Concurrency factor

The value of P for different values of 0 and a are given in Table 5.2. The size of the design space for a single resource type i with allocation a is 5 = P( 0, a). Note that the size remains low for modest values of 0 and a. 

Before describing the design space exploration strategy, we introduce first the concept of concurrency factor to measure of the degree of parallelism among subsets of shareable operations Q Q V. This concept is used extensively in resource allocation and heuristic exploration of the design space. 

Let Gm V E, 5) be the sequencing graph for a model M. We define the concurrency factor for a subset of shareable operations Q C F as follows. 

Definition 5.2.1 Given a cf-Merarchy G%, the concurrency factor of a subset of shareable operations Q C V is the maximum number of vertices in Q that can be executing simultaneously in It is denoted by cfactor Gni, Q)- 

The concurrency factor can range in value from 1 to Q. If the factor is equal to 1, then elements of Q must be temporally disjoint. If the factor is equal to Q, then elements of Q can all execute in parallel. 

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The other side of the psychological coin from cognition is emotion, and caffeine has also been shown to have important effects on this aspect of functioning as well. Depending on dosage level and concurrent factors, caffeine can result in either positive or negative mood changes. 

Direct determination of portal blood flow rate is difficult and would generally require placement of an electronic flow probe in each animal. However the technique proposed by Hoffman et al. utilised tritiated water as an absorption probe (i.e. internal standard) [89], By dosing and sampling drug/ absorption probe concurrently, factors such as variable portal blood flow rate are normalised between experiments. 

Neonatal Cocaine freely crosses the placental barrier, and prenatal exposure to cocaine alters neurobehavioral development in rat pups (Sobrian et al. 1990). The effects on humans exposed prenatally to cocaine is a complicated matter, because so many other concurrent factors contribute to development. Common confounds are prenatal care and maternal polydrug use. Prenatal cocaine use is associated with reduced gestational age, birth weight, body length, and head circumference (Richardson et al. 1999). In children exposed prenatally to cocaine, some studies have shown behavioral differences evident at 1 to 3 years of age (Richardson et al. 1993 Richardson 1998). Associations are also made with impulsivity and attention deficits at age 6 (Leech et al. 1999). 

Another concurrent factor not considered in the risk analysis was the toxicology of residual amounts of the disinfectant species including hypochlorous acid and chloramines related to chlorine that would normally be present as residuals in chlorinated water. The in vivo toxicology of hypochlorite now indicates the formation of haloforms and halonitriles and thus additional risks (34). 

The counting efficiencies obtained with in situ methods depend on various concurrent factors. The metabolites separated on the gel column must be fixed in a spaced way to minimize self-absorption. The spaced fixation must not be disturbed during dehydration and the following procedures. However, the counting efficiencies of the in situ method depends on the concentration of the gel. This is even more obvious when compared with those obtained with the in situ method described earlier (Gezelius, 1977) for a mixed gel of 2% polyacrylamide and 0.5% (Table 2 ). The reduction of the counting efficiencies in more concentrated gels is probably due to increased selfabsorption. 

Most areas of modern research contain two concurrent factors a basic aspect, which somehow aims at an improvement in scientific understanding, and an applied aspect, which is directed toward a practical goal such as a new or improved material or process. While one or the other factor frequently predominates in a given piece of work, sometimes the two are inextricable. 

The high extent of sorption of certain mannans onto TMP indicates that mannans are also able to sorb onto unclean surfaces, i.e. without exposed cellulose surfaces, or that the sorption may indeed be a deposition on the fiber surface after self-association, which can happen with galactose-poor GMs (33), or partial crystallization after deacetylation of GGMs, or simply due to concurrent factors affecting the mannan solubility upon mixing of mannans with fibers. 

In the remainder of this section, we describe the computation strategy for the concurrency factor. Section S.2.1 considers the simple case where G f consists of a single graph, and Section 5.2.2 considers the genial case where consists... 

The concurrency factor of Q can be computed by first constructing an undirected disjoint compatibility graph. Vertices of the disjoint compatibility graph, denoted by Gq = Q,Eq), correspond to elements of Q. Undirected edges Eq indicate when vertices are temporally disjoint, i.e. they cannot execute in parallel. 

Lemma 5,2.1 Given a sequencing graph Gm without conditional and loop vertices, the concurrency factor of a subset cf shareable operations Q CV is equal to the clique cover number of the corresponding disjoint compatibility graph Gq. 

Proof An edge in the disjoint compatibility graph Gq implies that two vertices cannot execute in parallel. Th fore, a clique in Gq corresponds to a subset of vertices that cannot execute in parallel. A clique cover partitions the elements of Q into cliques, and therefore the clique cover number for Gq is equal to the maximum number of elements in Q that can execute in parallel, which is in turn equal to the concurrency factor of Q. ... 

Returning to the example of Figure 5.6, the disjoint compatibility graph G q for the set Q = vi, V2, v, V4 is shown in Figure 5.7(b). A minimum clique cover is shown in (c) where operations within a clique are enclosed in ovals. The concurrency factor is equal to the clique cover number, which is 2. 

Figure 5.8 Example of computing the concurrency factor by finding clique cover number.

To compute the concurrency factor of Q for this general case, we traverse G m in a bottom-up fashion starting from the leaf graphs upwards to the root... 

The concurrency factor cfactor Guaf,Q) can be computed using the strategy presented in the previous section by first constructing a disjoint compatibility gr h and then finding its clique cover number. 

Loop vertex If v is a loop vertex with loop body Gjoop. then the weight is equal to the concurrency factor of the loop body ... 

Proof The weight for a vertex is equal to the maximum number of vertices in Q that can be activated if the vertex is executed. Consider a vertex v and its mapping >l(v). In the augmented disjoint compatibility graph, no path exists between the elements of, 4(v). Furthermore, for any vertex w that is disjoint with respect to v, all elements in the mapping are also disjoint w.r.t. all elements in -4(u). Therefore, Gq captures the degree of parallelism among elements of Q, and we conclude that its clique cover number is equal to the concurrency factor cfactor Gm, 0)- II... 

Since Gq is a comparability graph, its clique cover number can be computed in polynomial time [GolSO]. We therefore conclude that given any arbitrary sequencing graph, the concurrency factor for any subset of shareable operations can be computed in polynomial-time. 

Concurrency factor can be used to determine the minimum resource allocation that is necessary to avoid resource conflicts, where we assume the worst case of all operations having unbounded execution delays. We first consider a sequencing graph G, and a resource binding 0 defined on G,. The resource binding partitions the shareable operations V into one or more instance operation sets where elements within an instance operation set all share the same hardware resource. We define the conflict degree of the binding 0 as follows. 

The above theorems imply that conflict-free allocation /(<) can be evaluated efficiently because the concurrency factor can be computed in polynomial time. This point is worth emphasizing. The conflict-free allocation indicates the degree of parallelism among the operations of a given resource type. It corresponds to the extent resource conflicts are present in a graph. If the conflict-free allocation satisfies the required resource constraints, then it is not necessary to allocate more resources than this amount to obtain an implementation that satisfies the timing constraints. 

From Section 5.2, the concurrency factor of an implementable instance operation set is 1 and the conflict degree of an implementable binding is 0. 

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