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

Performance analysis of time-dependent queueing systems: survey and classification

Many queueing systems are subject to time-dependent changes in system parameters, such as the arrival rate or number of servers. Examples include time-dependent call volumes and agents at inbound call centers, time-varying air traffic at airports, time-dependent truck arrival rates at seaports, and cyclic message volumes in computer systems.There are several approaches for the performance analysis of queueing systems with deterministic parameter changes over time. In this survey, we develop a classification scheme that groups these approaches according to their underlying key ideas into (i) numerical and analytical solutions,(ii)approaches based on models with piecewise constant parameters, and (iii) approaches based on mod-ified system characteristics. Additionally, we identify links between the different approaches and provide a survey of applications that are categorized into service, road and air traffic, and IT systems

Sharing delay information in service systems: a literature survey

Service providers routinely share information about upcoming waiting times with their customers, through delay announcements. The need to effectively manage the provision of these announcements has led to a substantial growth in the body of literature which is devoted to that topic. In this survey paper, we systematically review the relevant literature, summarize some of its key ideas and findings, describe the main challenges that the different approaches to the problem entail, and formulate research directions that would be interesting to consider in future work

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

Analysis of buffer allocations in time-dependent and stochastic flow lines

This thesis reviews and classifies the literature on the Buffer Allocation Problem under steady-state conditions and on performance evaluation approaches for queueing systems with time-dependent parameters. Subsequently, new performance evaluation approaches are developed. Finally, a local search algorithm for the derivation of time-dependent buffer allocations is proposed. The algorithm is based on numerically observed monotonicity properties of the system performance in the time-dependent buffer allocations. Numerical examples illustrate that time-dependent buffer allocations represent an adequate way of minimizing the average WIP in the flow line while achieving a desired service level

Empirical Studies in Hospital Emergency Departments

This dissertation focuses on the operational impacts of crowding in hospital emergency departments. The body of this work is comprised of three essays. In the first essay, Waiting Patiently: An Empirical Study of Queue Abandonment in an Emergency Department, we study queue abandonment, or left without being seen. We show that abandonment is not only influenced by wait time, but also by the queue length and the observable queue flows during the waiting exposure. We 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 study shows that managers have an opportunity to impact abandonment behavior by altering what information is available to waiting customers. In the second essay, Doctors Under Load: An Empirical Study of State-Dependent Service Times in Emergency Care, we show that when crowded, multiple mechanisms in the emergency department act to retard patient treatment, but care providers adjust their clinical behavior to accelerate the service. We identify two mechanisms that providers use to accelerate the system: early task initiation and task reduction. In contrast to other recent works, we find the net effect of these countervailing forces to be an increase in service time when the system is crowded. Further, we use simulation to show that ignoring state-dependent service times leads to modeling errors that could cause hospitals to overinvest in human and physical resources. In the final essay, The Financial Consequences of Lost Demand and Reducing Boarding in Hospital Emergency Departments, we use discrete event simulation to estimate the number of patients lost to Left Without Being Seen and ambulance diversion as a result of patients waiting in the emergency department for an inpatient bed (known as boarding). These lost patients represent both a failure of the emergency department to meet the needs of those seeking care and lost revenue for the hospital. We show that dynamic bed management policies that proactively cancel some non-emergency patients when the hospital is near capacity can lead to reduced boarding, increased number of patients served, and increased hospital revenue

QUEUING SYSTEMS WITH STRATEGIC AND LEARNING CUSTOMERS

In many service systems customers are strategic and can make their own decisions. In particular, customers can be delay-sensitive and they will leave the system if they think the waiting time is too long. For the service provider, it is important to understand customers’ behaviors and choose the appropriate system design. This dissertation consists of two research projects. The first project studies the pooling decision when customers are strategic. It is generally accepted that operating with a combined (i.e., pooled) queue rather than separate (i.e., dedicated) queues is beneficial mainly because pooling queues reduces long-run average sojourn time. In fact, this is a well-established result in the literature when jobs cannot make decisions and servers and jobs are identical. An important corollary of this finding is that pooling queues improves social welfare in the aforementioned setting. We consider an observable multi-server queueing system which can be operated with either dedicated queues or a pooled one. Customers are delay-sensitive and they decide to join or balk based on queue length information upon arrival. In this setting, we prove that, contrary to the common understanding, pooling queues can considerably increase the long-run average sojourn time so that the pooled system results in strictly smaller social welfare (and strictly smaller consumer surplus) than the dedicated system under certain conditions. Specifically, pooling queues leads to performance loss when the arrival-rate-to-service-rate ratio and the relative benefit of service are both large. We also prove that performance loss due to pooling queues can be significant. Our numerical studies demonstrate that pooling queues can decrease the social welfare (and the consumer surplus) by more than 95%. The benefit of pooling is commonly believed to increase with the system size. In contrast to this belief, our analysis shows that when delay-sensitive customers make rational joining decisions, the magnitude of the performance loss due to pooling can strictly increase with the system size. The second project studies the learning behavior when customers don’t have full information of the service speed. We consider a single-server queueing system where customers make join- ing and abandonment decisions when the service rate is unknown. We study different ways in which customers process service-related information, and how these impact the performance of a service provider. Specifically, we analyze forward-looking, myopic and naive information process- ing behaviors by customers. Forward-looking customers learn about the service rate in a Bayesian framework by using their observations while waiting in the queue. Moreover, they make their abandonment decisions considering both belief and potential future payoffs. On the other hand, naive customers ignore the available information when they make their decisions. We prove that regardless of the way in which the information is processed by customers, a customer’s optimal joining and abandonment policy is of threshold-type. There is a rich literature that establishes that forward-looking customers are detrimental to a firm in settings different than queueing. In contrast to this common understanding, we prove that for service systems, forward-looking customers are beneficial to the firm under certain conditions.Doctor of Philosoph

