Decision Support Puzzles for Applied Mathematicians
June 2009 - Grocery Queuing
Grocery stores are one of the last commercial entities that continue to use the multiple-queue, multiple-server method for providing service to their customers. Banks and post offices have adopted a single-queue, multiple-server method that not only services customers faster but is a fairer system for providing service.
Let's compare these two queueing methods to see how much extra time a customer spends in a traditional queue versus the proposed queue.
Traditional queueing: On a typically busy day at the local grocery store, customers enter one of three queues in order to check out and pay for their groceries. They arrive at the checkout lanes with an inter-arrival time of two minutes, exponentially distributed. A cashier can scan a customer's groceries and finish on average in five minutes, also exponentially distributed. When selecting a queue, the customer will always choose the line with the fewest people waiting. For simplicity, let's assume that once a customer chooses a lane for checkout, he cannot move to another line.
Proposed queueing: The newly proposed queueing system (Figure 1) has the same arrival and checkout times as the traditional queueing model. But now there is only a single queue where customers wait until a checkout station is empty. Once a checkout station is empty, the customer proceeds immediately to that open station.
Question: How much less time on average (in seconds) will a customer have to wait in line if the grocery store switches from the traditional to the proposed queueing system?