Multidelay Differential Equations: A Taylor Expansion Approach

Abstract

It is already well-understood that many delay differential equations with only a single constant delay exhibit a change in stability according to the value of the delay in relation to a critical delay value. Finding a formula for the critical delay is important to understanding the dynamics of delayed systems and is often simple to obtain when the system only has a single constant delay. However, if we consider a system with multiple constant delays, there is no known way to obtain such a formula that determines for what values of the delays a change in stability occurs. In this paper, we present some single-delay approximations to a multidelay system obtained via a Taylor expansion as well as formulas for their critical delays which are used to approximate where the change in stability occurs in the multidelay system. We determine when our approximations perform well and we give extra analytical and numerical attention to the two-delay and three-delay settings.

Queues with Delayed Information: Analyzing the Impact of the Choice Model Function

“…This delay in information can be caused by factors such as the fact that it can take time to process and send information about queue lengths to the customers and thus the information that customers actually receive describes the queue lengths from some amount of time in the past. Queueing systems where customers are provided with delayed information have been studied extensively, see for example Novitzky et al [16], Doldo and Pender [3,2], Doldo et al [6], Novitzky and Pender [13], Novitzky et al [15], Atar and Lipshutz [1], Whitt [23].…”

“…For example, a queueing model that uses delayed updating information is examined in Doldo and Pender [5], Novitzky and Pender [14]. Additionally, queueing models with distributed delays have been studied by considering distributed delay equation models in Doldo and Pender [2], Novitzky and Pender [13], though it is worth noting that distributed delay models exhibit different dynamics in general than simply replacing the constant delay with a random variable in standard models, as discussed in Doldo and Pender [4].…”