Mode Locking and Chaos in a Deterministic Queueing Model with Feedback

Abstract

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

A discrete time single-server queue with balking: economic applications

“…Although chaotic systems obey certain rules that can be described by mathematical equations, chaos theory shows the difficulty of predicting their long-range behaviour. This queueing system could be tackled in an alternative way using the chaos theory [6,10,13]; specifically, we should follow the two crucial steps given by Borovkov [6]. Of course, when the parametric values tend to the ergodicity condition, the system becomes unstable and tends to the disintegration.…”

Modelling Decisions Under Uncertainty In A Behavioural Queuing System

“…Some models were proposed by Edelson (1971), Agnew (1976), Haxholdt et al (2003), van Ackere and Larsen (2004), van Ackere et al (2006). Haxholdt et al (2003) and van Ackere et al (2006), using system dynamics, have included feedback into their model to look at the return rates of customers to the service facility. van Ackere and Larsen (2004) have used a one-dimensional cellular automata (CA) model in order to study formation of customers' expectations about congestion on a three-road system.…”

Circulation of autonomous agents in production and service networks

“…As discussed in Law et al (2004), the waiting time also influences significantly the satisfaction and return decision of customers visiting fast-food facilities. General models of recurrent services where customer short-term satisfaction is driven solely by the perceived waiting time while queueing are presented in Haxholdt et al (2003), van Ackere and Larsen (2004) and van Ackere et al (2006). In complete analogy with the present modeling framework, customers base their service quality measure on the waiting time and will adapt their visit frequency to that particular measure.…”

“…In complete analogy with the present modeling framework, customers base their service quality measure on the waiting time and will adapt their visit frequency to that particular measure. The models introduced in Haxholdt et al (2003), van Ackere and Larsen (2004) and van Ackere et al (2006) find natural applications in sports clubs, supermarkets and internet access management. The theme park industry is another sector to which the models presented in this paper are closely related (Kataoka et al, 2005;.…”