  Machine type communications (MTC) in 3G/4G networks is getting more attention
recently due to bursty nature of traffic characteristics in contrast to Poisson
type H2H traffic. A large number of methods have been suggested in the
literature. In this paper we give a mathematical model on performance analysis
of Disjoint Allocation (DA) and Joint Allocation (JA) methods for allocating
preambles to M2M and H2H users. In an earlier work we had investigated the
performance of two possible splitting preamble methods on collision probability
and energy reduction for MTC subscribers. In this paper we develop a stochastic
model for JA/DA method in RACH procedure using a $K$th order Markov chain
approach and carry out performance analysis in terms of collision probability,
access success probability, average access delay, statistics of preamble
transmissions and statistics of access delay for M2M users. The optimal number
of reserved preamble set is derived based on the given success access delay
threshold. Numerical results verifies the simulation results and demonstrates
that this model can estimate the performance of the JA/DA method accurately.
