Achievable region with impatient customers

We consider a queueing system with heterogeneous agents. One class of agents demand immediate service, would leave the system if not provided. The second class of customers have longer job requirements and can wait for their turn. We discuss the achievable region of such a two-class system, which is the region of all possible pairs of performance metrics. Blocking probability is the relevant performance for eager/impatient class while the expected sojourn time is appropriate for the tolerant class. We obtain the achievable region, considering static policies that do not depend upon the state of the second class, using a conjecture of a pseudo conservation law. This law relates the blocking probability of eager customers with the expected sojourn time of the tolerant customers, in a fluid limit for eager customers. We validate the pseudo conservation law using two example families of static schedulers, by deriving their performance. Along the way, we obtain smooth control (sharing) of resources between such heterogeneous classes (e.g., voice and data calls of a communication system). We also demonstrate that the dynamic achievable region (obtained using state dependent policies) is strictly bigger than the static region, by deriving the performance of an example dynamic policy.