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In computer networks including the next generation network (NGN), both computing and network resources are required to be allocated simultaneously to each service request. It is possible to improve the efficiency of resource use if we delay the start of resource allocation rather than rejecting the request if required resources arc not available when the request occurs. This paper assumes that both computing and network resources are allocated per service request, and proposes an new joint resource allocation method that permits the start of resource allocation to be delayed if the required resources are not available, on condition that the service shall be completed within a maximum permissible service completion time. It is demonstrated by simulation evaluations that the proposed method could reduce the total amount of resource required by up to 25% regardless of request generation patterns, compared with the conventional method without delayed resource allocation.