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17.4 TRANSIENT ANALYSIS
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information is I 0 j; x j1 ; . . . ; jn ; x1 ; . . . ; xn and the conditional aggregate mean is E B t jI 0  M tjj; x
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From Eq. (17.5), we see that we need to compute the conditional distribution of i the level, that is, the probabilities Pjk tjx , for each source i. However, we can nd i relatively simple expressions for the Laplace transform of Pjk tjx with respect to time because the level process of each source has been assumed to be a semi-Markov process. We now consider a single source and assume that its required bandwidth process is a semi-Markov process (SMP). (We now have no within-level variation process.) We now omit the superscript i. Let L t and B t be the level and required bandwidth, respectively, at time t as in Eq. (17.4). The process fL t : t ! 0g is assumed to be an SMP, while the process fB t : t ! 0g is a function of an SMP, that is B t bL t , where bj is the required bandwidth in level j. If bj T bk for j T k, then fB t : t ! 0g itself is an SMP, but if bj bk for some j T k, then in general fB t : t ! 0g is not an SMP. 17.4.2 Laplace Transform Analysis
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Let A t be the age of the level holding time at time t. We are interested in calculating Pjk tjx  P L t kjL 0 j; A 0 x 17:6
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as a function of j; k; x, and t. The state information at time 0 is the pair j; x . Let P be the transition matrix of the discrete-time Markov chain governing level transitions and let Fjk t be the holding-time cdf given that there is a transition from level j to c level k. For simplicity, we assume that Fjk t 1 Fjk t > 0 for all j; k, and t, so that all positive x can be level holding times. Let P tjx be the matrix with elements ^ Pjk tjx and let P sjx be the Laplace transform (LT) of P tjx , that is, the matrix with elements that are the Laplace transforms of Pjk tjx with respect to time: ^ Pjk sjx I
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^ We can obtain a convenient explicit expression for P sjx . For this purpose, let Gj be the holding-time cdf in level j, unconditional on the next level, that is, Gj x P
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For any cdf G, let Gc be the complementary cdf, that is, Gc x 1 G x . Also, let Hjk tjx Pjk Fjk t x Gjc x and Gj tjx P
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^ ^ for Gj in Eq. (17.8). Then let hjk sjx and gj sjx be the associated Laplace Stieltjes transforms (LSTs): ^ hjk sjx I
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^ ^ ^ Let h sjx be the matrix with elements hjk sjx . Let q s be the matrix with elements ^ qjk s , where Qjk t Pjk Fjk t and ^ qjk s I
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^ ^ Let D sjx and D s be the diagonal matrices with diagonal elements ^ ^ Djj sjx  1 gj sjx =s; ^ ^ Djj s  1 gj s =s; 17:12
^ where gj s is the LST of the cdf Gj in Eq. (17.8). Theorem 17.4.1. The transient probabilities for a single SMP source have the matrix of Laplace transforms ^ ^ ^ ^ P sjx D sjx h sjx P sj0 ; where ^ ^ ^ P sj0 I q s 1 D s : Proof. In the time domain, condition on the rst transition. For j T k, Pjk tjx so that ^ Pjk sjx P^ ^ hjl sjx Plk sj0 ;
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