The relaxation time of two queueing systems in series

Abstract

This paper deals with the time-dependent behaviour of two queueing systems in series, the simplest example of a Jackson network. The Laplace transform of the probability p 0(t) that the tandem system is empty at time t is obtained by reducing the functional equation for the generating function of the joint queue length distribution to a Riemann-Hilbert boundary value problem. From this Laplace transform the relaxation time of p 0(t) is determined for all cases, and the first term of the asymptotic expansion ofp 0 (t)-p 0(oo) as too is found in the ergodic and in the null recurrent cases.

On the convergence to stationarity of birth-death processes

“…Since, as we will show, the integral (1) can be evaluated explicitly in terms of the birth and death rates of X it may be an attractive alternative to r(X ) as a one-parameter characterization of the speed of convergence. Rather than (1), however, we propose its normalized…”

“…(which is independent of j), or its reciprocal r(X ) ≡ 1/γ(X ), the relaxation time (see, for example, [1] and [12]). If M ≡ lim t→∞ E(X(t)) < ∞ we also have…”

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

“…For this problemθ is real, so that −θ = r(β, η). Our definition of the relaxation time in (7) assumes the initial condition p(x, 0) = δ(x − x 0 ) in (4), and then the approach to equilibrium is governed by λ 1 = r. But we could certainly have initial conditions that would lead to a faster approach. For example, if p(x, 0) = p(x, ∞) then p(x, t) = p(x, ∞) for all t and equilibrium is attained instantaneously.…”

“…Main results. The diffusion process (X(t)) t≥0 is a Markov process on the real line with continuous paths and density p = p(x, t) = p(x, t; x 0 ; β, η) that satisfies the forward Kolmogorov equation p(x, 0) = δ(x − x 0 ), (4) p(0 + , t) = p(0 − , t), p x (0 + , t) = p x (0 − , t), (5) and p(x, t) must decay as x → ±∞. The limiting distribution of the diffusion process is (see [15]) (6) p(x, ∞; x 0 ; β, η) = C e − 1 2 ηx 2 e −βx , x > 0, e − 1 2 x 2 e −βx , x < 0, where C −1 = ∞ 0 e − 1 2 ηx 2 e −βx dx + 0 −∞ e − 1 2 x 2 e −βx dx.…”

“…Denote this dominant singularity byθ and the spectral gap by r(β, η). The relaxation time, which measures the time it takes for the system to approach its steadystate behavior, is defined as (see [4,7]) (7) τ = inf{T : p(x, t; x 0 ; β, η) − p(x, ∞; x 0 ; β, η) = O(e −t/T )}, Downloaded from informs.org by [34.210.69.67] on 11 May 2018, at 04:05 . For personal use only, all rights reserved.…”