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iteration DG cannot be constructed This fact is noteworthy, especially in conjunction with the observation made at the end of Section 252: the known techniques of dependence analysis can only compute in reasonable time the distance vectors for dependence relations based on simple, albeit common, types of subscript functions Not to forget that it is sometimes even impossible to determine the dependence relations, for example, when array subscripts are functions of input variables of the program The only solution is then to create a dependence edge for every possible dependence relation in order to obtain a conservative dependence graph, that is, a graph that includes all edges that the true dependence graph would have 332 Summary As the name indicates, the dependence graph is a representation of the dependence structure of a program In its general form, as in De nition 310, the DG is not bound to any particular computation type or granularity This general theoretical form is widely employed, whenever it is necessary to reason about the dependence structure of a program, the feasibility of computations, or the validity of program transformations (see the literature referenced in Section 251) In practical usage, the DG is limited to a coarse grained or partial representation of a program, because its size (and with it its computer representation) grows with the number of tasks The dependence structure of cyclic computations is represented by the iteration dependence graph, where every node of the graph is associated with coordinates of the iteration space spanned by the index variables of the loop nest For dependence analysis of loops and, above all, for loop transformations, the iteration DG is a valuable instrument Banerjee et al [17] survey many of the existing techniques Although several of them do not require the explicit construction of the graph, the underlying theory of these techniques is based on the dependence graph Granularity in loops is typically small to medium, which is therefore the common granularity of iteration dependence graphs The size issue of the general DG is evaded by bene tting from the regular iterative structure of the computation A compact computer representation, given that the dependence distances are uniform, comprises only the tasks of the loop kernel, the intraiteration dependence relations, and the distance vectors of the interiteration dependence relations Iteration DGs are employed not only in parallelizing compilers but also in VLSI array processor design (Kung [109]) There, the iteration DG, constructed from an initial application speci cation typically signal processing serves as an intermediate representation for the mapping and scheduling of the application to array processors Those graphs possess two distinctive attributes: (1) a node represents one iteration, not only a part of it; and (2) the dependence relations are local, that is, the absolute values of the distance vectors elements are typically 1 or 0 (Karp et al [101]) The mapping and scheduling process transforms the iteration DG into a ow graph, which is treated in the next section While the dependence graph has no af nity for any particular parallel architecture, the iteration dependence graph, owing to its regular iterative structure, is usually employed for homogeneous systems, with SIMD or SPMD streams.



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We summarize various facts concerning the busy server process of a discrete-time M =G=I system; details are available in Parulekar [27]. 9.2.1 The Model





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Consider a system with in nitely many servers. During time slot t; t 1 ; bt 1 new customers enter the system. Customer i; i 1; . . . ; bt 1, is presented to its own server and begins service by the start of slot t 1; t 2 ; its service time has duration st 1;i (expressed in number of slots). Let bt denote the number of busy servers or, equivalently, of customers still present in the system, at the beginning of slot t; t 1 . We assume that b servers are initially present in the system at t 0 (i.e., at the beginning of slot 0; 1 ) with customer i; i 1; . . . ; b, requiring an amount of work of duration s0;i from its own server. The busy server process fbt ; t 0; 1; . . .g is what we refer to as the M =G=I input process.

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The following assumptions are enforced on the R-valued random variables (rvs) b, fbt 1 ; t 0; 1; . . .g and fst;i ; t 0; 1; . . . ; i 1; 2; . . .g: (1) The rvs are mutually independent; (2) The rvs fbt 1 ; t 0; 1; . . .g are i.i.d. Poisson rvs with parameter l > 0; (3) The rvs fst;i ; t 1; 2; . . . ; i 1; 2; . . .g are i.i.d. with common pmf G on f1; 2; . . .g. We denote by s a generic R-valued rv distributed according to the pmf G. Throughout we assume this pmf G to have a nite rst moment, or equivalently, E s < I. At this point, no additional assumptions are made on the rvs fs0;i ; i 1; 2; . . .g. For each t 0; 1; . . . , we note the decomposition bt b 0 b a ; t t 9:6

34 FLOW GRAPH (FG) The iteration dependence graph, as discussed in the previous section, is not the only graph model for the representation of cyclic computations Intuitively, a model for such computations can bene t from the inherent regular structure, as insinuated during the discussion of the computer representation of an iteration dependence graph in the previous section It seems to be suf cient for a model to re ect the tasks of one iteration with their intra- and interiteration communications1 The ow graph (FG) achieves a concise representation of the structure of an iterative computation by introducing timing information into the graph, which allows one to distinguish between intra- and interiteration communication This is illustrated with Example 10

where the rvs b 0 and b a describe the contributions to the number of customers in t t the system at the beginning of slot t; t 1 from those initially present (at t 0) and from the new arrivals in the interval [0, t], respectively. Under the enforced operational assumptions, we readily check that b a t

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1 s0;i > t :

9:7

Example 10 Flow Graph for i = 1 to N do S: A(i) = C(i) D(i+2) T: B(i+1) = A(i) + 10 U: C(i+2) = A(i) + B(i) end for

The rv b a can also be interpreted as the number of busy servers in the system at the t beginning of slot t; t 1 given that the system was initially empty (i.e., b 0). 9.2.2 The Stationary Version

Although the busy server process fbt ; t 0; 1; . . .g is in general not a (strictly) stationary process, it does admit a stationary and ergodic version fb*; t 0; 1; . . .g. t This stationary version satis es the decomposition (9.6) with the portion in (9.7) due to the initial condition replaced by b 0 t

^ 1 sn > t ;

Analyzing the data dependence of its loop (Section 252), the following communications can be identi ed Considering the three statements S, T, and U as tasks, there are two intraiteration communications the output of S is the input of T and U, caused by A(i) and two interiteration communications the output of T(i) is the input of U(i + 1) (array B) and the output of U(i) is the input of S(i + 2) (array C) The ow graph for this computation is then as depicted in Figure 311

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