# Limiting the Oscillations in Queues with Delayed Information Through a Novel Type of Delay Announcement

## TL;DR

This paper models queue dynamics with delayed information and velocity signals, showing how optimal velocity information can reduce oscillations and improve system stability, with practical bounds for implementation.

## Contribution

It introduces a neutral delay differential equation model incorporating velocity information and derives bounds for optimal velocity information use.

## Key findings

- Velocity information can eliminate or reduce oscillations.
- Too much velocity information can induce oscillations.
- Derived bounds guide optimal velocity information usage.

## Abstract

Many service systems use technology to notify customers about their expected waiting times or queue lengths via delay announcements. However, in many cases, either the information might be delayed or customers might require time to travel to the queue of their choice, thus causing a lag in information. In this paper, we construct a neutral delay differential equation (NDDE) model for the queue length process and explore the use of velocity information in our delay announcement. Our results illustrate that using velocity information can have either a beneficial or detrimental impact on the system. Thus, it is important to understand how much velocity information a manager should use. In some parameter settings, we show that velocity information can eliminate oscillations created by delays in information. We derive a fixed point equation for determining the optimal amount of velocity information that should be used and find closed form upper and lower bounds on its value. When the oscillations cannot be eliminated altogether, we identify the amount of velocity information that minimizes the amplitude of the oscillations. However, we also find that using too much velocity information can create oscillations in the queue lengths that would otherwise be stable.