The following queuing system is considered:Two independent recurrent input streams (streams 1 and 2) arrive at a server. It is assumed that stream 1 is of Poisson type. Three priority disciplines are studied in case that these customers have priority:head-of-the-line, preemptive-resume, and preemptive-repeat discipline. Formulas derived for the limiting distribution functions of the actual and the virtual waiting time of low priority customers and of the number of these customers in the system, by using of independences of certain random processes when the time tends to infinity.