Carrier sense multiple access (CSMA) provides a feasible way to support machine type communications (MTC). However, in CSMA how to guarantee the diverse delay QoS efficiently is unsolved because the relationship between the back-off factor and delay QoS is unexploited. In this paper, we propose a new CSMA -type scheme named QoS-aware Local-Information-driven Multiple Access (LIMA) to handle this problem. The LIMA is formulated as a throughput maximization problem subject to the statistical delay QoS including the delay bound and delay bound violation probability. The diverse statistical delay QoS constraints are decomposed constraint into a series of the parameter adjusting period (PAP)-progressive QoS constraints in the time scale. Based on the effective capacity, the relationship between the back-off factor (defined as the average of back-off duration) and statistical delay QoS is studied. Then, each PAP-progressive QoS constraint is explicitly expressed in a simple form. Finally, a distributed algorithm is developed to find the optimal back-off factors based on local information. The global optimality of the proposed solution is theoretically proved. The salient feature of LIMA is that it can satisfy the diverse delay QoS requirements in a distributed way without signaling overhead for traffic information of other devises and scheduling information from the coordinator/access point (AP).