50 research outputs found
Large deviations analysis for the queue in the Halfin-Whitt regime
We consider the FCFS 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 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
Fluid Models of Many-server Queues with Abandonment
We study many-server queues with abandonment in which customers have general
service and patience time distributions. The dynamics of the system are modeled
using measure- valued processes, to keep track of the residual service and
patience times of each customer. Deterministic fluid models are established to
provide first-order approximation for this model. The fluid model solution,
which is proved to uniquely exists, serves as the fluid limit of the
many-server queue, as the number of servers becomes large. Based on the fluid
model solution, first-order approximations for various performance quantities
are proposed
Recommended from our members
Asymptotic Analysis of Service Systems with Congestion-Sensitive Customers
Many systems in services, manufacturing, and technology, feature users or customers sharing a limited number of resources, and which suffer some form of congestion when the number of users exceeds the number of resources. In such settings, queueing models are a common tool for describing the dynamics of the system and quantifying the congestion that results from the aggregated effects of individuals joining and leaving the system. Additionally, the customers themselves may be sensitive to congestion and react to the performance of the system, creating feedback and interaction between individual customer behavior and aggregate system dynamics.This dissertation focuses on the modeling and performance of service systems with congestion-sensitive customers using large-scale asymptotic analyses of queueing models. This work extends the theoretical literature on congestion-sensitive customers in queues in the settings of service differentiation and observational learning and abandonment. Chapter 2 considers the problem of a service provider facing a heterogeneous market of customers who differ based on their value for service and delay sensitivity. The service provider seeks to find the revenue maximizing level of service differentiation (offering different price-delay combinations). We show that the optimal policy places the system in heavy traffic, but at substantially different levels of congestion depending on the degree of service differentiation. Moreover, in a differentiated offering, the level of congestion will vary substantially between service classes. Chapter 3 presents a new model of customer abandonment in which congestion-sensitive customers observe the queue length, but do not know the service rate. Instead, they join the queue and observe their progress in order to estimate their wait times and make abandonment decisions. We show that an overloaded queue with observational learning and abandonment stabilizes at a queue length whose scale depends on the tail of the service time distribution. Methodologically, our asymptotic approach leverages stochastic limit theory to provide simple and intuitive results for optimizing or characterizing system performance. In particular, we use the analysis of deterministic fluid-type queues to provide a first-order characterization of the stochastic system dynamics, which is demonstrated by the convergence of the stochastic system to the fluid model. This also allows us to crisply illustrate and quantify the relative contributions of system or customer characteristics to overall system performance
Call Center Experience Optimization: A Case for a Virtual Predictive Queue
The evolution of the call center into contact centers and the growth of their use in providing customer-facing service by many companies has brought considerable capabilities in maintaining customer relationships but it also has brought challenges in providing quality service when call volumes are high. Limited in their ability to provide service at all times to all customers, companies are forced to balance the costs associated with hiring more customer service representatives and the quality of service provided by a fewer number. A primary challenge when there are not enough customer service representatives to engage the volume of callers in a timely manner is the significant wait times that can be experienced by many customers. Normally, callers are handled in accordance with a first-come, first-served policy with exceptions being skill-based routing to those customer service representatives with specialized skills. A proposed call center infrastructure framework called a Virtual Predictive Queue (VPQ) can allow some customers to benefit from a shorter call queue wait time. This proposed system can be implemented within a call center’s Automatic Call Distribution (ACD) device associated with computer telephony integration (CTI) and theoretically will not violate a first-come, first served policy
Waiting Patiently: An Empirical Study of Queue Abandonment in an Emergency Department
We study queue abandonment from a hospital emergency department. We show that abandonment is influenced by the queue length and the observable queue flows during the waiting exposure, even after controlling for wait time. For example, observing an additional person in the queue or an additional arrival to the queue leads to an increase in abandonment probability equivalent to a 25-minute or 5-minute increase in wait time, respectively. We also show that patients are sensitive to being “jumped” in the line and that patients respond differently to people more sick and less sick moving through the system. This customer response to visual queue elements is not currently accounted for in most queuing models. Additionally, to the extent the visual queue information is misleading or does not lead to the desired behavior, managers have an opportunity to intervene by altering what information is available to waiting customers
Estimating customer impatience in a service system with unobserved balking
This paper studies a service system in which arriving customers are provided
with information about the delay they will experience. Based on this
information they decide to wait for service or to leave the system. The main
objective is to estimate the customers' patience-level distribution and the
corresponding potential arrival rate, using knowledge of the actual
queue-length process only. The main complication, and distinguishing feature of
our setup, lies in the fact that customers who decide not to join are not
observed, but, remarkably, we manage to devise a procedure to estimate the load
they would generate. We express our system in terms of a multi-server queue
with a Poisson stream of customers, which allows us to evaluate the
corresponding likelihood function. Estimating the unknown parameters relying on
a maximum likelihood procedure, we prove strong consistency and derive the
asymptotic distribution of the estimation error. Several applications and
extensions of the method are discussed. The performance of our approach is
further assessed through a series of numerical experiments. By fitting
parameters of hyperexponential and generalized-hyperexponential distributions
our method provides a robust estimation framework for any continuous
patience-level distribution