The problem of maximizing lifetime of a sensor network is still challenging mainly due to the stringent delay-deadline of real-time applications and heterogeneity of sensor devices. The problem is further complicated when the network contains many obstacles. In maximizing network lifetime, existing literature works either merely address issues of application delay-deadline and presence of obstacles, or analyze primitive data collection approaches for such an environment. In this paper, we formulate optimal data collection schedule of a mobile sink in an obstructed sensor network as a mixed-integer linear programming (MILP) problem. The proposed data collection scheduling finds an optimal set of rendezvous nodes over a preformed Starfish routing backbone, and corresponding sojourn duration so as to maximize the network lifetime while maintaining delay-deadline constraint in an obstructed network. The proposed Starfish-scheduling ensures a loop-free traveling path for a mobile sink across the network. The results of performance evaluation, performed in network simulator-2, depict the suitability of Starfish scheduling as it outperforms state-of-the-art-works in terms of extending network lifetime and data delivery throughput as well as reducing average end-to-end delay.