Moving beyond just interconnected devices, the increasing interplay between communication and computation has fed the vision of real-time networked control systems. To obtain timely situational awareness, IoT devices continuously sample computation-intensive status updates, generate perception tasks and offload them to edge servers for processing. In this sense, the timeliness of information is considered as one major contextual attribute of status updates. In this paper, we derive the closed-form expressions of timeliness of information for computation offloading at both edge tier and fog tier, where two stage tandem queues are exploited to abstract the transmission and computation process. Moreover, we exploit the statistical structure of Gauss-Markov process, which is widely adopted to model temporal dynamics of system states, and derive the closed-form expression for process-related timeliness of information. The obtained analytical formulas explicitly characterize the dependency among task generation, transmission and execution, which can serve as objective functions for system optimization. Based on the theoretical results, we formulate a computation offloading optimization problem at edge tier, where the timeliness of status updates is minimized among multiple devices by joint optimization of task generation, bandwidth allocation, and computation resource allocation. An iterative solution procedure is proposed to solve the formulated problem. Numerical results reveal the intertwined relationship among transmission and computation stages, and verify the necessity of factoring in the task generation process for computation offloading strategy design.