Traffic congestion is a serious problem in this modern society. To consider the trade off between the traffic congestion induced by private cars and waiting times for public transportation, we propose a model in which customers can strategically select a single or batch service, and then receive a common service in a single-server queue with general service time distribution. Customers arrive at the system according to a Poisson process and decide whether to join the queue directly (i.e., selecting private cars) or wait for a batch service (i.e., selecting public transportation). The batch service commences periodically according to a Poisson process and the capacity of the batch is determined by a geometric distribution. The general service time distribution of the single-server expresses the service time of the service station on the road, which differs depending on various situations. The social optimization problem for such a mode choice model is considered in this paper. We derive the unique Nash equilibrium strategy of customers, socially optimal strategy, and socially optimal relationship of the fees for both services. Furthermore, we show that these socially optimal fees exhibit linear relationships and show some insights into the socially optimal relationship between fees depending on the service time distribution of the single-server.