Performance Analysis of a Queue with Congestion-Based Staffing Policy

This paper analyzes a waiting line system that is motivated by the operations of border-crossing stations between the United States and Canada. There are two main conflicting goals in such a system: high security level, which often leads to a longer line; and good customer service, which requires a shorter line. Thus, unlike other queueing systems, maintaining the average queue length within a certain range is the primary objective. This is achieved using a staffing policy, called "congestion-based staffing," or CBS, where the number of servers (inspection booths) is adjusted according to the queue length during a planning period. We first present an exact benchmark model of Markovian type based on the matrix-geometric solution. For practical CBS policies, we develop a set of closed-form formulas for the major performance measures based on regenerative cycle analysis and fluid limit approximation. Numerical examples show that these approximation formulas are simple, accurate, and robust for practitioners to use in designing CBS policies.