We construct an optimal execution strategy for the purchase of a large number of shares of a financial asset over a fixed interval of time. Purchases of the asset have a nonlinear impact on price, and this is moderated over time by resilience in the limit-order book that determines the price. The limit-order book is permitted to have arbitrary shape. The form of the optimal execution strategy is to make an initial lump purchase and then purchase continuously for some period of time during which the rate of purchase is set to match the order book resiliency. At the end of this period, another lump purchase is made, and following that there is again a period of purchasing continuously at a rate set to match the order book resiliency. At the end of this second period, there is a final lump purchase. Any of the lump purchases could be of size zero. A simple condition is provided that guarantees that the intermediate lump purchase is of size zero.

A queueing model has J ≥ 2 heterogeneous service stations, each consisting of many independent servers with identical capabilities. Customers of I ≥ 2 classes can be served at these stations at different rates, that depend on both the class and the station. A system administrator dynamically controls scheduling and routing. We study this model in the central limit theorem (or heavy traffic) regime proposed by Halfin and Whitt. We derive a diffusion model on R I with a singular control term that describes the scaling limit of the queueing model. The singular term may be used to constrain the diffusion to lie in certain subsets of R I at all times t > 0. We say that the diffusion is null-controllable if it can be constrained to X−, the minimal closed subset of R I containing all states of the prelimit queueing model for which all queues are empty. We give sufficient conditions for null controllability of the diffusion. Under these conditions we also show that an analogous, asymptotic result holds for the queueing model, by constructing control policies under which, for any given 0 < ε < T < ∞, all queues in the system are kept empty on the time interval [ε, T ], with probability approaching one. This introduces a new, unusual heavy traffic "behavior": On one hand, the system is critically loaded, in the sense that an increase in any of the external arrival rates at the "fluid level" results with an overloaded system. On the other hand, as far as queue lengths are concerned, the system behaves as if it is underloaded.

A Markovian queueing network is considered with d independent customer classes and d server pools in Halfin-Whitt regime. Class i customers has priority for service in pool i for i 1,. .. , d, and may access some other pool if the pool has an idle server and all the servers in pool i are busy. We formulate an ergodic control problem where the running cost is given by a non-negative convex function with polynomial growth. We show that the limiting controlled diffusion is modelled by an action space which depends on the state variable. We provide a complete analysis for the limiting ergodic control problem and establish asymptotic convergence of the value functions for the queueing model.

This paper introduces and analyzes the notion of throughput suboptimality for many-server queueing systems in heavy traffic. The queueing model under consideration has multiple customer classes, indexed by a finite set $\mathcal{I}$, and heterogenous, exponential servers. Servers are dynamically chosen to serve customers, and buffers are available for customers waiting to be served. The arrival rates and the number of servers are scaled up in such a way that the processes representing the number of class-$i$ customers in the system, $i\in\mathcal{I}$, fluctuate about a static fluid model, that is assumed to be critically loaded in a standard sense. At the same time, the fluid model is assumed to be throughput suboptimal. Roughly, this means that the servers can be allocated so as to achieve a total processing rate that is greater than the total arrival rate. We show that there exists a dynamic control policy for the queueing model that is efficient in the following strong sense: Under this policy, for every finite $T$, the measure of the set of times prior to $T$, at which at least one customer is in the buffer, converges to zero in probability as the arrival rates and number of servers go to infinity. On the way to prove our main result, we provide a characterization of throughput suboptimality in terms of properties of the buffer-station graph.Comment: Published in at http://dx.doi.org/10.1214/08-AAP551 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

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