Choosing the Best Queue Through Math

• A very serious "queuing theory" is used to mathematically model this process, according to our partner The Conversation.

• This theory makes it possible to calculate the average wait that a customer should experience according to parameters such as the average service time, the average time between two arrivals or the number of agents in service.

• This analysis was conducted by Benjamin Legros, Professor of Finance at the Normandy School of Management.

To consult a doctor, go shopping, or call a company, we have all already lost precious minutes in a queue.

The painful experience of waiting has led many services to reflect on its causes and possibly find solutions to reduce it.

To understand waiting, the queuing theory, born in 1917 following the work of Erlang, was used to mathematically model this process, which leads individuals not to be served directly.

So let's start with a simple question: why are we waiting?

Imagine a supermarket checkout with an agent who serves every customer in five minutes and where a customer arrives every 6 minutes.

This system will not induce a priori any waiting as each customer will see the free agent at the time of his arrival at the checkout.

Now imagine that a customer arrives every 4 minutes.

The first customer of the day will not wait, the second customer will wait 1 minute, the third 2 minutes, the fourth 3 minutes and so on, so the queue will grow indefinitely.

In such a situation, the system becomes unstable and does not allow to reach a steady state.

These two examples suggest that in the long term, we should only see infinitely long or completely empty queues… which is far from being the case!

​At the source of expectation: variability

Both of these examples lack the main ingredient of expectation: variability.

The service and inter-arrival times are in fact random, which induces at times concentrated arrivals, at times long services, and leads to alternations between periods of empty queue and periods of congested queue.

Thus, even if the service time is shorter on average than the interarrival time, some customers will wait.

The queuing theory makes it possible to model this hazard and to calculate the average wait that a customer should experience according to parameters such as the average service time, the average time between two arrivals or the number of agents in service. .

These formulas make it possible to better understand the causes of waiting and to help decision-making when intuition is lacking.

Which box to choose?

The one for less than 10 items!

Among these questions, you may have already wondered which checkout to go to in a supermarket.

If the two checkouts are almost identical, the choice has no impact on the wait, but you may find yourself in the situation where one of the checkouts is specialized "less than 10 items", and the other is a normal checkout.

In this case, the estimated average waiting time, which is equal to the number of customers in front of you multiplied by their service time, is not necessarily the only indicator.

There is also the variability of services that matters.

Mathematical formulas show that a series of short serves is less variable than a few longer serves.

So, even if the average wait is the same, it is better to choose the checkout for less than 10 items, because this checkout gives you the highest chance of getting out of the store quickly.

​Be few but fast or many and slower?

Another question: is it better to hire few fast agents or a lot of slow agents (without taking into account salary costs)?

You might think it's the same thing to have one agent serving every customer in one minute or 10 agents each serving one customer in ten minutes.

However, it is not the case.

It is better to hire few fast agents.

Queuing theory models situations where some agents are not working due to no customer in the queue, which happens more often when there are many agents, as more customers can be served simultaneously .

In these situations, the performance of the queue is not optimal, which leads to degraded performance with many slow agents compared to a situation with few fast agents.

​Should the flow of customers entering the supermarket be regulated?

It is clear that the wait increases with the volume of arrivals and with the service time.

One might wonder which is worse: seeing an increase in arrivals or experiencing a slowdown in services?

The two phenomena seem quite equivalent.

However, the formulas show that for the same occupancy rate, defined as the proportion of time during which an agent serves customers, it is worse to have slow services.

There is no clear intuition behind this result, however the managerial consequences are significant, leading companies to put more effort into maintaining productivity at work, more than into regulating incoming flows.

Our “MATHEMATICS” file

In summary, queuing theory is a powerful tool for making decisions about routing customers, how many agents to hire, and designing a service system.

As waiting concerns most services, from the most everyday such as supermarkets, call centers and bike-sharing systems, to the most basic such as hospital emergencies, the tools of this field of study are constantly being develop to aid in more effective decision-making.

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This analysis was written by Benjamin Legros, Professor of Finance at the Normandy School of Management.

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