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Some M2M applications such as those found in a Smart Grid environment generate event driven and delay sensitive uplink traffic. Wide area cellular systems such as LTE are usually not optimized for such traffic. In this paper, we design an LTE scheduler with the main objective of maximizing the percentage of uplink packets that satisfy their individual delay budgets. In order to do this accurately, we allow devices to notify the eNodeB of the age of the oldest packet in their buffers via a new MAC control element in the uplink MPDU. This information is used by the eNodeB to calculate an absolute deadline for each packet request individually, and the eNodeB scheduler ranks requests according to an urgency metric that depends upon the time remaining to the deadline and other factors such as the volume of pending data in the device buffers. Using an OPNET simulation model of an LTE TDD system, we show that our proposed scheduler can satisfy the uplink delay budget for more than 99% of packets for bursty delay sensitive M2M traffic even when the system is fully loaded with regard to the data channel utilization.