Equilibrium Arrivals to Preemptive Queueing System with Fixed and Random Population Size

A single-server queueing system with preemptive access is considered. Each customer has one attempt to enter the system at its working interval [0, T]. As soon as the customer request enters the system, the server immediately starts the service. But when the next request arrives in the system, the previous one leaves the system even he has not finished his service yet. We study a non-cooperative game in which the customers wish to maximize their probability of obtaining service within a certain period of time. We characterize the Nash equilibrium and the price of anarchy, which is defined as the ratio between the optimal and equilibrium social utility. Two models are considered. In the first model the number of players is fixed, while in the second it is random and obeys the Poisson distribution. We demonstrate that there exists a unique symmetric equilibrium for both models. Finally, we calculate the price of anarchy for both models and show that the price of anarchy is not monotone with respect to the number of customers.