A hierarchical approach for bounding the completion time distribution of stochastic task graphs

The analytical evaluation of the completion time distribution of a general directed acyclic graph (DAG) is known to be an NP-complete problem. In this paper we present a new algorithm, named Tree_Bound, for the evaluation of bounds on the completion time of stochastic graphs assuming ideal conditions for shared resources and independent random variables as task execution times. The Tree_Bound method uses a hierarchical approach that first gives a tree-like representation of the graph, and then evaluates lower and upper bounds through a single visit of the tree. As lower bound the method takes the distribution of an embedded series–parallel graph which is evaluated by means of a simple recursion. The upper bound is based on a hierarchical application of other bounding techniques. In this paper, we use the Shogan algorithm because its determinism allows us to demonstrate some interesting properties of the Tree_Bound method. , through stochastic ordering and stochastic comparison techniques, we demonstrate analytically that our approach provides tighter bounds than Shogan’s and Yazici-Pekergin’s bounds. On the other hand, we cannot compare formally the Tree_Bound accuracy to that of other important methods, such as Kleinöder and Dodin, because of their non-determinism. Various empiric comparisons show that the Tree_Bound algorithm provides analogous or superior results than heuristics derived from main non-deterministic methods. Moreover, the Tree_Bound algorithm keeps linear complexity and avoids non-determinism. Finally, it represents a useful basis for the combination of different bounding techniques which seems the only way to achieve even tighter bounds on the completion time distribution of stochastic graphs.