# It’s 5:30 pm, it looks like everyone has left work and you are stuck in a long line at the supermarket cash register. Are you surprised that the store management still does not realize that it needs a few cashiers during rush hour?

f you’re surprised, you probably understand the importance of queues.

Whether it’s waiting to use a copier or landing a plane, or waiting for parts to enter the assembly line, queues are an obvious and often frustrating part of our lives. Waiting queues affect people’s lives every day, which is why one of the main goals of many businesses is to provide the best level of service. Minimizing queues is an important part of creating a positive customer experience.

But how can you achieve this in your organization? Well, a great deal of mathematical knowledge is devoted to the study, simulation, and analysis of waiting times. This knowledge is called queuing theory and helps reduce the cost of waiting lines in your business.

Queue theory helps you determine the best way to employ staff and other resources and reduce your customers’ waiting time.

Queue models show you how you can maintain your level of profitability by eliminating inefficient people who only work on time and make sure you need a few employees to get the job done in the best possible time.

To use queuing models, you should always consider the following factors:

• Average customer entry rate.
• Average customer service rates
• Costs due to customer dissatisfaction due to waiting time.
• Costs of providing service points.

## Little Law

Most queuing models have the same basic structure: Customers arrive to receive services , enter the queue, and wait for their turn. To see if there is a bottleneck or problem in your queue, you need to know what is happening in the queue. falls down. Little Law helps you do this. According to this theory, the average queue length (L) is equal to the average entry rate (λ) multiplied by the average waiting time (W).

Consider this example: Suppose your call center receives 8,000 (L) calls in each season of the year. You need to provide the best and most effective way to provide telephone services to your customers. So you have to calculate using Little’s law, as follows:

L = λW
8,000 = λ (0.25)
λ = 32,000 calls per year

If you have two employees at the call center, each working eight-hour shifts and 250 days a year, then you have 4,000 hours to serve customers.

The number of calls that need to be processed per hour are:

λ = 32,000 / 4,000 = 8 calls per hour

Although Little’s model gives us very useful information about what happens in the queue, it is not enough to study and optimize the queue. Service demand rates and other variables that cause inefficiencies are largely unpredictable. Instead, Little Law helps you more in defining the data you need to use in more complex queues.

To analyze the efficiency of your queue model, start by analyzing the queue characteristics.

## Queue model features

To build a queue model, you must first understand the main queue system. In the queuing system, customers enter through a process, then wait in line for the service provider. When the service provider is ready, the customer is selected according to a predetermined queue rule. The customer leaves the queue system as soon as the service is complete.

Therefore, the queuing system is determined using three main factors:

1. How customers enter.
2. Queue rules.
3. How services are provided.

Now let’s take a closer look at each of these factors.

### 1. How customers enter

In general, you have no control over the entry of customers. In our example , callers can log in whenever the call center opens. However, you can calculate the average entry rate, as we did above. But to create a queue model, you need to analyze how customers arrive more carefully. To do this, follow these steps:

• Follow the inputs. Get the arrival time of calls for a certain period of time.
• Draw a diagram. Show the number of calls received at different times; Daily calls, calls per shift, calls per hour and…
• Determine the pattern of distribution of inputs. How are your inputs or calls distributed throughout the day?

In our example, when you chart incoming calls, you may find that the calls are evenly distributed throughout the day. Therefore, eight calls per hour is a reasonable number to use in determining the required resource analysis.

#### With two employees, your call center must work effectively, and each employee is required to answer four calls per hour.

When you look at random events at intervals, the events you are researching on the chart take the form of a Poisson distribution. For large numbers of events, the Poisson distribution is usually in the form of a normal curve or a bell curve.

For smaller events, this curve is usually tilted to the right. When analyzing a queue using mathematical equations, we usually consider the Poisson distribution.

Assume your customers’ arrivals are uneven throughout the day. So you have to use simulation to create the queue model. If your analysis shows that more than half of the calls were received between 10 a.m. and 2 p.m., then you cannot say that the call center received eight calls per hour; This is not what really happened.

