from the theory of Renewal Reward Processes (e.g. [14], p. 279) it can be seen that t ,oo t C = w.p.1. n= EM Combining these equations, we obtain EC= n=l EM (In some references, such as in [15] Wald s Theorem is used improperly to obtain the same result. By using similar arguments, we obtain from which it follows that E[Qii(z) rx ii) ....
M.S. Ross, Introduction to Probability Models (Academic Press; 3rd ed.).
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....rate from other VLR areas and 1=m is the mean VLR residence time. Similarly, treating class 1 and class 2 users separately, K 1 and K 2 are Poisson random variables with means u;1 =m and u;2 =m , respectively. When u;1 m and u;2 m are both of order N , from the Chebyshev s inequality [15], we have N = K 1 = and K 2 = where notation O p ( Delta) is defined [8] as X n = O p (f(n) def = 8 ffl 0, 9 m 0 such that Pr fi fi fi fi X n f(n) fi fi fi m 1 Gamma ffl for all n If N is sufficiently large, K 1 (32) 23 Consider the loss ....
Ross, S.M. Introduction to Probability Models, 5th ed. Academic Press, 1993.
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