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Consider the nonpreemptive priority queueing system with two classes of packets. Class-1 packets have priority over class-2 packets. The packets of class 1 (2) arrive into the buffer according to the Poisson process with rate λ1 (λ2), respectively. The service time has the exponential distribution with the same rate μ for each class. The service times are independent of the arrival processes. The buffer has a finite size N and it is shared by both types of customers. If the buffer is full, a new incoming customer of class 1 can push out of the buffer a customer of class 2 with probability α. Note that if α = 1, we retrieve the standard nonrandomized push-out mechanism. The infinite buffer priority queueing has been thoroughly studied in [4, 7, 8]. The case of finite buffer priority queueing has received considerably less attention. The M | M | C | K type finite-buffer nonpreemptive priority queueing with nonrandomized push-out mechanism is analyzed in [5, 6]. Bondi, in [1], considers the M | M | 1 | K type preemptive and nonpreemptive priority queueing with the following buffer management schemes: complete partitioning, complete sharing, and sharing with minimum allocation. Wagner and Krieger, in [9], analyze the M | M | 1 | K type nonpreemptive priority queueing with complete sharing buffer management scheme and with class-dependent service rates. In [2], Cheng and Akyildiz consider the priority queueing with general service-time distributions and a general service discipline function. Most of the above works use recursive relations to solve steady-state Kolmogorov equations. We use a generating function approach, which only requires the solution of a linear system ofN equations. As far as we know, the randomized push-out mechanism is analyzed for the first time.