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In this paper, we study the delay-constrained message delivery capacity formulation in mobile wireless networks. The message delivery capacity specifies the maximum percentage of messages that can be successfully delivered to base stations within a given time constraint. By taking full advantage of node mobility and rendezvous during mobile node encounter, messages can be delivered to a base station either directly or through relays by other nodes. We first identify a number of unique challenges involved in such systems including message relay and buffer replacement mechanisms and then derive the capacity bound under perfect message relay and buffer replacement mechanisms. We next prove that this problem is NP-hard. We proceed to propose a practical algorithm to approximate the maximal message delivery capacity. Finally, we evaluate the algorithm and examine the sensitivity with delay constraint, buffer size and message relay and replacement schemes.