At the Department of Motor Vehicles (DMV), customers wait in a single queue to be served by the first DMV officer that is available to serve them. There are 4 officers, and each takes on average 15 minutes to serve a customer, with a standard deviation of service time of 20 minutes. Customers arrive on average every 4 minutes, with the standard deviation of the time between arrivals of 3 minutes.

From the above, we can see that the arrival rate is 15 customers/hour, the service rate (for each officer) is 4 customers/hour, and the utilization is 93.75%.

Question:

The manager at the DMV wishes to improve the waiting time at the DMV. She proposes that in addition to serving as manager, she will also spend half of her time helping customers (i.e., filling the role of a DMV officer). On average, the manager expects she can serve a customer in 15 minutes, just like the other officers. However, the manager's service time has higher variability than the other DMV officers, as she is very skilled at some tasks and less skilled at others. She isn't quite sure how variable her service times will be, however.

What can you tell this manager? Is she likely to improve waiting times by assisting the DMV officers? Or, should the manager leave the job of helping customers to the officers? Hint: How high can the service time standard deviation be before the manager is hurting the wait time instead of helping it?