1. L/O/G/O Equilibrium prices for providers in Queueing System Petrozavodsk Nina Plaksina Petrozavodsk State University 2011 year

2. Introduction «Parking-provider» «Service-provider 1» «Service-provider 2» A. Veltman, R. Hassin [Queuing System 50, 2005] The model R – a reward of each customer – prices charged by the service-providers, i =1, 2 – price per time unit charged by the parking-provider

3. Game with two stages A Nash Equilibrium is a set of strategies, one for each player, such that no player has incentive to unilaterally change his action. Providers Customers Providers Service-provider 1 Service-provider 2 Parking-provider

4. Profit For customers For providers i =1, 2

5. Model with fixed p where p - affect only the magnitude of profit of service-providers There exist a unique equilibrium Theorem

6. Model with elastic p Reward of customer to visit 1 Reward of customer to visit 2

7. Model with elastic p Prices charged by service-providers for customers: Equilibrium:

8. Model with elastic p

9. Simulation model Profit Poisson stream of customers Customers choice Utility Principles of modeling system 5. U ≥ 0 1. Max 2. A stationary Poisson stream of customers arrives in the system 3. Customers choose service with some probability Example simulation model System with losses 4. Unserved customers leave system

10. References Hassin R., Haviv M. Equilibrium of Customers Behavior in Queueing Systems: To Queue or Not to Queue. Boston: Kluwer, 2003. Ivchenko G., Kashtanov V., Kovalenko I. Queueing theory. М.: «High school», 1982. Petrosyan L., Zenkevich N., Semina E. The game theory. M., 1998. Veltman A., Hassin R. Equilibrium in Queueing Systems with Complementary Products // Queueing Systems, Netherlands: Springer Science + Business Media, 2005. P.325-342.

11. Thank You for your attention