Diffusion models and steady-state approximations for exponentially ergodic Markovian queues

Motivated by queues with many servers, we study Brownian steady-state approximations for continuous time Markov chains (CTMCs). Our approximations are based on diffusion models (rather than a diffusion limit) whose steady-state, we prove, approximates that of the Markov chain with notable precision. Strong approximations provide such "limitless" approximations for process dynamics. Our focus here is on steady-state distributions, and the diffusion model that we propose is tractable relative to strong approximations. Within an asymptotic framework, in which a scale parameter $n$ is taken large, a uniform (in the scale parameter) Lyapunov condition imposed on the sequence of diffusion models guarantees that the gap between the steady-state moments of the diffusion and those of the properly centered and scaled CTMCs shrinks at a rate of $\sqrt{n}$. Our proofs build on gradient estimates for solutions of the Poisson equations associated with the (sequence of) diffusion models and on elementary martingale arguments. As a by-product of our analysis, we explore connections between Lyapunov functions for the fluid model, the diffusion model and the CTMC.Comment: Published in at http://dx.doi.org/10.1214/13-AAP984 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Coupled queues with customer impatience

Motivated by assembly processes, we consider a Markovian queueing system with multiple coupled queues and customer impatience. Coupling means that departures from all constituent queues are synchronised and that service is interrupted whenever any of the queues is empty and only resumes when all queues are non-empty again. Even under Markovian assumptions, the state space grows exponentially with the number of queues involved. To cope with this inherent state space explosion problem, we investigate performance by means of two numerical approximation techniques based on series expansions, as well as by deriving the fluid limit. In addition, we provide closed-form expressions for the first terms in the series expansion of the mean queue content for the symmetric coupled queueing system. By an extensive set of numerical experiments, we show that the approximation methods complement each other, each one being accurate in a particular subset of the parameter space. (C) 2017 Elsevier B.V. All rights reserved

Large deviations analysis for the M/H2/n+MM/H_2/n + M queue in the Halfin-Whitt regime

We consider the FCFS $M/H_2/n + M$ queue in the Halfin-Whitt heavy traffic regime. It is known that the normalized sequence of steady-state queue length distributions is tight and converges weakly to a limiting random variable W. However, those works only describe W implicitly as the invariant measure of a complicated diffusion. Although it was proven by Gamarnik and Stolyar that the tail of W is sub-Gaussian, the actual value of $\lim_{x \rightarrow \infty}x^{-2}\log(P(W >x))$ was left open. In subsequent work, Dai and He conjectured an explicit form for this exponent, which was insensitive to the higher moments of the service distribution. We explicitly compute the true large deviations exponent for W when the abandonment rate is less than the minimum service rate, the first such result for non-Markovian queues with abandonments. Interestingly, our results resolve the conjecture of Dai and He in the negative. Our main approach is to extend the stochastic comparison framework of Gamarnik and Goldberg to the setting of abandonments, requiring several novel and non-trivial contributions. Our approach sheds light on several novel ways to think about multi-server queues with abandonments in the Halfin-Whitt regime, which should hold in considerable generality and provide new tools for analyzing these systems

A Fixed-Point Algorithm for Closed Queueing Networks

In this paper we propose a new efficient iterative scheme for solving closed queueing networks with phase-type service time distributions. The method is especially efficient and accurate in case of large numbers of nodes and large customer populations. We present the method, put it in perspective, and validate it through a large number of test scenarios. In most cases, the method provides accuracies within 5% relative error (in comparison to discrete-event simulation)

Simple and explicit bounds for multi-server queues with 1/(1−ρ)1/(1 - \rho) (and sometimes better) scaling

We consider the FCFS $GI/GI/n$ queue, and prove the first simple and explicit bounds that scale as $\frac{1}{1-\rho}$ (and sometimes better). Here $\rho$ denotes the corresponding traffic intensity. Conceptually, our results can be viewed as a multi-server analogue of Kingman's bound. Our main results are bounds for the tail of the steady-state queue length and the steady-state probability of delay. The strength of our bounds (e.g. in the form of tail decay rate) is a function of how many moments of the inter-arrival and service distributions are assumed finite. More formally, suppose that the inter-arrival and service times (distributed as random variables $A$ and $S$ respectively) have finite $r$th moment for some $r > 2.$ Let $\mu_A$ (respectively $\mu_S$) denote $\frac{1}{\mathbb{E}[A]}$ (respectively $\frac{1}{\mathbb{E}[S]}$). Then our bounds (also for higher moments) are simple and explicit functions of $\mathbb{E}\big[(A \mu_A)^r\big], \mathbb{E}\big[(S \mu_S)^r\big], r$, and $\frac{1}{1-\rho}$ only. Our bounds scale gracefully even when the number of servers grows large and the traffic intensity converges to unity simultaneously, as in the Halfin-Whitt scaling regime. Some of our bounds scale better than $\frac{1}{1-\rho}$ in certain asymptotic regimes. More precisely, they scale as $\frac{1}{1-\rho}$ multiplied by an inverse polynomial in $n(1 - \rho)^2.$ These results formalize the intuition that bounds should be tighter in light traffic as well as certain heavy-traffic regimes (e.g. with $\rho$ fixed and $n$ large). In these same asymptotic regimes we also prove bounds for the tail of the steady-state number in service. Our main proofs proceed by explicitly analyzing the bounding process which arises in the stochastic comparison bounds of amarnik and Goldberg for multi-server queues. Along the way we derive several novel results for suprema of random walks and pooled renewal processes which may be of independent interest. We also prove several additional bounds using drift arguments (which have much smaller pre-factors), and make several conjectures which would imply further related bounds and generalizations