Amy R. Ward

A Diffusion Approximation for a Markovian Queue with Reneging

Consider a single-server queue with a Poisson arrival process and exponential processing times in which each customer independently reneges after an exponentially distributed amount of time. We establish that this system can be approximated by either a reflected Ornstein–Uhlenbeck process or a reflected affine diffusion when the arrival rate exceeds or is close to the processing rate and the reneging rate is close to 0. We further compare the quality of the steady-state distribution approximations suggested by each diffusion.

A Diffusion Approximation for a GI/GI/1 Queue with Balking or Reneging

Consider a single-server queue with a renewal arrival process and generally distributed processing times in which each customer independently reneges if service has not begun within a generally distributed amount of time. We establish that both the workload and queue-length processes in this system can be approximated by a regulated Ornstein-Uhlenbeck (ROU) process when the arrival rate is close to the processing rate and reneging times are large. We further show that a ROU process also approximates the queue-length process, under the same parameter assumptions, in a balking model. Our balking model assumes the queue-length is observable to arriving customers, and that each customer balks if his or her conditional expected waiting time is too large.

Properties of the Reflected Ornstein–Uhlenbeck Process

Consider an Ornstein–Uhlenbeck process with reflection at the origin. Such a process arises as an approximating process both for queueing systems with reneging or state-dependent balking and for multi-server loss models. Consequently, it becomes important to understand its basic properties. In this paper, we show that both the steady-state and transient behavior of the reflected Ornstein–Uhlenbeck process is reasonably tractable. Specifically, we (1) provide an approximation for its transient moments, (2) compute a perturbation expansion for its transition density, (3) give an approximation for the distribution of level crossing times, and (4) establish the growth rate of the maximum process.

Optimal Control of a High-Volume Assemble-to-Order System

We consider an assemble-to-order system with a high volume of prospective customers arriving per unit time. Our objective is to maximize expected infinite-horizon discounted profit by choosing product prices, component production capacities, and a dynamic policy for sequencing customer orders for assembly. We prove that a myopic discrete-review sequencing policy, which allocates scarce components among orders for different products to minimize instantaneous physical and financial holding costs, is asymptotically optimal. Furthermore, we prove that optimal prices and production capacity nearly balance the supply and demand for components (i.e., it is economically optimal to operate the system in heavy traffic), so system performance is characterized by a diffusion approximation. The diffusion approximation exhibits state-space collapse: Its dimension equals the number of components (rather than the number of components plus the number of products). These results complement the existing assemble-to-order literature, which focuses on managing component inventory and assumes FIFO sequencing of orders for assembly.

Asymptotically Optimal Admission Control of a Queue with Impatient Customers

We consider a GI/GI/1 queue with impatient customers in heavy traffic. We use the solution of an approximating singular diffusion control problem to construct an admission control policy for the queue. The approximating control problem does not admit a so-called pathwise solution. Hence, the resulting admission control policy depends on second-moment data. We prove asymptotic optimality of the constructed policy using weak-convergence methods.rh

Internet service performance failure detection

The increasing complexity of computer networks and our increasing dependence on them means enforcing reliability requirements is both more challenging and more critical. The expansion of network services to include both traditional interconnect services and user-oriented services such as the web and email has guaranteed both the increased complexity of networks and the increased importance of their performance. The first step toward increasing reliability is early detection of network performance failures. Here we consider the applicability of statistical model frameworks under the most general assumptions possible. Using measurements from corporate proxy servers, we test the framework against real world failures. The results of these experiments show we can detect failures, but with some tradeoff questions. The pull is in the warning time: either we miss early warning signs or we report some false warnings. Finally, we offer insight into the problem of failure diagnosis.

