Equilibrium prices for providers in Queueing System

1. Equilibrium pricesfor providersin Queueing System Nina Plaksina Petrozavodsk State University 2011 year Petrozavodsk

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

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

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

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

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

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 Youfor your attention