Strategic equilibrium for a pair of competing servers with convex cost and balking

A two-person game is formulated for a queueing situation involving a pair of exponential servers competing for arriving customers. The servers have identical characteristics except for their service rates. Each server is free to select its own service rate. The objective of each server is to select a service rate that will maximize its own profit. Arrivals are Poisson. The probability that an arriving customer enters the queue is allowed to depend on the queue length at the time of arrival. The proportion of arrivals to a given server is shown to be strictly concave in the server's own service rate and decreasing in the other service rate. Furthermore, we show that when the cost function is convex and increasing, there exists a unique pure strategy Nash equilibrium point for the resulting game.