We consider resource allocation in a system which must process and subsequently deliver jobs from N input terminals to N output terminals as per matching constraints. Jobs are stored in a common memory before processing and transfer. The resource allocation objective is to minimize the job loss due to memory overflow. We present optimal scheduling and memory management policies for attaining this objectives for N = 2 and symmetric traffic. We identify certain characteristics of the optimal strategy for N = 2 and asymmetric traffic, and present a near optimal heuristic for this case. We obtain a heavy traffic lower bound for the optimal discounted cost function for the general case of arbitrary N and arbitrary traffic. We use this lower bound to propose a heuristic strategy whose performance is close to the optimal strategy in this case. The policies proposed in this paper substantially outperform the existing strategy for the system, which was designed and proved to be optimal under the assumption of infinite storage.