Using a recent result of Blanchet and Wallwater for exactly simulating the maximum of a negative drift random walk queue endowed with iid increments, we extend it to a multidmensional setting and then we give a new algorithm for simulating exactly the stationary distribution of a stable FIFO GI/GI/c queue (c>1),  as well as dealing with (for the first time) some other models. Our method for the FIFO GI/GI/c queue utilizes dominated coupling from the past (DCFP) as well as the Random Assignment (RA) discipline, and compliments the earlier works of Sigman (2011) in which Poisson arrivals were assumed,  as well as the recent work of Connor and Kendall (2015) who (also assuming Poisson arrivals) extended the DCFP method and time reversibility used by Sigman to allow for rho < c by also using the RA model. It also contrasts with a recent method used in Blanchet, Dong and Pei in which a vacation model was used as an upper bound. We also consider the models in continuous-time and show that with mild further assumptions, the exact simulation of those stationary distributions can also be achieved.  In addition we show how to handle models with different service disciplines, such as LIFO and “randomly choose next”.  We also give, using our FIFO algorithm, a new exact simulation algorithm for the stationary distribution of the GI /GI /inf model. Finally, we discuss further applications.
Joint work with Jose Blanchet and Yanan Pei.