Optimization problems in simple competitive model

Abstract

We consider here the optimization problems of simple competitive model. There are two servers providing the some service. Each server fix the price and the rate of service. The rate of service defines the customer losses waiting in line for the service. The customer go to the server with lesser total service cost. The total cost includes the service price plus waiting losses. A customer goes away, if the total cost exceeds some critical level. The flow of customers and the service time both are stochastic. There is no known analytical solution for this model. We get the results by Monte Carlo simulation. We get the analytical solution of the simplyfied model.

We use the model as an illustration to show the possibilities and limitations of optimization theory and numerical techniques in the competitive models.

We consider optimization in two different mathematical frameworks: the fixed point and the Lagrange multipliers. We consider two different economic and social objectives, too: the equilibrium and the social cost minimization.

We use the model teaching Operations Research. The simple model may help to design more realistic models describing the processes of competition.