A critically loaded multirate link with trunk reservation

We consider a loss system model of interest in telecommunications. There is a single service facility with N servers and no waiting room. There are K types of customers, with type ί customers requiring Aί servers simultaneously. Arrival processes are Poisson and service times are exponential. An arriving type ί customer is accepted only if there are Rί(⩾Aί ) idle servers. We examine the asymptotic behavior of the above system in the regime known as critical loading where both N and the offered load are large and almost equal. We also assume that R1,..., RK-1 remain bounded, while RKN←∞ and RKN/√N ← 0 as N ← ∞. Our main result is that the K dimensional “queue length” process converges, under the appropriate normalization, to a particular K dimensional diffusion. We show that a related system with preemption has the same limit process. For the associated optimization problem where accepted customers pay, we show that our trunk reservation policy is asymptotically optimal when the parameters satisfy a certain relation.