Queues with Updating Information: Finding the Amplitude of Oscillations

TL;DR

This paper addresses the open problem of quantifying oscillation amplitudes in queue systems with periodically updated information, providing exact calculations and accurate approximations to understand the impact of updates.

Contribution

It introduces a method to explicitly calculate the oscillation amplitude using a fixed point equation and derives accurate Taylor expansion approximations for large update sizes.

Findings

Exact fixed point equation for oscillation amplitude

Closed-form Taylor approximations are highly accurate for large 

Provides new insights into information dissemination effects in queue systems

Abstract

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 $\Delta$. 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 $\Delta$ is large. Our analysis provides new insight for systems that use updates as a way of disseminating information to customers.