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where j j1 ; . . . ; jn is the vector of levels and x x1 ; . . . ; xn is the vector of elapsed holding times for the n sources with the single-source means and variances as in Eqs. (17.19) and (17.21). It is signi cant that we can calculate the conditional aggregate mean at any time t by performing a single numerical inversion, for example, by using the Euler algorithm in Abate and Whitt [2]. We summarize this elementary but important consequence as a theorem. Theorem 17.4.2. The Laplace transform of the n-source conditional mean aggregate required bandwidth as a function of time is ^ M sjj; x  I
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^ where the single-source transform Pj i i sjxi is given in Theorem 17.4.1. ik Unlike for the aggregate mean, for the aggregate variance we evidently need to perform n separate inversions to calculate viji tjxi for each i and then add to calculate V tjj; x in Eq. (17.24). (We assume that the within-level variances wiji tjxi , if included, are speci ed directly.) Hence, we suggest calculating only the conditional mean in real time to perform control, and occasionally calculating the conditional variance to evaluate the accuracy of the conditional mean.
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^ The most complicated part of the conditional aggregate mean transform M sjj; x in ^ Eq. (17.25) is the matrix inverse I q s 1 in the transform of the single-source transition probability in Eq. (17.14). Since the matrix inverse calculation can be a computational burden when the number of levels is large, it is natural to seek approximations that avoid this matrix inverse. We describe such approximations in this section. The matrix inverse I q s 1 is a compact representation for the series PI n n 0 q s . For P tjx , it captures the possibility of any number of transitions up to time t. However, if the holding times in the levels are relatively long in the time scales relevant for control, then the mean for times t of interest will only be signi cantly affected by a very few transitions. Indeed, often only a single transition need be considered. The single-transition approximation is obtained by making the Markov chain absorbing after one transition. Hence, the single-transition approximation is simply Pjk tjx % Hjk tjx ; j T k; and Pjj tjx % Gjc tjx Hjj tjx 17:26
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for Hjk tjx in Eq. (17.9) and Gj tjx in Eq. (17.9). From Eq. (17.26) we see that no inversion is needed. Alternatively, we can develop a two-transition approximation. (Extensions to higher numbers are straightforward.) Modifying the proof of Theorem 17.4.1 in a straightforward manner, we obtain Pjk tjx % for j T k and Pjj tjx % Gjc tjx P
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Expressed in the form of transforms, Eqs. (17.27) and (17.28) become ^ 1 gk s ^ ^ Pjk sjx % hjk sjx s for j T k and ^ Pjj sjx % ^ 1 gj sjx P f^lj s ^ hjl sjx Plj : s s l 17:30 P^ f^ s hjl sjx Plk lk s l 17:29
17.6 THE VALUE OF INFORMATION
Numerical inversion can easily be applied with Eqs. (17.29) and (17.30). However, since the time-domain formulas (17.27) and (17.28) involve single convolution integrals, numerical computation of Eqs. (17.27) and (17.28) in the time domain is also a feasible alternative. Moreover, if the underlying distributions have special structure, then the integrals in Eqs. (17.27) and (17.28) can be calculated analytically. For example, analytical integration can easily be done when all holding-time distributions are hyperexponential. In Duf eld and Whitt [15, Section 3] we give a numerical example illustrating how the two approximations compare to the exact conditional mean for a single source with four levels. 17.6 THE VALUE OF INFORMATION
We can use the source model to investigate the value of information. We can consider how prediction is improved when we condition rst, on, only the level and, second, on both level and age. The reference case is the steady-state mean M