skip to main content

This content will become publicly available on March 15, 2023

Title: Queues with Updating Information: Finding the Amplitude of Oscillations
Many service systems provide customers with information about the system so that customers can make an informed decision about whether to join or not. Many of these systems provide information in the form of an update. Thus, the information about the system is updated periodically in increments of size [Formula: see text]. It is known that these updates can cause oscillations in the resulting dynamics. However, it is an open problem to explicitly characterize the size of these oscillations when they occur. In this paper, we solve this open problem and show how to exactly calculate the amplitude of these oscillations via a fixed point equation. We also calculate closed form approximations via Taylor expansions of the fixed point equation and show that these approximations are very accurate, especially when [Formula: see text] is large. Our analysis provides new insight for systems that use updates as a way of disseminating information to customers.
Authors:
;
Award ID(s):
1751975
Publication Date:
NSF-PAR ID:
10335427
Journal Name:
International Journal of Bifurcation and Chaos
Volume:
32
Issue:
03
ISSN:
0218-1274
Sponsoring Org:
National Science Foundation
More Like this
  1. The replenishment storage problem (RSP) is to minimize the storage capacity requirement for a deterministic demand, multi-item inventory system, where each item has a given reorder size and cycle length. We consider the discrete RSP, where reorders can only take place at an integer time unit within the cycle. Discrete RSP was shown to be NP-hard for constant joint cycle length (the least common multiple of the length of all individual cycles). We show here that discrete RSP is weakly NP-hard for constant joint cycle length and prove that it is strongly NP-hard for nonconstant joint cycle length. For constant joint cycle-length discrete RSP, we further present a pseudopolynomial time algorithm that solves the problem optimally and the first known fully polynomial time approximation scheme (FPTAS) for the single-cycle RSP. The scheme is utilizing a new integer programming formulation of the problem that is introduced here. For the strongly NP-hard RSP with nonconstant joint cycle length, we provide a polynomial time approximation scheme (PTAS), which for any fixed [Formula: see text], provides a linear time [Formula: see text] approximate solution. The continuous RSP, where reorders can take place at any time within a cycle, seems (with our results) to bemore »easier than the respective discrete problem. We narrow the previously known complexity gap between the continuous and discrete versions of RSP for the multi-cycle RSP (with either constant or nonconstant cycle length) and the single-cycle RSP with constant cycle length and widen the gap for single-cycle RSP with nonconstant cycle length. For the multi-cycle case and constant joint cycle length, the complexity status of continuous RSP is open, whereas it is proved here that the discrete RSP is weakly NP-hard. Under our conjecture that the continuous RSP is easier than the discrete one, this implies that continuous RSP on multi-cycle and constant joint cycle length (currently of unknown complexity status) is at most weakly NP-hard.« less
  2. Many service systems provide queue length information to customers, thereby allowing customers to choose among many options of service. However, queue length information is often delayed, and it is often not provided in real time. Recent work by Dong et al. [Dong J, Yom-Tov E, Yom-Tov GB (2018) The impact of delay announcements on hospital network coordination and waiting times. Management Sci. 65(5):1969–1994.] explores the impact of these delays in an empirical study in U.S. hospitals. Work by Pender et al. [Pender J, Rand RH, Wesson E (2017) Queues with choice via delay differential equations. Internat. J. Bifurcation Chaos Appl. Sci. Engrg. 27(4):1730016-1–1730016-20.] uses a two-dimensional fluid model to study the impact of delayed information and determine the exact threshold under which delayed information can cause oscillations in the dynamics of the queue length. In this work, we confirm that the fluid model analyzed by Pender et al. [Pender J, Rand RH, Wesson E (2017) Queues with choice via delay differential equations. Internat. J. Bifurcation Chaos Appl. Sci. Engrg. 27(4):1730016-1–1730016-20.] can be rigorously obtained as a functional law of large numbers limit of a stochastic queueing process, and we generalize their threshold analysis to arbitrary dimensions. Moreover, we prove amore »functional central limit theorem for the queue length process and show that the scaled queue length converges to a stochastic delay differential equation. Thus, our analysis sheds new insight on how delayed information can produce unexpected system dynamics.« less
