Stability and Chaos in Input Pricing for a Service Facility with Adaptive Customer Response to Congestion
研究服务设施中客户根据拥堵情况自适应调整行为时，均衡到达率和定价的稳定性，发现不稳定的均衡可能导致混沌行为，为排队系统到达率波动提供新解释。
We consider the stability of the equilibrium arrival rate and equilibrium admission price at a service facility, using a generalization of an input-pricing model introduced by Dewan and Mendelson and further examined by Stidham. At the equilibrium, the marginal value of service equals the admission price, that is, the sum of the admission fee and the expected delay cost. Stability means (roughly) that the system returns to the equilibrium after a perturbation, assuming the customers base their join/balk decisions on previous prices. We extend the discrete-time, dynamic-system pricing model of Stidham to allow adaptive expectations in which customers predict the future price based on a convex combination of the current price and the previous prediction. We show that this can lead to chaotic behavior when the equilibrium is unstable. That is, the price and arrival rate can follow aperiodic orbits, which appear to be completely random. Our results suggest an alternative explanation for observed variations in the mean arrival rate to a queueing system, which are often modeled by means of a random exogenous (e.g., Markovian) environment process.