"Surrogate" Services

An incoming call gets a regular service by an agent if the actual or estimated waiting time is less then $\tau_w$ sec. Otherways, a call gets a "surrogate" service by voice-mail. For simplicity replace this service system by a system with $r=\tau_w / \tau_s$ where the expected service time is defined as $\tau_s=1 / m \mu$. Now we can use all the expressions for the queuing system with $r$ waiting places regarding the call served by voice-mail as the "lost" call. We may consider calls of different type separately, because they are served by different agents.