The Passport Control Problem or How to Keep an Unstable Service System Load Balanced ?

In many real life situations (such as the passport control system in Tel Aviv airport, department stores, and more) parallel queues are formed in front of control stations. Typically, some of the stations are manned while others are not. As the queues build up, management assigns additional officers to the unmanned stations. When this happens—some people move to the newly manned queues from nearby busy queues. In anticipation, people may prefer to line up in busy queues next to unmanned ones. Mathematically we discuss the problem of dynamic arrangement of the queues in a service system where at any time each server can be in either active or inactive mode. We seek a partition of customers to queues, that minimizes the expected wait time of a customer in each of the active stations, thereby keeping the system balanced at all times. We study two balancing algorithms which we call Split and Trim. For the Split algorithm we discuss a special case (the stations are ordered on a line and a single unmanned station is at one end). We show how an optimal partition can be calculated recursively. We then give partitions that approximate the minimal expected wait time within a factor of 1 + ", for some small 0 < " < 1, under each of these algorithms.