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In this paper, we study the message delivery capacity problem in delay-constrained mobile sensor networks. The message delivery capacity specifies the maximum percentage of sensing messages that can be successfully delivered to sink nodes within a given time constraint. This captures the overall system capacity in terms of successful sensing message delivery. By taking full advantage of sensor mobility and rendezvous during senor node encounters, messages can be delivered to a sink node either directly or through relays by other sensor nodes. In this paper, for the first time, we present the delay-constrained message delivery capacity formulation in mobile sensor networks. The objective is to maximize the message delivery capacity subject to the delay and buffering constraints. We first identify a number of unique challenges involved in such systems including message relay and buffer replacement mechanisms, and we derive the capacity bound under perfect message relay and buffer replacement mechanisms. Due to an unrealistic assumption about the foreknowledge of sensor moving trajectories, we next proceed to propose a practical algorithm to approximate the maximal message delivery capacity based on current global network knowledge. Furthermore, a distributed algorithm is proposed to reduce the control overhead for information exchange. Finally, we evaluate the algorithms and examine their sensitivity with respect to delay constraints, buffer size and message relay, and replacement schemes.