Pseudo conservation for partially fluid, partially lossy queueing systems

We consider a queueing system with heterogeneous customers. One class of customers is eager; these customers are impatient and leave the system if service does not commence immediately upon arrival. Customers of the second class are tolerant; these customers have larger service requirements and can wait for service. In this paper, we establish pseudo-conservation laws relating the performance of the eager class (measured in terms of the long run fraction of customers blocked) and the tolerant class (measured in terms of the steady state performance, e.g., sojourn time, number in the system, workload) under a certain partial fluid limit. This fluid limit involves scaling the arrival rate as well as the service rate of the eager class proportionately to infinity, such that the offered load corresponding to the eager class remains constant. The workload of the tolerant class remains unscaled. Interestingly, our pseudo-conservation laws hold for a broad class of admission control and scheduling policies. This means that under the aforementioned fluid limit, the performance of the tolerant class depends only on the blocking probability of the eager class, and not on the specific admission control policy that produced that blocking probability. Our psuedo-conservation laws also characterize the achievable region for our system, which captures the space of feasible tradeoffs between the performance experienced by the two classes. We also provide two families of complete scheduling policies, which span the achievable region over their parameter space. Finally, we show that our pseudo-conservation laws also apply in several scenarios where eager customers have a limited waiting area and exhibit balking and/or reneging behaviour.