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Several authors [10, 16, 19] have derived asymptotics such as (9.1) by means of large deviations estimates associated with the sequence ft 1 Stb ct ; t 0; 1; . . .g. These results (and their necessary extensions) are summarized below as they apply to the present context. 9.6.1 A General Setup



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With a given R-valued sequence fxt 1 ; t 0; 1; . . .g, we associate the R -valued rv qI given by qI sup St ; t 0; 1; . . . ; where S0 0; St x1 xt ; t 1; 2; . . . : 9:33 9:32

If the sequence fxt 1 ; t 0; 1; . . .g is assumed stationary and ergodic with E x1 < 0, then qI is a.s. nite. We are interested in characterizing the asymptotic behavior of the tail probability P qI > b for large b.

Figure 314 The ow graph of Figure 311 with queues and tokens (initial data items) on the edges; a line through an arrow denotes a place in a queue and a dot denotes an initial data item





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To x the terminology, a scaling sequence is any monotone increasing R-valued sequence fvt ; t 0; 1; . . .g such that limt3I vt I. The sequence ft 1 St , t 1; 2; . . .g is said to satisfy the large deviations principle under scaling vt if there exists a lower-semicontinuous function I : R 3 0; I such that for every open set G, inf I x

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9:34

inf I x :

FLOW GRAPH (FG)

9:35

We refer to Eqs. (9.34) and (9.35) as the large deviations lower and upper bounds, respectively. The rate function I is said to be good if for each r > 0, the level set fx P R : I x rg is a compact subset of R. Additional information on large deviations can be found in Dembo and Zeitouni [9]. We consider a given scaling sequence fvt ; t 0; 1; . . .g with the property that there exist functions g, h : R 3 R such that h is monotone increasing with limb3I h b I, and the limits

9:36

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One of the earliest ow graphs for parallel computations is the computation graph (CG) (Karp and Miller [100], Reiter [162]) Each edge has a FIFO queue without restriction of its length and a weight is associated with each edge indicating the initial number of data words in the queue The token concept is also used in data ow languages (Ackerman [1], Davis and Keller [53]), a graphical approach to software development Data ow program graphs allow the programming of data ow computers, since both share the self-timed and data-driven nature The data ow graph (DFG) (Kung [109], Parhi [145]) bene ts particularly from the hardware-oriented view of the data-driven execution model, as it is used in VLSI array processor design Enhanced with two additional weight types, compared to the simple ow graph, the DFG also re ects the computation time of the nodes and the capacity of the FIFO queues Restricting the queue capacity can lead to deadlocks, which is of real concern in VLSI array processors DFGs are used for the mapping of parallel computations on VLSI wavefront array processors (Kung [109]) The data-driven execution model is a valuable mechanism whenever an accurate view of the data ow in space and time is required, as, for example, in hardwareoriented parallelization For many purposes, however, such a detailed view of the data ow is not necessary and the iteration-driven execution model is appropriate and suf cient In any case, the ow graph de ned in De nition 312 is suitable for a datadriven view as soon as the edge delays D(e) are considered initial tokens (Kung [109]), as described earlier The iteration-driven execution model simply presupposes a queue mechanism for the delayed communication that is suf ciently large, and therefore no further consideration is needed

The following theorem is essentially due to Duf eld and O'Connell [10]; a proof is included here for the sake of completeness. Proposition 9.6.1. If the process ft 1 St ; t 1; 2; . . .g satis es the large deviations lower bound (9.34) with good rate function I : R 3 0; I (under scaling vt ), then we have Eq. (9.4) with g inf

7

g y inf I x :

9:37

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