Approximation of Chaotic Dynamics for Input Pricing at Service Facilities Based on the GP and the Control of Chaos

摘要

The paper deals with the estimation method of system equations of dynamic behavior of an input-pricing mechanism by using the Genetic Programming (GP) and its applications. The scheme is similar to recent noise reduction method in noisy speech which is based on the adaptive digital signal processing for system identification and subtraction estimated noise. We consider the dynamic behavior of an input-pricing mechanism for a service facility in which heterogeneous self-optimizing customers base their future join/balk decisions on their previous experiences of congestion. In the GP, the system equations are represented by parse trees and the performance (fitness) of each individual is defined as the inversion of the root mean square error between the observed data and the output of the system equation. By selecting a pair of individuals having higher fitness, the crossover operation is applied to generate new individuals. The string Used for the GP is extended to treat the rational form of system functions. The condition for the Li-Yorke chaos is exploited to ensure the chaoticity of the approximated functions. In our control, since the system equations are estimated, we only need to change the input incrementally so that the system moves to the stable region. By assuming the targeted dynamic system f(x(t)) with input u(t) + 0 is estimated by using the GP (denoted f(x(t))), then we impose the input u(t) so that x_f = x(t + 1) = f(x(t)) + u(t) where x_f is the fixed point. Then, the next state x(t + 1) of targeted dynamic system f(x(t)) is replaced by x(t + 1) + u(t). We extend ordinary control method based on the GP by imposing the input u(t) so that the deviation from the targeted level x_L becomes small enough after the control. The approximation and control method are applied to the chaotic dynamics generating various time series based on several queuing models and real world data. Using the GP, the control of chaos is straightforward, and we show some example of stabilizing the price expectation in the service queue.