Revisiting queueing output processes: a point process viewpoint

After some historical notes concerning queueing output processes Ndep , the paper discusses methods for establishing asymptotic linear relations for varNdep (0, t], whether in the crude form B1t or the more detailed form B1t + B0 + o(1) for t → ∞. The crude form holds whenever the process Nadm of customers admitted to service has a linear asymptote, and then (varNdep (0, t])/t and (varNadm(0, t])/t share a common limit (that may be infinite) in stationary G/G/k/K systems. A standard integral formula for the variance of a stationary orderly point process shows that, if Ndep is a renewal process whose generic lifetime X has finite second moment, then B1 = (varX)/([E(X)]2), and the more detailed linear asymptote holds when E(X3) is finite. Geometric ergodicity of the queue size process Q(·) in stationary M/M/k/K systems establishes that the more detailed linear asymptote is true for them. It is conjectured that varN(0, t] ∼ B1t for any stationary point process N possessing an embedded regenerative structure.