Approximating the GI/GI/1+GI Queue with a Nonlinear Drift Diffusion: Hazard Rate Scaling in Heavy Traffic

We study a single-server queue, operating under the first-in-first-out (FIFO) service discipline, in which each customer independently abandons the queue if his service has not begun within a generally distributed amount of time. Under some mild conditions on the abandonment distribution, we identify a limiting heavy-traffic regime in which the resulting diffusion approximation for both the offered waiting time process (the process that tracks the amount of time an infinitely patient arriving customer would wait for service) and the queue-length process contain the entire abandonment distribution. To use a continuous mapping approach to establish our weak convergence results, we additionally develop existence, uniqueness, and continuity results for nonlinear generalized regulator mappings that are of independent interest. We further perform a simulation study to evaluate the quality of the proposed approximations for the steady-state mean queue length and the steady-state probability of abandonment suggested by the limiting diffusion process.

A Separation Principle for a Class of Assemble-to-Order Systems with Expediting

In an assemble-to-order system, a wide variety of products are rapidly assembled from component inventories in response to customer orders. We assume that orders must be filled within a product-specific target lead time. In the event that some of the components required to fill an order are out of stock, these components must be expedited. The objective is to minimize the expected infinite-horizon discounted cost of primary component production and expediting. Our formulation captures financial holding costs but implicitly assumes that physical holding costs are negligible. The controls are (1) sequencing orders for assembly, (2) primary component production, and (3) component expediting. We prove that the multidimensional assemble-to-order control problem separates into single-item inventory control problems. In particular, under an optimal policy for assembly sequencing, the optimal production and expediting policy for each component is independent of all other components. Hence, the literature on single-item inventory management with expediting or lost sales is directly relevant to the control of assemble-to-order systems.

On the dynamic control of matching queues

We consider the optimal control of matching queues with random arrivals. In this model, items arrive to dedicated queues, and wait to be matched with items from other (possibly multiple) queues. A match type corresponds to the set of item classes required for a match. Once a decision has been made to perform a match, the matching itself is instantaneous and the matched items depart from the system. We consider the problem of minimizing finite-horizon cumulative holding costs. The controller must decide which matchings to execute given multiple options. In principle, the controller may choose to wait until some “inventory” of items builds up to facilitate more profitable matches in the future. We introduce a multi-dimensional imbalance process, that at each time t , is given by a linear function of the cumulative arrivals to each of the item classes. A non-zero value of the imbalance at time t means that no control could have matched all the items that arrived by time t . A lower bound based on the imbalance process can be specified, at each time point, by a solution to an optimization problem with linear constraints.While not achievable in general, this lower bound can be asymptotically approached under a dedicated item condition (an analogue of the local traffic condition in bandwidth sharing networks). We devise a myopic discrete-review matching control that asymptotically–as the arrival rates become large–achieves the imbalance-based lower bound.

Blind Fair Routing in Large-Scale Service Systems with Heterogeneous Customers and Servers

In a call center, arriving customers must be routed to available servers, and servers that have just become available must be scheduled to help waiting customers. These dynamic routing and scheduling decisions are very difficult, because customers have different needs and servers have different skill levels. A further complication is that it is preferable that these decisions are made blindly; that is, they depend only on the system state and not on system parameter information such as call arrival rates and service speeds. This is because this information is generally not known with certainty. Ideally, a dynamic control policy for making routing and scheduling decisions balances customer and server needs by keeping customer delays low but still fairly dividing the workload amongst the various servers. In this paper, we propose a blind dynamic control policy for parallel-server systems with multiple customer classes and server pools that is based on the number of customers waiting and the number of agents idling. We show that in the Halfin-Whitt many-server heavy-traffic limiting regime, our proposed blind policy performs extremely well when the objective is to minimize customer holding costs subject to “server fairness,” as defined by how the system idleness is divided among servers. To do this, we formulate an approximating diffusion control problem DCP and compare the performance of the nonblind DCP solution to a feasible policy for the DCP that is blind. We establish that the increase in the DCP objective function value is small over a wide range of parameter values. We then use simulation to validate that a small increase in the DCP objective function value is indicative of our proposed blind policy performing very well.