On the rates of convergence of Erlang's model

Abstract

The convergence to equilibrium of the renormalized M/M/N/N queue is analysed. Upper bounds on the distance to equilibrium are obtained and the cut-off property for two regimes of this queue is proved. Simple probabilistic methods, such as coupling techniques and martingales, are used to obtain these results.

On the speed of convergence to stationarity of the Erlang loss system

“…The following result, which was stated in [6] (without proof) and proven in [19], establishes that, actually, both sides of (30) are asymptotically equal.…”

“…As an aside we note that the total variation distance between p(t) and π may exhibit very interesting behaviour if t and N tend to infinity simultaneously. A discussion of these issues is outside the scope of this paper (but see, for example, [6,21] and [20]). …”

Spectral Gap of the Erlang A Model in the Halfin-Whitt Regime

“…which establishes (12). Then since by definition r = r(β, η) is the minimal negative root of V = 0, the right-hand side of (96) has a root where −θ/η = r(−β/ √ η, 1/η) and then (13) follows immediately.…”

“…As η → ∞, the process will spend all its time below zero, and hence (X(t)) t≥0 reduces to a reflected OU process (see Ward and Glynn [35], Linetsky [24] and Fricker et al [12]). In this limit we have…”

Untitled

“…Usually it is assumed that the intensities are constant (see references in [18] and follow up papers [7,24,23]). The number of customers X(t) for this queue is a birth-death process with a finite phase space E = {0, 1, .…”