The Delay Time Profile of Multistage Networks with Synchronization

Abstract

The interaction between projects and servers has grown significantly in complexity; thus, applying parallel calculations increases dramatically. However, it should not be ignored that parallel processing gives rise to synchronization constraints and delays, generating penalty costs that may overshadow the savings obtained from parallel processing. Motivated by this trade-off, this study investigates two special and symmetric systems of split–join structures: (i) parallel structure and (ii) serial structure. In a parallel structure, the project arrives, splits into m parallel groups (subprojects), each comprising n subsequent stages, and ends after all groups are completed. In the serial structure, the project requires synchronization after each stage. Employing a numerical study, we investigates the time profile of the project by focusing on two types of delays: delay due to synchronization overhead (occurring due to the parallel structure), and delay due to overloaded servers (occurring due to the serial structure). In particular, the author studies the effect of the number of stages, the number of groups, and the utilization of the servers on the time profile and performance of the system. Further, this study shows the efficiency of lower and upper bounds for the mean sojourn time. The results show that the added time grows logarithmically with m (parallelism) and linearly with n (seriality) in both structures. However, comparing the two types of split–join structures shows that the synchronization overhead grows logarithmically undr both parallelism and seriality; this yields an unexpected duality property of the added time to the serial system.