  3. The prevalence of e-commerce has made customers’ detailed personal information readily accessible to retailers, and this information has been widely used in pricing decisions. When using personalized information, the question of how to protect the privacy of such information becomes a critical issue in practice. In this paper, we consider a dynamic pricing problem over T time periods with an unknown demand function of posted price and personalized information. At each time t, the retailer observes an arriving customer’s personal information and offers a price. The customer then makes the purchase decision, which will be utilized by the retailer to learn the underlying demand function. There is potentially a serious privacy concern during this process: a third-party agent might infer the personalized information and purchase decisions from price changes in the pricing system. Using the fundamental framework of differential privacy from computer science, we develop a privacy-preserving dynamic pricing policy, which tries to maximize the retailer revenue while avoiding information leakage of individual customer’s information and purchasing decisions. To this end, we first introduce a notion of anticipating [Formula: see text]-differential privacy that is tailored to the dynamic pricing problem. Our policy achieves both the privacy guarantee and the performancemore »guarantee in terms of regret. Roughly speaking, for d-dimensional personalized information, our algorithm achieves the expected regret at the order of [Formula: see text] when the customers’ information is adversarially chosen. For stochastic personalized information, the regret bound can be further improved to [Formula: see text]. This paper was accepted by J. George Shanthikumar, big data analytics.« less
  4. In mechanism design, the firm has an advantage over its customers in its knowledge of the state of the system, which can affect the utilities of all players. This poses the question: how can the firm utilize that information (and not additional financial incentives) to persuade customers to take actions that lead to higher revenue (or other firm utility)? When the firm is constrained to "cheap talk," and cannot credibly commit to a manner of signaling, the firm cannot change customer behavior in a meaningful way. Instead, we allow firm to commit to how they will signal in advance. Customers can then trust the signals they receive and act on their realization. This thesis contains the work of three papers, each of which applies information design to service systems and online markets. We begin by examining how a firm could signal a queue's length to arriving, impatient customers in a service system. We show that the choice of an optimal signaling mechanism can be written as a infinite linear program and then show an intuitive form for its optimal solution. We show that with the optimal fixed price and optimal signaling, a firm can generate the same revenue as itmore »could with an observable queue and length-dependent variable prices. Next, we study demand and inventory signaling in online markets: customers make strategic purchasing decisions, knowing the price will decrease if an item does not sell out. The firm aims to convince customers to buy now at a higher price. We show that the optimal signaling mechanism is public, and sends all customers the same information. Finally, we consider customers whose ex ante utility is not simply their expected ex post utility, but instead a function of its distribution. We bound the number of signals needed for the firm to generate their optimal utility and provide a convex program reduction of the firm's problem.« less
  5. A great number of robotics applications demand the rearrangement of many mobile objects, for example, organizing products on store shelves, shuffling containers at shipping ports, reconfiguring fleets of mobile robots, and so on. To boost the efficiency/throughput in systems designed for solving these rearrangement problems, it is essential to minimize the number of atomic operations that are involved, for example, the pick-n-places of individual objects. However, this optimization task poses a rather difficult challenge due to the complex inter-dependency between the objects, especially when they are tightly packed together. In this work, in tackling the aforementioned challenges, we have developed a novel algorithmic tool, called Rubik Tables, that provides a clean abstraction of object rearrangement problems as the proxy problem of shuffling items stored in a table or lattice. In its basic form, a Rubik Table is an n × n table containing n2items. We show that the reconfiguration of items in such a Rubik Table can be achieved using at most n column and n row shuffles in the partially labeled setting, where each column (resp., row) shuffle may arbitrarily permute the items stored in a column (resp., row) of the table. When items are fully distinguishable, additional nmore »shuffles are needed. Rubik Tables allow many generalizations, for example, adding an additional depth dimension or extending to higher dimensions. Using Rubik Table results, we have designed a first constant-factor optimal algorithm for stack rearrangement problems where items are stored in stacks, accessible only from the top. We show that, for nd items stored in n stacks of depth d each, using one empty stack as the swap space, O( nd) stack pop-push operations are sufficient for an arbitrary reconfiguration of the stacks where [Formula: see text] for arbitrary fixed m > 0. Rubik Table results also allow the development of constant-factor optimal solutions for solving multi-robot motion planning problems under extreme robot density. These algorithms based on Rubik Table results run in low-polynomial time.

    « less