Many-server queues with customer abandonment: numerical analysis of their diffusion models

We use multidimensional diffusion processes to approximate the dynamics of a queue served by many parallel servers. The queue is served in the first-in-first-out (FIFO) order and the customers waiting in queue may abandon the system without service. Two diffusion models are proposed in this paper. They differ in how the patience time distribution is built into them. The first diffusion model uses the patience time density at zero and the second one uses the entire patience time distribution. To analyze these diffusion models, we develop a numerical algorithm for computing the stationary distribution of such a diffusion process. A crucial part of the algorithm is to choose an appropriate reference density. Using a conjecture on the tail behavior of a limit queue length process, we propose a systematic approach to constructing a reference density. With the proposed reference density, the algorithm is shown to converge quickly in numerical experiments. These experiments also show that the diffusion models are good approximations for many-server queues, sometimes for queues with as few as twenty servers

Critically loaded multi-server queues with abandonments, retrials, and time-varying parameters

In this paper, we consider modeling time-dependent multi-server queues that include abandonments and retrials. For the performance analysis of those, fluid and diffusion models called "strong approximations" have been widely used in the literature. Although they are proven to be asymptotically exact, their effectiveness as approximations in critically loaded regimes needs to be investigated. To that end, we find that existing fluid and diffusion approximations might be either inaccurate under simplifying assumptions or computationally intractable. To address that concern, this paper focuses on developing a methodology by adjusting the fluid and diffusion models so that they significantly improve the estimation accuracy. We illustrate the accuracy of our adjusted models by performing a number of numerical experiments

Many-server queues with customer abandonment

Customer call centers with hundreds of agents working in parallel are ubiquitous in many industries. These systems have a large amount of daily traffic that is stochastic in nature. It becomes more and more difficult to manage a call center because of its increasingly large scale and the stochastic variability in arrival and service processes. In call center operations, customer abandonment is a key factor and may significantly impact the system performance. It must be modeled explicitly in order for an operational model to be relevant for decision making. In this thesis, a large-scale call center is modeled as a queue with many parallel servers. To model the customer abandonment, each customer is assigned a patience time. When his waiting time for service exceeds his patience time, a customer abandons the system without service. We develop analytical and numerical tools for analyzing such a queue. We first study a sequence of G/G/n+GI queues, where the customer patience times are independent and identically distributed (iid) following a general distribution. The focus is the abandonment and the queue length processes. We prove that under certain conditions, a deterministic relationship holds asymptotically in diffusion scaling between these two stochastic processes, as the number of servers goes to infinity. Next, we restrict the service time distribution to be a phase-type distribution with d phases. Using the aforementioned asymptotic relationship, we prove limit theorems for G/Ph/n+GI queues in the quality- and efficiency-driven (QED) regime. In particular, the limit process for the customer number in each phase is a d-dimensional piecewise Ornstein-Uhlenbeck (OU) process. Motivated by the diffusion limit process, we propose two approximate models for a GI/Ph/n+GI queue. In each model, a d-dimensional diffusion process is used to approximate the dynamics of the queue. These two models differ in how the patience time distribution is built into them. The first diffusion model uses the patience time density at zero and the second one uses the entire patience time distribution. We also develop a numerical algorithm to analyze these diffusion models. The algorithm solves the stationary distribution of each model. The computed stationary distribution is used to estimate the queue's performance. A crucial part of this algorithm is to choose an appropriate reference density that controls the convergence of the algorithm. We develop a systematic approach to constructing a reference density. With the proposed reference density, the algorithm is shown to converge quickly in numerical experiments. These experiments also show that the diffusion models are good approximations of queues with a moderate to large number of servers.Ph.D.Committee Chair: Dai, Jiangang; Committee Member: Ayhan, Hayriye; Committee Member: Foley, Robert; Committee Member: Kleywegt, Anton; Committee Member: Tezcan, Tolg

Diffusion Models for Double-ended Queues with Renewal Arrival Processes

We study a double-ended queue where buyers and sellers arrive to conduct trades. When there is a pair of buyer and seller in the system, they immediately transact a trade and leave. Thus there cannot be non-zero number of buyers and sellers simultaneously in the system. We assume that sellers and buyers arrive at the system according to independent renewal processes, and they would leave the system after independent exponential patience times. We establish fluid and diffusion approximations for the queue length process under a suitable asymptotic regime. The fluid limit is the solution of an ordinary differential equation, and the diffusion limit is a time-inhomogeneous asymmetric Ornstein-Uhlenbeck process (O-U process). A heavy traffic analysis is also developed, and the diffusion limit in the stronger heavy traffic regime is a time-homogeneous asymmetric O-U process. The limiting distributions of both diffusion limits are obtained. We also show the interchange of the heavy traffic and steady state limits

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