We present performance analysis of a wireless communication network system using multi-rate transmission with (automatic repeat request) ARQ-based error control and adaptive modulation and coding (AMC) scheme. The system is set up as a Markov chain. Arrivals and service are both in batches, and the arrival process is modeled as a batch Markovian arrival process (BMAP).  We obtain the distribution of the number of packets in the system and study other performance measures, such as buffer overflows for the finite buffer case and mean waiting time for both finite and infinite buffer cases. Finally we discuss how this idea can be extended to other wireless networks with reasonably similar features.