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This structure suffices for problems requiring known numbers of iterations However, many problems require checks for convergence or some other global property between iterations (Conversely, in a few chaotic relaxation algorithms you don't even need a barrier after each iteration, but can instead let segments free-run for a while between barriers and/or checks) One way to provide convergence checks is to rework the CyclicBarrier class to optionally run a supplied Runnable command whenever a barrier is about to be reset A more classic approach, which illustrates a technique useful in other contexts as well, is to rely on the index returned by barrier The caller obtaining index zero (as an arbitrary, but always legal choice) can perform the check while all others are quietly waiting for a second barrier For example, here a a barrier-based version of a segment class for the Jacobi problem described in 442 Collections of JacobiSegment objects can be initialized and run by a driver of the generic form given above

The Lindberg Feller central limit theorem (CLT) for non-identically-distributed summands can be applied to generate a normal approximation characterized by the conditional mean and conditional variance; see Feller [18, p. 262]. For the normal approximation to be appropriate, we should check that the aggregate is not dominated by only a few sources. Let B t denote the (random) aggregate required bandwidth at time t, and let I 0 denote the (known deterministic) state information at time 0. Let B t jI 0 represent a random variable with the conditional distribution of B t given the information I 0 . By the CLT, the normalized random variable B t jI 0 E B t jI 0 p Var B t jI 0 17:2

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is approximately normally distributed with mean 0 and variance 1 when the number of sources is suitably large. Since the conditional mean alone tends to be very descriptive, we use the approximation B t jI 0 % E B t jI 0 ; 17:3

class JacobiSegment implements Runnable { // Incomplete // These are same as in Leaf class version: double[][] A; double[][] B; final int firstRow; final int lastRow; final int firstCol; final int lastCol; volatile double maxDiff; int steps = 0; void update() { /* Nearly same as Leafrun */ } final CyclicBarrier bar; final JacobiSegment[] allSegments; // for convergence check volatile boolean converged = false; JacobiSegment(double[][] A, double[][] B, int firstRow, int lastRow,

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which can be justi ed by a (weaker) law of large numbers instead of the CLT. We will show that the conditional mean in Eq. (17.3) can be ef ciently computed, so that it can be used for real-time control. From Eq. (17.2), we see that the error in the approximation (17.3) is approximately characterized by the conditional standard p deviation Var B t jI 0 . We also will show how to compute this conditional standard deviation, although the required computation is more dif cult. If there are n sources that have roughly equal rates, then the conditional standard deviation will be p O n , while the conditional mean is O n . Given that our approximation is the conditional mean, and given that our state information does not include the state of the within-level variation process, the within-level variation process plays no role because it has zero mean. Let i index the source. Since the required bandwidths need not have integer values, we index the level by the integer j; 1 j Ji, and indicate the associated required bandwidths in the level by bij . Hence, instead of Eq. (17.1), the required bandwidth for source i can be expressed as Bi t biLi t WLi t t ; t ! 0: 17:4

i Let Pjk tjx be the probability that the source-i level process is in level k at time t given that time 0 it was in level j and had been so for a period x (i.e., the age or elapsed level holding time at time 0 is x). If j j1 ; . . . ; jn and x x1 ; . . . ; xn are the vectors of levels and ages of the n source level processes at time 0, then the state

int firstCol, int lastCol, CyclicBarrier b, JacobiSegment[] allSegments) { thisA = A; thisB = B; thisfirstRow = firstRow; thislastRow = lastRow; thisfirstCol = firstCol; thislastCol = lastCol; thisbar = b; thisallSegments = allSegments;