STABILITY AND CHAOS IN INPUT PRICING FOR A SERVICE FACILITY WITH ADAPTIVE CUSTOMER RESPONSE TO CONGESTION

We consider the stability of the equilibrium arrival rate and equilibr ium admission price at a service facility, using a generalization of a n input-pricing model introduced by Dewan and Mendelson and further ex amined by Stidham. At the equilibrium, the marginal value of service e quals the admission price, that is, the sum of the admission fee and t he expected delay cost. Stability means (roughly) that the system retu rns to the equilibrium after a perturbation, assuming the customers ba se their join/balk decisions on previous prices. We extend the discret e-time, dynamic-system pricing model of Stidham to allow adaptive expe ctations 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 unstab le. That is, the price and arrival rate can follow aperiodic orbits, w hich appear to be completely random. Our results suggest an alternativ e explanation for observed variations in the mean arrival rate to a qu eueing system, which are often modeled by means of a random exogenous (e.g., Markovian) environment process.