The adaptive modulation and coding (AMC) technique, which has been adopted by advanced mobile telecommunication systems, supports a flexible response to the random radio behaviour. As a result, the attained transmission rate over a wireless link is time varying. Hence, resource demands are not deterministic but fluctuating even for calls with constant bit rate service requirements. Consequently, constant bit rate calls are susceptible to a forced call termination because of insufficient resources not only in a target cell during inter-cell handoffs but also in a serving cell during radio link deterioration. Furthermore, call blocking and dropping probabilities depend on radio propagation conditions among other factors and therefore they are dissimilar throughout a service area. The latter leads to unfairness problems. We analytically measure the impact of AMC on fixed-rate service with hard delay constraints such as voice for different signal, mobility and traffic conditions. We consider a reference case (call requests are admitted into the system provided there are enough free resources) and two classes of admission control approaches: traditional (only inter-cell handoffs are prioritised) and modified (all ongoing calls are prioritised). The reported results reveal conditions for which AMC affects voice call performance and can serve as guidelines on admission control design.