Revisiting queueing output processes: a point process viewpoint

Abstract

After some historical notes concerning queueing output processes N _dep, the paper discusses methods for establishing asymptotic linear relations for var  N _dep(0,t], whether in the crude form B ₁ t or the more detailed form B ₁ t+B ₀+o(1) for t→∞. The crude form holds whenever the process N _adm of customers admitted to service has a linear asymptote, and then (var  N _dep(0,t])/t and (var  N _adm(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 N _dep is a renewal process whose generic lifetime X has finite second moment, then B ₁=(var  X)/([E(X)]²), and the more detailed linear asymptote holds when E(X ³) 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 var  N(0,t]∼B ₁ t for any stationary point process N possessing an embedded regenerative structure.