Decomposition Theory Meets Reliability Analysis: Processing of Computation-Intensive Dependent Tasks over Vehicular Clouds with Dynamic Resources

Vehicular cloud (VC) is a promising technology for processing computation-intensive applications (CI-Apps) on smart vehicles. Implementing VCs over the network edge faces two key challenges: (C1) On-board computing resources of a single vehicle are often insufficient to process a CI-App; (C2) The dynamics of available resources, caused by vehicles' mobility, hinder reliable CI-App processing. This work is among the first to jointly address (C1) and (C2), while considering two common CI-App graph representations, directed acyclic graph (DAG) and undirected graph (UG). To address (C1), we consider partitioning a CI-App with $m$ dependent (sub-)tasks into $k\le m$ groups, which are dispersed across vehicles. To address (C2), we introduce a generalized reliability metric called conditional mean time to failure (C-MTTF). Subsequently, we increase the C-MTTF of dependent sub-tasks processing via introducing a general framework of redundancy-based processing of dependent sub-tasks over semi-dynamic VCs (RP-VC). We demonstrate that RP-VC can be modeled as a non-trivial semi-Markov process (SMP). To analyze this SMP model and its reliability, we develop a novel mathematical framework, called event stochastic algebra ($\langle e\rangle$-algebra). Based on $\langle e\rangle$-algebra, we propose decomposition theorem (DT) to transform the presented SMP to a decomposed SMP (D-SMP). We subsequently calculate the C-MTTF of our methodology. We demonstrate that $\langle e\rangle$-algebra and DT are general mathematical tools that can be used to analyze other cloud-based networks. Simulation results reveal the exactness of our analytical results and the efficiency of our methodology in terms of acceptance and success rates of CI-App processing.