  In this paper, we consider co-existing two different types of devices in
machine-type communication (MTC), namely type-1 and type-2 devices, where
type-1 devices need short access delay for low-latency requirements, while
type-2 devices are delay-tolerant. For short access delay, we study the use of
fast retrial in preamble transmissions when a group of preambles is divided
into two subsets to support two different types of devices. Stability
conditions are derived using Foster-Lyapunov criteria in terms of arrival
rates, the number of preambles, and the number of type-1 devices. We also
propose an adaptive algorithm that dynamically decides the minimum number of
preambles for type- 1 devices under stability conditions.
