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Note that blocking probabilities and data loss rates are insensitive to the distribution of ow duration A reasonable approximation for the blocking probability of a ow with peak rate pi when c is large with respect to the ei is given by Bi % where a P ei li =mi , d P pi E a=d; c=d ; d 16:1.

.

e2 li =mi =a and i E a; n an . P ai n! i n i!

DECLARE_STANDARD_LDD() { return new DSimpleSerialLDD; }

is Erlang's formula. Formula (16.1) is a simpli cation of the formulas given by Lindberger [20]. It is less accurate but more clearly demonstrates the structural relationship between performance and traf c characteristics. Instead of identifying traf c classes with

Notice that KernelModuleEntry does not provide any initialization hooks. As we shall see later, LDDs and PDDs are polymorphic DLLs which are dynamically loaded after the kernel has booted. Instead, this macro de nes the rst export to represent the DLL factory function.

common traf c characteristics, it may prove more practical to estimate the essential parameters a and d directly. It is well known that application of Erlang's formula leads to scale economies: to achieve a low blocking probability and high utilization (a=c), it is necessary to have a large capacity c. For multirate traf c with blocking probabilities given by Eq. (16.1), the same requirement implies a high value of c=d. The line labeled ``stream'' in Fig. 16.3 shows how achievable utilization a=c in a simple Erlang loss system varies with c for a target blocking probability of 0.01. 16.6.2 Provisioning for Elastic Traf c

Normally, to save space, two different matrices are swapped as newMatrix and oldMatrix across successive steps Algorithms requiring that all tasks periodically wait for all others to complete do not always scale quite as well as more loosely coupled fork/join designs Even so, these algorithms are common, efficient, and amenable to significant parallel speedups 4421 Building and using trees It would be inefficient to repeatedly apply fork/join decomposition in iterative designs in order to update sections in parallel Because the sections are the same across iterations, they can be constructed just once and then repeatedly invoked so that on each iteration, the corresponding updates execute in the same order as would be produced by a recursive solution Computation trees are explicit representations of the tree-structured computations implicitly arising in fork/join recursion These trees have two kinds of nodes, internal nodes and leaf nodes, corresponding to the recursive and base cases of a recursive solution They can be constructed and used for iterative update problems via the following steps: 1 Create a tree of task objects representing the recursive partitions, where: o Each internal node contains references to subpartitions, and has an update method that performs fork/join processing of each of them o Each leaf node represents a finest-granularity partition, and has an update method that operates directly on it 2 For a fixed number of steps, or until convergence, do: o Execute the task performing the root partition's update method For example, the following code illustrates the highlights of a set of classes that perform Jacobi iteration using the averaging formula shown above In addition to updating, this version also keeps track of the differences among computed cell values across iterations, and stops when the maximum difference is within a constant EPSILON Also, like many programs of this form, this code assumes that the matrices have been set up with extra edge cells that are never updated, so boundary conditions never need to be checked (Alternatives include recomputing edge values using special edge formulas after each pass, and treating edges as toroidally wrapping around the mesh) The recursive decomposition strategy used here is to divide the mesh into quadrants, stopping when the number of cells is at most leafCells, which serves as the granularity threshold This strategy works well so long as the numbers of rows and columns in the matrix are approximately equal If they are not, additional classes and methods could be defined to divide across only one dimension at a time The approach here assumes that the matrix as a whole already exists, so rather than actually dividing up cells, task nodes just keep track of the row and column offsets of this matrix that each partition is working on The subclass-based design used here reflects the different structure and behavior of internal versus leaf nodes Both are subclasses of abstract base JTree:.