In this technical note, we consider closed tandem queueing networks with finite buffers and blocking. We assume that at least one station has subexponential service time distribution. We analyze this system under communication blocking and manufacturing blocking rules. Our objective is to derive expressions for the tail asymptotics of transient cycle times and waiting times. Furthermore, we study under which conditions on system parameters these tail asymptotics also hold for their stationary counter parts. Finally, we provide numerical examples to understand the convergence behavior of the tail asymptotics.