Josh Reed

The G/GI/N queue in the Halfin–Whitt regime

In this paper, we study the

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

G/\mathit{GI}/N

On many-server queues in heavy traffic

queue in the Halfin--Whitt regime. Our first result is to obtain a deterministic fluid limit for the properly centered and scaled number of customers in the system which may be used to provide a first-order approximation to the queue length process. Our second result is to obtain a second-order stochastic approximation to the number of customers in the system in the Halfin--Whitt regime. This is accomplished by first centering the queue length process by its deterministic fluid limit and then normalizing by an appropriate factor. We then proceed to obtain an alternative but equivalent characterization of our limiting approximation which involves the renewal function associated with the service time distribution. This alternative characterization reduces to the diffusion process obtained by Halfin and Whitt [Oper. Res. 29 (1981) 567--588] in the case of exponentially distributed service times.

Forecasting Prices from Level-I Quotes in the Presence of Hidden Liquidity

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.

High Frequency Asymptotics for the Limit Order Book

We establish a heavy-traffic limit theorem on convergence in distribution for the number of customers in a many-server queue when the number of servers tends to infinity. No critical loading condition is assumed. Generally, the limit process does not have trajectories in the Skorohod space. We give conditions for the convergence to hold in the topology of compact convergence. Some new results for an infinite server are also provided.

Hazard Rate Scaling of the Abandonment Distribution for the GI/M/n + GI Queue in Heavy Traffic

Bid and ask sizes at the top of the order book provide information on short-term price moves. Drawing from classical descriptions of the order book in terms of queues and order-arrival rates (Smith et al., 2003), we consider a diffusion model for the evolution of the best bid/ask queues. We compute the probability that the next price move is upward, conditional on the best bid/ask sizes, the hidden liquidity in the market and the correlation between changes in the bid/ask sizes. The model can be useful, among other things, to rank trading venues in terms of the “information content” of their quotes and to estimate hidden liquidity in a market based on high-frequency data. We illustrate the approach with an empirical study of a few stocks using quotes from various exchanges

An Overloaded Multiclass FIFO Queue with Abandonments

We study the one-sided limit order book corresponding to limit sell orders and model it as a measure-valued process. Limit orders arrive to the book according to a Poisson process and are placed on the book according to a distribution which varies depending on the current best price. Market orders to buy periodically arrive to the book according to a second, independent Poisson process and remove from the book the order corresponding to the current best price. We consider the above described limit order book in a high frequency regime in which the rate of incoming limit and market orders is large and traders place their limit sell orders close to the current best price. Our first set of results provide weak limits for the unscaled price process and the properly scaled measure-valued limit order book process in the high frequency regime. In particular, we characterize the limiting measure-valued limit order book process as the solution to a measure-valued stochastic differential equation. We then provide an analysis of both the transient and long-run behavior of the limiting limit order book process.

Distribution-valued heavy-traffic limits for the

We obtain a heavy traffic limit for the GI/M/n + GI queue, which includes the entire patience time distribution. Our main approach is to scale the hazard rate function of the patience time distribution in such a way that our resulting diffusion approximation contains the entire hazard rate function. We then show through numerical studies that for various performance measures, our approximations tend to outperform those commonly used in practice. The robustness of our results is also demonstrated by applying them to solving constraint satisfaction problems arising in the context of telephone call centers.

\mathbf{{G/\mathit{GI}/\infty}}

In this paper we consider a single-server queue fed by K independent renewal arrival streams, each representing a different job class. Jobs are processed in a FIFO fashion, regardless of class. The total amount of work arriving to the system exceeds the servers capacity. That is, the nominal traffic intensity of the system is assumed to be greater than one. Jobs arriving to the system grow impatient and abandon the queue after a random amount of time if service has not yet begun. Interarrival, service, and abandonment times are assumed to be generally distributed and class specific. We approximate this system using both fluid and diffusion limits. To this end, we consider a sequence of systems indexed by n in which the arrival and service rates are proportional to n; the abandonment distribution remains fixed across the sequence. In our first main result, we show that in the limit as n tends to infinity, the virtual waiting time process converges to a limiting deterministic process. This limit may be characterized as the solution to a first-order ordinary differential equation ODE. Specific examples are then presented for which the ODE may be explicitly solved. In our second main result, we refine the deterministic fluid approximation by showing that the fluid-centered and diffusion-scaled virtual waiting time process weakly converges to an Ornstein-Uhlenbeck process whose drift and infinitesimal variance both vary over time. This process may also be solved for explicitly, thus yielding approximations to the transient as well as steady-state behavior of the virtual waiting time process.

queue

We study the