PERFORMANCE CONSIDERATIONS ON A RANDOM GRAPH MODEL FOR PARALLEL-PROCESSING

Consider a random directed acyclic graph (dag) with nodes 1, 2, ..., n , and an edge from node i to node j (only if i >j) with fixed probabil ity p. Such a graph can be thought of as the task graph associated wit h a job and thus it serves as a parallel processing model; the vertice s correspond to tasks and the edges correspond to precedence constrain ts between tasks. In this case, the length of the graph corresponds to the parallel processing time of the job (an infinite number of availa ble processors is assumed) and the width of the graph corresponds to t he parallelism of the job. We estimate here the average length of the random dag (that is, the average processing time of the job) as a func tion of the probability p and the number of tasks n hy establishing ti ght lower and upper bounds. The lower (resp. upper) bound is determine d as being equal to the average length of a random dag considerably si mpler to manipulate than the original one. Furthermore, the asymptotic behaviour of the average length is studied and the results obtained i mprove previously published results. Finally, asymptotic results are o btained concerning the average width of the task graph; it is shown th at the average width tends to 1/p as n --> infinity.