Ap Bert Zwart

Simultaneously Learning and Optimizing Using Controlled Variance Pricing

Price experimentation is an important tool for firms to find the optimal selling price of their products. It should be conducted properly, since experimenting with selling prices can be costly. A firm, therefore, needs to find a pricing policy that optimally balances between learning the optimal price and gaining revenue. In this paper, we propose such a pricing policy, called controlled variance pricing CVP. The key idea of the policy is to enhance the certainty equivalent pricing policy with a taboo interval around the average of previously chosen prices. The width of the taboo interval shrinks at an appropriate rate as the amount of data gathered gets large; this guarantees sufficient price dispersion. For a large class of demand models, we show that this procedure is strongly consistent, which means that eventually the value of the optimal price will be learned, and derive upper bounds on the regret, which is the expected amount of money lost due to not using the optimal price. Numerical tests indicate that CVP performs well on different demand models and time scales. This paper was accepted by Assaf Zeevi, stochastic models and simulation.

Tails in scheduling

This paper gives an overview of recent research on the impact of scheduling on the tail behavior of the response time of a job. We cover preemptive and non-preemptive scheduling disciplines, consider light-tailed and heavy-tailed distributions, and discuss optimality properties. The focus is on results, intuition and insight rather than methods and techniques.

Refining Square-Root Safety Staffing by Expanding Erlang C

We apply a new corrected diffusion approximation for the Erlang C formula to determine staffing levels in cost minimization and constraint satisfaction problems. These problems are motivated by large customer contact centers that are modeled as an M/M/s queue with s the number of servers or agents. The proposed staffing levels are refinements of the celebrated square-root safety-staffing rule and have the appealing property that they are as simple as the conventional square-root safety-staffing rule. In addition, we provide theoretical support for the empirical fact that square-root safety-staffing works well for moderate-sized systems.

Gaussian expansions and bounds for the Poisson distribution applied to the Erlang B formula

This paper presents new Gaussian approximations for the cumulative distribution function P(A λ ≤ s) of a Poisson random variable A λ with mean λ. Using an integral transformation, we first bring the Poisson distribution into quasi-Gaussian form, which permits evaluation in terms of the normal distribution function Φ. The quasi-Gaussian form contains an implicitly defined function y, which is closely related to the Lambert W-function. A detailed analysis of y leads to a powerful asymptotic expansion and sharp bounds on P(A λ ≤ s). The results for P(A λ ≤ s) differ from most classical results related to the central limit theorem in that the leading term Φ(β), with is replaced by Φ(α), where α is a simple function of s that converges to β as s tends to ∞. Changing β into α turns out to increase precision for small and moderately large values of s. The results for P(A λ ≤ s) lead to similar results related to the Erlang B formula. The asymptotic expansion for Erlangs B is shown to give rise to accurate approximations; the obtained bounds seem to be the sharpest in the literature thus far.

Large deviations of sojourn times in processor sharing queues

This paper presents a large deviation analysis of the steady-state sojourn time distribution in the GI/G/1 PS queue. Logarithmic estimates are obtained under the assumption of the service time distribution having a light tail, thus supplementing recent results for the heavy-tailed setting. Our proof gives insight into the way a large sojourn time occurs, enabling the construction of an (asymptotically efficient) importance sampling algorithm. Finally our results for PS are compared to a number of other service disciplines, such as FCFS, LCFS, and SRPT.

Is Tail-Optimal Scheduling Possible?

This paper focuses on the competitive analysis of scheduling disciplines in a large deviations setting. Although there are policies that are known to optimize the sojourn time tail under a large class of heavy-tailed job sizes e.g., processor sharing and shortest remaining processing time and there are policies known to optimize the sojourn time tail in the case of light-tailed job sizes e.g., first come first served, no policies are known that can optimize the sojourn time tail across both light-and heavy-tailed job size distributions. We prove that no such work-conserving, nonanticipatory, nonlearning policy exists, and thus that a policy must learn or know the job size distribution in order to optimize the sojourn time tail.

Sojourn Time Tails In The M / D /1 Processor Sharing Queue

We consider the sojourn time V in the M/D/1 processor sharing (PS) queue and show that P(V > x) is of the form Ce−gx as x becomes large. The proof involves a geometric random sum representation of V and a connection with Yule processes, which also enables us to simplify Otts [21] derivation of the Laplace transform of V. Numerical experiments show that the approximation P(V > x) ≈ Ce−gx is excellent even for moderate values of x.

Dynamic Pricing and Learning with Finite Inventories

We study a dynamic pricing problem with finite inventory and parametric uncertainty on the demand distribution. Products are sold during selling seasons of finite length, and inventory that is unsold at the end of a selling season perishes. The goal of the seller is to determine a pricing strategy that maximizes the expected revenue. Inference on the unknown parameters is made by maximum-likelihood estimation. We show that this problem satisfies an endogenous learning property, which means that the unknown parameters are learned on the fly if the chosen selling prices are sufficiently close to the optimal ones. We show that a small modification to the certainty equivalent pricing strategy-which always chooses the optimal price w.r.t. current parameter estimates-satisfies RegretT = Olog2T, where RegretT measures the expected cumulative revenue loss w.r.t. a clairvoyant who knows the demand distribution. We complement this upper bound by showing an instance for which the regret of any pricing policy satisfies I©log T.

Staffing Call Centers with Impatient Customers: Refinements to Many-Server Asymptotics

In call centers it is crucial to staff the right number of agents so that the targeted service levels are met. These staffing problems typically lead to constraint satisfaction problems that are hard to solve. During the last decade, a beautiful many-server asymptotic theory has been developed to solve such problems for large call centers, and optimal staffing rules are known to obey the square-root staffing principle. This paper presents refinements to many-server asymptotics and this staffing principle for a Markovian queueing model with impatient customers.

Corrected asymptotics for a multi-server queue in the Halfin-Whitt regime

To investigate the quality of heavy-traffic approximations for queues with many servers, we consider the steady-state number of waiting customers in an M/D/s queue as s→∞. In the Halfin-Whitt regime, it is well known that this random variable converges to the supremum of a Gaussian random walk. This paper develops methods that yield more accurate results in terms of series expansions and inequalities for the probability of an empty queue, and the mean and variance of the queue length distribution. This quantifies the relationship between the limiting system and the queue with a small or moderate number of servers. The main idea is to view the M/D/s queue through the prism of the Gaussian random walk: as for the standard Gaussian random walk, we provide scalable series expansions involving terms that include the Riemann zeta function.