  Understanding how delayed information impacts queueing systems is an
important area of research. However, much of the current literature neglects
one important feature of many queueing systems, namely non-stationary arrivals.
Non-stationary arrivals model the fact that customers tend to access services
during certain times of the day and not at a constant rate. In this paper, we
analyze two two-dimensional deterministic fluid models that incorporate
customer choice behavior based on delayed queue length information with time
varying arrivals. In the first model, customers receive queue length
information that is delayed by a constant Delta. In the second model, customers
receive information about the queue length through a moving average of the
queue length where the moving average window is Delta. We analyze the impact of
the time varying arrival rate and show using asymptotic analysis that the time
varying arrival rate does not impact the critical delay unless the frequency of
the time varying arrival rate is twice that of the critical delay. When the
frequency of the arrival rate is twice that of the critical delay, then the
stability is enlarged by a wedge that is determined by the model parameters. As
a result, this problem allows us to combine the theory of nonlinear dynamics,
parametric excitation, delays, and time varying queues together to provide
insight on the impact of information in queueing systems.
