# Insights From Queueing Theory

## A mathematical explanation for why reducing variation and adding servers is good for operational efficiency

Welcome to a user-friendly overview of the general purpose queuing formula, which is written as a supplement to my piece The Social Consequences of Operational Efficiency.

On a personal note, I have always shied away from quantitative methods, and still struggle with mathematical intuition. So if you’re like me, don’t fret, because I’ve tried to break this down into the basic concepts, within the realm of my knowledge.

Queuing theory is a mathematical abstraction that helps illustrate the phenomenon of congestion and waiting in a line. Inputs step into a queue, and if resources and demand are mismatched, there will be a wait. To learn the general purpose queueing formula, we’ll look at the process of wrapping gifts to start to define our terms.

The arrival rate of the packages and the time it takes to wrap them can help us understand the utilized capacity of our process. In addition, the variation or probability distribution of the arrival and service rates play a role. If half the time the wrapping takes 50 seconds, and half the time the wrapping takes 10 seconds, it will affect our ability to handle the flow of packages. We’ll define those terms in the next graphic.

Last, let’s plug these concepts into the General Purpose Queuing Formula, which lets us calculate the expected length of the queue. Please note that this formula is only for processes that are not over capacity, in other words <1.

Not that scary, right? The goal of this outline is not to perfectly understand the formulas, but to glean a little insight into *what matters* to reducing the queue.

- If there is a lot of variance in the arrival times, we expect to get a queue
- If there is a lot of variance in the service time, we expect to get a queue
- If we increase the number of workers, N, then we expect the queue to decrease exponentially (because service utilization is less than 1)

With this simplistic model in mind, we can now identify the key areas that operations leaders look to in order to effect the efficiency of their operations. Armed with these insights, we can look the choices companies make, and how those choices can have social consequences.