The advance reservation technique has been widely applied in many grid systems to provide end-to-end quality of service (QoS). However, it will result in low resource utilization rate and high rejection rate when the reservation rate is high. To mitigate these negative effects brought about by advance reservation, a relaxed advance reservation policy is proposed, which allows accepting new reservation requests that overlap the existing reservations under certain conditions. Both the benefits and the risks of the proposed policy are presented theoretically. The experimental results show that the policy can achieve a higher resource utilization rate and lower rejection rate compared to the conventional reservation policy and backfilling technique. In addition, the policy shows better adaptation when the grid systems are in the presence of a high reservation rate.