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The ow graph FG does not contain information about the number of iterations This also means that the number of iterations does not have to be known when constructing the graph, as is the case, for instance, with a loop whose index bound is a variable (N in Example 10) It should be mentioned, that all communications must be uniform, in order to represent their communication distance as one integer value Nonuniform communication, as found in the loop of Figure 39, cannot be expressed as one integer value Yet, a ow graph can still be constructed in such cases using a conservative approximation of the distance, for example, +1 to denote an unknown dependence distance greater than or equal to one For certain loop parallelization algorithms, such information can be suf cient If the accurate representation of the dependence structure is indispensable, a ow graph can only be used with uniform communication This establishes an important limitation of the ow graph, which relates to the representable computation types In comparison to the DG or the iteration DG (Section 33), the ow graph has two essential characteristics First, nodes in the ow graph represent tasks that can be executed several times, and not instances of tasks as in the DG, where each represented node is executed exactly once Second, the ow graph is, in general, not acyclic Cycles can arise in connection with interiteration dependence, but in contrast to the DG, the represented computation is still feasible When reading the code of Example 10, one concludes that it is a feasible program, but in fact its FG in Figure 311 contains two cycles S, U, S and S, T, U, S The ow graph breaks a limitation imposed on the dependence graph: in contrast to the DG, a ow graph can include cycles As expressed by Lemma 34, a DG must be acyclic in order to represent a feasible computation So, how can a ow graph, based on the same general graph model of De nition 37, be a valid representation of a feasible program According to De nition 38 (node strictness), the nodes must be strict regarding their input and output Even though an FG models the data ow among nodes, and not as the DG the dependence relations, communications among nodes create ( ow) data dependence relations (see Section 251) In other words, the edges implicitly represent dependence relations; thus, a communication cycle would lead to a dependence cycle The reason an FG remains feasible, despite the cycles, lies in the introduction of delays on the edges, which prevent cycles in the graph from turning into dependence cycles In the ow graph, every cycle must contain at least one delayed edge, breaking the dependence chain Seen the other way around, paths in ow graphs are only closed to cycles by interiteration communications, which by de nition are delayed Lemma 35 (Feasible Flow Graph) A ow graph FG = (V, E, D) represents a feasible iterative computation P if and only if any cycle pc in FG contains at least one edge e with a nonzero delay D(e) = 0: pc FG e E : e pc D(e) = 0 (33).



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2:15

Stationary M =G=I processes being time-reversible, we have the representation qb st sup Stb ct; t 0; 1; . . . I for the steady-state buffer content qb with I

b S0 0;

FLOW GRAPH (FG)

9:12

t 1; 2; . . . :

9:13

4

.

Hereafter, by an M =G=I input process we mean its stationary version fb*; t 0; 1; . . .g, which is fully characterized by the pair l; G . Moreover, from t now on, we always assume the stability condition rin lE s < c: 9:14

Before discussing the asymptotics associated with buffer over ow induced by M =G=I input processes, we make a slight detour to explore the correlation structure of such input processes. 9.4.1 Correlation Properties





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In view of Proposition 9.2.1, the stationary version fb*; t 0; 1; . . .g has a wellt de ned (auto)covariance function G : R 3 R, say, * G h Cov b*; bt h ; t Proposition 9.4.1. We have h 0; 1; . . . : 9:16 t; h 0; 1; . . . : 9:15

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Proof : Suppose FG contains at least one cycle whose edges all have zero delay This corresponds to a cycle in the DG, which, however, must be acyclic to represent a feasible program (Lemma 34) : The program s feasibility is shown by demonstrating that the DG corresponding to the ow graph has no cycles (Lemma 34) One node in DG represents one instance of a node in FG; that is, DG consists of i-times the nodes of FG, therefore |VDG | = i |VFG |, with VDG and VFG being the node sets of DG and FG, respectively, and i the number of iterations (Figure 313 shows the DG of the ow graph in Figure 311 assuming three iterations N = 3 in the underlying program of Example 10 and each iteration comprises the nodes S, T, and U) Communication edges in FG with zero delay {e EFG : D(e) = 0}, correspond to dependence edges in DG between the nodes of the same iteration; thus, there are i |{e EFG : D(e) = 0}| edges of this type (These are three times the edges eSU and eST in Figure 313) As every cycle in FG contains at least one nonzero edge, there cannot be a cycle in DG among the nodes of one iteration Edges with nonzero delay are directed from an earlier to a later iteration, never the other way around, because the weight is nonnegative, D(e) 0 e VFG (The graph of Figure 313 has three such edges: eT(1)U(2) , eT(2)U(3) , and eU(1)S(3) ) This means that the entering edges of a node always have their origin nodes in the same or a previous iteration, while leaving edges have their destination node in the same or a subsequent iteration But then there is also no dependence cycle in DG across iterations, as no path can return to its origin node.

^ G h lE s h lE s P s > h ;

The rst equality in Eq. (9.16) is established in Cox and Isham [7] and the second equality follows readily from the de nition (9.9). From Eq. (9.16) we nd the autocorrelation function g : R 3 R of the M=G=I process l; s to be given by g h G h ^ P s > h ; G 0 h 0; 1; . . . : 9:17

Proposition 9.4.1 shows that the correlation structure of the stationary M =G=I ^ input process l; s is completely determined by the pmf of s (thus of s). It turns out

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