We consider a simple, deterministic queueing system with feedback, which exhibits the phenomena of sustained oscillation, mode locking, quasi-periodic behaviour, and chaos. This implies that a fully deterministic queueing system can exhibit seemingly unpredictable behaviour. We ignore variability, and focus on two forms of feedback: (i) the service rate increases as queue length increases, and (ii) the arrival rate depends on customers' perception of past waiting times. We model a customer's decision to seek service as a two stage process: (i) deciding whether or not to use a facility, and (ii) deciding the frequency of visit (daily, weekly, monthly, etc.). This frequency is initially constant, and later on replaced by a deterministic, time-dependent pattern. Although highly stylised, this model captures the essential features of many real-life systems whose average arrival rate varies over time. Reducing the amplitude of cycles in demand makes the system more predictable and thus easier to manage. Although we represent this model as a queue of customers waiting for service, the model can be interpreted more generally as any situation where an increase in demand lowers the quality of service.Deterministic Queueing, Simulation, Mode Locking, Chaos

We analyse a simple, deterministic queueing system with feedback. We model a customer's decision to seek service as a two-stage process: (1) deciding whether or not to use a facility (become a customer), and (2) deciding the frequency of use (daily, weekly, monthly, etc.). We consider both a constant and a deterministic, time-dependent usage pattern. We ignore variability, and focus on three forms of feedback: (1) the service rate increases as queue length increases; (2) the frequency with which customers use the service depends on customers' perception of recent waiting times; and (3) the number of customers depends on their perception of long-term average waiting time. Although highly stylised, this model captures the essential features of many real-life systems whose average arrival rate varies over time. The main conclusions are that more capacity can make the system less manageable, and a stronger external cycle can be a stabilising factor. The model turns out to be remarkably robust to external disturbances.

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