So you may want to do more analysis over that four-hour interval to determine if these contacts are evenly distributed over that time. In other words, have they followed the Poisson distribution?

Another hypothesis for queue entries is that when a person enters the queue, the customer waits for the service to be completed. In fact, this may not always be the case.

Therefore, the model should consider this issue. Usually two terms are used to describe exceptions to this hypothesis:

If customers leave before entering the queue, it is said to have “turned around”. If they enter the queue, but before the service ends, they change their mind (for reasons such as the queue is crowded) and leave the queue, they are “knocked out”.

### 2. Queue rules

This section refers to the longest possible queue and order of services used.

#### Maximum possible queue length

Some queues have a limit for the maximum length. As soon as they reach that value, other customers are prevented from joining the queue. If you have a waiting room, you should see how many people it can accommodate? If you use telephone lines, how many calls can you make at one time?

Specify the capacity of your services. What is the maximum number of services you can provide in the analysis interval?

#### Sequence of services

This model assumes that queues follow the FIFO rule (first input, first output). Other queue rules include LIFO (last input, first output), first shortest task, random selection, and order of priority.

Analyze the nature of your services. In which sequence do you process customer input? Is this sequence efficient? How can this sequence affect efficiency, customer satisfaction , and resource utilization?

Let’s go back to our own example. Your analysis shows that customers’ phone calls for services are usually divided into two categories:

1. Quick and easy, like asking a customer to press a button.
2. It is complex and time consuming and involves difficult activities performed by the customer. For example, in this case, about 60% of calls are fast and easy.

You are now using the LIFO method to analyze customers. As a result, many fast and easy customers have to wait in long lines for complex calls. If you hire an employee to answer complex calls and another employee for quick and easy calls, you can potentially reduce the waiting time and not need to hire a new employee to answer calls.

When an employee is not in charge of complex calls, he or she can help his or her co-worker by receiving the rest of the calls and providing customer service.

### 3. How services are provided

Here, you should consider the following:

#### Number of service providers or employees

Some queues have only one service provider and others employ more. By determining the right number of service providers, you can make the best use of resources and optimize the services provided to customers, these achievements are clear results in the queue model.

#### Number of interruptions in the process

The number of queues in the queue is one of the issues that should be considered when analyzing the best way to provide services. In the production process, one of the ways to reduce interruptions, and shorten different queues, is to limit the frequency of product movement.

Many call centers use automated systems to reduce the number of options available to customers. But if calls are answered by one of your employees, it can add more time to the service process.

#### Distribution of services

In some queues, service time is essentially the same for all customers, so you can logically calculate the average service time. For example, the average call time for a heater repair service may be 15 minutes.

No matter how many people wait for the service, each call takes about a quarter of an hour. In these cases, depending on the number of your service providers, you can predict exactly how long the customer will have to wait in line.

But in other cases, sometimes the total service time depends on the number of people in the queue. This usually happens when you only have one service provider.

The more people in the queue, the longer the last person to enter the queue will have to wait. Restaurant is one example of this type of service distribution.

The chef usually prepares two meals much faster than six, so the smaller the party, the less time you will have to wait. Or in the fast food queue, if you are the fifth person to order a very tasty pizza, you should wait longer for your meal, while if you choose a delicious but unpopular meal, this time will be shorter.

In our example, some calls are very short and some are very long. When you have a large standard deviation in service time, your waiting time will increase. So the key strategy is to look for ways to determine the distribution of services.

## performance analysis

Many calculations can be used to measure queue performance. Since a more detailed discussion of this topic is beyond the scope of this article, here are just a few of the key standards:

The formulas used to calculate these values ​​depend on the type of queue and the type of distribution. More complex queue models, for example, queues with multiple servers, require more types of these values.

The overall analysis process is very complex, and even after inferring the standards, the desired answers are not black and white.

#### To better understand your queues, you can use spreadsheets (such as Excel software) to simulate queues at specific times.

How to do this simulation is beyond the scope of this article, however you can use software to create your queue model and determine the optimal status for your queuing system. Searching for the term queuing software on Google will give you many results to get you started.