We study queueing systems where customers have strict deadlines until the beginning of their service. An analytic method is given for the analysis of a class of such queues, namely, $$\begin{gathered} + \hfill \\ M\left( n \right)/M/m/FCFS + G \hfill \\ \end{gathered}$$ models. These are queues with state-dependent Poisson arrival process, exponential service times, multiple servers, FCFS service discipline, and general customer impatience. The state of the system is viewed to be the number of customers in the system. The principal measure of performance is the probability measure induced by the offered waiting time. Other measures of interest are the probability of missing deadline and the probability of blocking. Closed-form solutions are derived for the steady-state probabilities of the state process and some important modeling variables and parameters. The efficacy of our method is illustrated through a numerical example.