In this paper, we consider a single-source multi-server generate-at-will discrete-time non-preemptive status update system where update packets are transmitted using {\em only one} of the available servers, according to a server selection policy. In particular, when a transmission is complete, the update system makes a threshold-based decision on whether to wait or transmit, and if latter, which server to use for transmissions, on the basis of the instantaneous value of the age of information (AoI) process. In our setting, servers have general heterogeneous discrete phase-type (DPH) distributed service times, and also heterogeneous transmission costs. The goal is to find an age-dependent multi-threshold policy that minimizes the AoI cost with a constraint on transmission costs, the former cost defined in terms of the time average of an arbitrary function of AoI. For this purpose, we propose a novel tool called \emph{multi-regime absorbing Markov chain} (MR-AMC) in discrete time. Using the MR-AMC framework, we exactly obtain the distribution of AoI, and subsequently the costs associated with AoI and transmissions. With the exact analysis in hand, optimum thresholds can be obtained in the case of a few servers, by exhaustive search. We validate the proposed analytical model, and also demonstrate the benefits of age-dependent server selection, with numerical examples.