In the paper we deal with a Markovian queueing system with two heterogeneous servers and
constant retrial rate. The system operates under a threshold policy which prescribes the activation of the
faster server whenever it is idle and a customer tries to occupy it. The slower server can be activated only
when the number of waiting customers exceeds a threshold level. The dynamic behaviour of the system is
described by a two-dimensional Markov process that can be seen as a quasi-birth-and-death process with
infinitesimal matrix depending on the threshold. Using a matrix-geometric approach we perform a
stationary analysis of the system and derive expressions for the Laplace transforms of the waiting time as
well as arbitrary moments. Illustrative numerical results are presented for the threshold policy that
minimizes the mean number of customers in the system and are compared with other heuristic control
policies.