Much of the research on network modeling and analysis has focused on the design of end controllers and network algorithms with the objective of stability and convergence of the transmission rate. However, a network is typically composed of a mixture of both controlled elastic flows and uncontrolled real-time flows. In this paper, we study the effects of marking elasticity (which characterizes how aggressively the marking function responds to congestion) on queue overflow probability for uncontrolled real-time flows. First, we derive lower and upper bounds on the queue overflow probability at a router of a single bottleneck system. Using this, we quantify the trade-off between stability for controlled flows and queue overflow probability for uncontrolled real-time flows as a function of marking elasticity. Next, we compare the capacity required at a router with only FIFO scheduling versus a router with priority scheduling (priority given to the real-time flows) for supporting a given queue overflow probability. We quantify the "scheduling-gain" of priority scheduling over FIFO scheduling, as a function of marking elasticity. We show that this scheduling gain decreases with more elastic marking functions.