A variety of scheduling strategies can be employed in wireless systems to satisfy different system objectives and to cater for different traffic types. Static persistent resource allocations can be employed to transfer small M2M data packets efficiently compared to dynamic packet-by-packet scheduling, even when the M2M traffic model is non-deterministic. Recently adaptive persistent allocations have been proposed in which the volume of allocated resources can change in sympathy with the instantaneous queue size at the M2M device and without expensive signaling on control channels. This increases the efficiency of resource usage at the expense of a (typically small) increased packet delay. In this paper, we derive a statistical model for the device queue size and packet delay in static and adaptive persistent allocations which can be used for any arrival process (i.e., Poisson or otherwise). The primary motivation is to assist with dimensioning of persistent allocations given a set of QoS requirements (such as a prescribed delay budget). We validate the statistical model via comparison with queue size and delay statistics obtained from a discrete event simulation of a persistent allocation system. The validation is performed for both exponential and gamma distributed packet inter-arrivals to demonstrate the model generality.