This paper presents a queueing model with a cost structure that considers the imposition of tolls on newly-arriving customers. It shows that tolls can be a rational strategy to achieve social optimality. The research is supported by the Office of Naval Research.
The model assumes a stationary Poisson stream of customers arriving at a single service station with exponentially distributed service times. Customers receive a reward R upon service completion and incur a cost C per unit time for waiting in the queue. Each customer chooses between joining the queue or not based on expected net gains. The model introduces two key conditions: (1) the existence of a public good, and (2) the possibility of diverting customers from the service station.
The model shows that the optimal strategy for the decision maker is to determine a critical queue size n, beyond which new customers are diverted. The critical size is determined by the ratio of reward R and cost C, adjusted by the service rate μ. The model demonstrates that the expected number of customers in the queue and the expected number of customers diverted must be equal under steady-state conditions.
The paper also discusses the concept of self-optimization, where customers act in their own self-interest, leading to a critical queue size n_s. However, this strategy may not be socially optimal. The paper then considers overall optimization, where the expected total net gain to customers is maximized. This leads to a different critical queue size n_o, which is determined by the ratio of R and C, adjusted by the service rate μ.
The paper further discusses the imposition of tolls to reduce congestion. The optimal toll θ* is determined by the difference between the critical queue size n_s and the optimal queue size n_o. The toll is set such that the expected net gain of customers is reduced to the level of the optimal queue size.
The paper also considers the case where the toll-collecting agency seeks to maximize its own revenue rather than optimize the system. In this case, the optimal toll is determined by the ratio of R and C, adjusted by the service rate μ.
The paper concludes that the model's results are independent of the specifics of the model. The key assumptions are the existence of a public good and the possibility of diverting customers from the service station. The model shows that tolls can be an effective tool for managing queue size and achieving social optimality. However, if the toll-collecting agency is not aligned with the public good, the optimal toll may not be achieved. The paper also notes that the model's results can be extended to more general cases where the reward depends on traffic density.This paper presents a queueing model with a cost structure that considers the imposition of tolls on newly-arriving customers. It shows that tolls can be a rational strategy to achieve social optimality. The research is supported by the Office of Naval Research.
The model assumes a stationary Poisson stream of customers arriving at a single service station with exponentially distributed service times. Customers receive a reward R upon service completion and incur a cost C per unit time for waiting in the queue. Each customer chooses between joining the queue or not based on expected net gains. The model introduces two key conditions: (1) the existence of a public good, and (2) the possibility of diverting customers from the service station.
The model shows that the optimal strategy for the decision maker is to determine a critical queue size n, beyond which new customers are diverted. The critical size is determined by the ratio of reward R and cost C, adjusted by the service rate μ. The model demonstrates that the expected number of customers in the queue and the expected number of customers diverted must be equal under steady-state conditions.
The paper also discusses the concept of self-optimization, where customers act in their own self-interest, leading to a critical queue size n_s. However, this strategy may not be socially optimal. The paper then considers overall optimization, where the expected total net gain to customers is maximized. This leads to a different critical queue size n_o, which is determined by the ratio of R and C, adjusted by the service rate μ.
The paper further discusses the imposition of tolls to reduce congestion. The optimal toll θ* is determined by the difference between the critical queue size n_s and the optimal queue size n_o. The toll is set such that the expected net gain of customers is reduced to the level of the optimal queue size.
The paper also considers the case where the toll-collecting agency seeks to maximize its own revenue rather than optimize the system. In this case, the optimal toll is determined by the ratio of R and C, adjusted by the service rate μ.
The paper concludes that the model's results are independent of the specifics of the model. The key assumptions are the existence of a public good and the possibility of diverting customers from the service station. The model shows that tolls can be an effective tool for managing queue size and achieving social optimality. However, if the toll-collecting agency is not aligned with the public good, the optimal toll may not be achieved. The paper also notes that the model's results can be extended to more general cases where the reward depends on traffic density.