On the rate of convergence to stationarity of the M/M/N queue in the Halfin–Whitt regime

Abstract

We prove several results about the rate of convergence to stationarity, that is, the spectral gap, for the M/M/n queue in the Halfin-Whitt regime. We identify the limiting rate of convergence to steady-state, and discover an asymptotic phase transition that occurs w.r.t. this rate. In particular, we demonstrate the existence of a constant B * ≈ 1.85772 s.t. when a certain excess parameter B ∈ (0, B * ], the error in the steady-state approximation converges exponentially fast to zero at rate B 2 4 . For B > B * , the error in the steady-state approximation converges exponentially fast to zero at a different rate, which is the solution to an explicit equation given in terms of special functions. This result may be interpreted as an asymptotic version of a phase transition proven to occur for any fixed n by van Doorn [Stochastic Monotonicity and Queueing Applications of Birth-death Processes (1981) Springer].We also prove explicit bounds on the distance to stationarity for the M/M/n queue in the Halfin-Whitt regime, when B < B * . Our bounds scale independently of n in the Halfin-Whitt regime, and do not follow from the weak-convergence theory.and Whitt [18], who studied the GI/M/n system for large n when the traffic intensity scales like 1− Bn −1/2 for some strictly positive excess parameter B. They proved that, under minor technical assumptions on the inter-arrival distribution, this sequence of GI/M/n queueing models has the following properties:(i) the steady-state probability that an arriving job has to wait for service has a nontrivial limit;(ii) the sequence of queueing processes, normalized by n 1/2 , converges weakly to a nontrivial positive recurrent diffusion, a.k.a. the HW diffusion;(iii) the sequence of steady-state queue length distributions, normalized by n 1/2 , is tight and converges distributionally to the mixture of a point mass at zero and an exponential distribution.Since the steady-state behavior of the M/M/n queue in the HW regime is quite simple [18], while the transient dynamics are more complicated [18], it is common to use the steady-state approximation to the transient distribution [16]. Thus it is important to understand the quality of the steady-state approximation. The only work along these lines seems to be the recent papers [38,39], in which the authors study the Laplace transform of the HW and related diffusions, and prove several results analogous to our own for these diffusions. The key difference is that in this paper we study the prelimit diffusion-scaled M/M/n queue, not the limiting diffusion. We note that the relevant transform functions were also studied in [1], although in a different context. Also, similar questions were studied for the associated sequence of fluid-scaled queues in [23].The question of how quickly the positive recurrent M/M/n queue approaches stationarity has a rich history in the queueing literature. In [28], Morse derives an explicit solution for the transient M/M/1 queue, and discusses implications for the exponential rate of convergence to stationarity...

Spectral gap of the Erlang A model in the Halfin-Whitt regime

“…By expanding (14) for β → −∞ and (167) as η → 0 we see that the two agree in this intermediate limit. Thus for β large and negative, r is exponentially close to η, as could be expected since then almost all of the probability mass in the model migrates to the range x > 0.…”

“…The mean hitting time of the Halfin-Whitt diffusion was obtained in Maglaras and Zeevi [25]. Gamarnik and Goldberg [14] were the first to identify the spectral gap of the M/M/s system, asymptotically in the Halfin-Whitt regime.…”

“…Theorem 12 (Gamarnik and Goldberg [14]). Let β * = 1.85722... represent the smallest positive solution to D ′ β 2 /4 (−β) = 0.…”

“…In [14] the starting point is the discrete M/M/s model, and its spectral gap is then analyzed in the Halfin-Whitt regime (1). An alternative proof of Theorem 12 was given by the authors in [23] by deriving the expression for the Laplace transformp of the transient density in the diffusion limit, which shows that the limits of large time and (1) may be, in this case, interchanged.…”

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

“…When β becomes small, both (14) and (18) become invalid, as the correction terms become larger than the leading term. Then a separate analysis leads to the following result.…”