Limiting distribution for the moment of first loss of a customer in a single-line service system with a limited number of positions in the queue

Abstract

We consider a single-line service system with a Palm arrival rate and exponential service time, with n-1 places in the queue. Let τ_n be the moment of first loss of a customer. It is assumed that$\alpha _0 = \int_0^\infty {e^{ - t_{dF (t) \to 0} } }$, where F(t) is the distribution function of the time interval between successive arrivals of customers. We shall study the class of limiting distributions of the quantity τ_nδ (α₀), where δ (α₀) is some normalizing factor. We shall obtain conditions for which Pτ_n/Mτ_n<→ 1 −e^−t.