The Operations and Design of Markets with Spatial and Incentive Considerations

Technology has greatly impacted how economic agents interact in various markets, including transportation and online display advertising. This calls for a better understanding of some of the key features of these marketplaces and the development of fundamental insights for this class of problems. In this thesis, we study markets for which spatial and incentive considerations are crucial factors for their operational and economic success. In particular, we study pricing and staffing decisions for ride-hailing platforms. We also consider the contract design problem faced by Ad Exchanges when buyers' strategic behavior and inherent business constraints limit these platforms' decisions. Firstly, we investigate the pricing challenges of ride-hailing platforms and propose a general measure-theoretical framework in which a platform selects prices for different locations, and drivers respond by choosing where to relocate based on prices, travel costs, and market congestion levels. Our results identify the revenue-maximizing pricing policy and showcase the importance of accounting for global network effects. Secondly, we develop a queuing approach to study the link between capacity and performance for a service firm with spatial operations. In a classical M/M/n queueing model, the square root safety (SRS) staffing rule balances server utilization and customer wait times. By contrast, we find that the SRS rule does not lead to such a balance in spatial systems. In these settings, a service firm should use a higher safety factor, proportional to the offered load to the power of 2/3. Lastly, motivated by the online display advertising market where publishers frequently use transaction-contingent fees instead of up-front fees, we study the classic sequential screening problem and isolate the impact of buyers' ex-post participation constraints. We characterize the optimal selling mechanism and provide an intuitive necessary and sufficient condition under which screening is better than pooling

Microstructure Analysis of Dynamic Markets: Limit Order Books and Dynamic Matching Markets

This thesis is concerned with addressing operational issues in two types of dynamic markets where queueing plays an important role: limit order books (financial industry), and dynamic matching markets (residential real estate). We first study the smart order routing decisions of investors in fragmented limit order book markets and the implications on the market dynamics. In modern equity markets, participants have a choice of many exchanges at which to trade. Exchanges typically operate as electronic limit order books operating under a “price-time” priority rule and, in turn, can be modeled as multi-class FIFO queueing systems. A market with multiple exchanges can be thought as a decentralized, parallel queueing system. Heterogeneous traders that submit limit orders select the exchange to place their orders by trading off delays until their order may fill against financial considerations. Simultaneously, traders that submit market orders select the exchange to direct their orders. These market orders trigger instantaneous service completions of queued limit orders. Taking into account the effect of investors’ order routing decisions, we find that the equilibrium of this decentralized market exhibits a state space collapse property. The predicted dimension reduction is the result of high-frequency order routing decisions that essentially couple the dynamics across exchanges. Analyzing a TAQ dataset for a sample of stocks over a one month period, we find empirical support for the predicted state space collapse. In the second part of this thesis, we model an electronic limit order book as a multi-class queueing system under fluid dynamics, and formulate and solve a problem of limit and market order placement to optimally buy a block of shares over a short, predetermined time horizon. Using the structure of the optimal execution policy, we identify microstructure variables that affect trading costs over short time horizons and propose a resulting microstructure-based model of market impact costs. We use a proprietary data set to estimate this cost model, and highlight its insightful structure and increased accuracy over conventional (macroscopic) market impact models that estimate the cost of a trade based on its normalized size but disregarding measurements of limit order book variables. In the third part of this thesis, we study the residential real estate markets as dynamic matching systems with an emphasis on their microstructure. We propose a stylized microstructure model and analyze the market dynamics and its equilibrium under the simplifying approximation where buyers and sellers use linear bidding strategies. We motivate and characterize this near closed-form approximation of the market equilibrium, and show that it is asymptotically accurate. We also provide numerical evidence in support of this approximation. Then with the gained tractability, we characterize steady-state properties such as market depth, price dispersion, and anticipated delays in selling or buying a unit. We characterize congestion and matching patterns for sellers and buyers, taking into account market dynamics, heterogeneity, and supply and demand imbalance manifested in the competition among buyers and sellers. Furthermore, we show the effects of market primitives with comparative statics results