Waiting Line Theory: Waiting Line Theory

Waiting line theory, also known as queuing theory, is the mathematical study of waiting lines. This theory can be used to model and predict wait times and number of customer arrivals. Wait lines are experienced in customer service industries such as banks, retail stores, and transportation, as well as manufacturing industries when items wait to be processed in assembly lines. This theory is even useful in information technology to predict the number of requests a computer server will receive. Companies can use this sort of information to allocate their resources most efficiently and ensure they can meet their needs of their customers.

Structure

All waiting lines are comprised of the following four parts:

1. Inputs - customer population (people or items) that are arriving to be processed

2. Waiting Line - where the customers wait after arriving but before being processed

3. Workstations - process the customers in the waiting line

4. Priority Rule - the method by which customers are chosen to be served

Inputs

The customer population can be catagorized as finite or infinite. A population is considered infinite if the number of customers in the waiting line does not effect the arrival rate. To simplify matters, many models use the assumption that the population is infinite.

Waiting line

Customers in the waiting line can be catagorized as patient or impatient. Impatient customers are more likely to not enter waiting lines (balk) or will not stay in the line to be served (reneging). In order to simplify the math, many models assume the customers to be patient.

Waiting lines can be configured as a single line such as in a bank, or they can have multiple lines like in a grocery store.

Workstations

Customers are processed at the workstations. Workstation can be configured as single or multiple channel, and single or multiple phase. The channel refers to how many ways to reach the workstation, while a phase refers to how many workstations are required to process the customers. A single channel, single phase workstation means there is only one waiting line and one station required to process the customers in the line. A multiple channel, multiple phase workstation means there are multiple waiting lines that must be processed by a series of workstations.

Priority Rule

The order in which customers are processed is given by the priority rule. The most common is the first come, first served in which the first person in line is served first. Another rule is the highest priority which ranks customers in priorities and the highest priorities are served first. This type of priority rule is used in hospital waiting rooms where the most urgent cases are seen first.

Little's Law

A simple method of determining the average number of customers in a stable system is Little's Law. This relationship was developed by John Little in 1961 at the MIT Sloan School of Management.

N is the average number of customers currently in the system
Lambda is the average number of customers that arrive during the time period
T is the average time spent by each customer in the system

Therefore, if the number of customers arriving doubles, so does the number of customers in the system. If the time spent in the system is halved, then the number of customers in the system is also halved.

eg. If 100 customers arrive per hour and spend an average of 10 minutes in the store, then the solution is as follows:

Note: all time units must be the same, so 10 min was changed to 0.1667 hours.

Poisson Process

Most mathematical models of waiting lines use Poisson processes to imitate a random arrivals based on an average arrival rate. As in real life, customers do not arrive exactly 1 min apart even though the average arrival rate may be 60 customers per hour. Therefore, a random arrival can be estimated using a mathematical formula derived by the French mathematician, Siméon-Denis Poisson.

The following figure compares a Poisson distribution to an Erlang distribution which is another popular distribution model.

The Poisson distribution is much more clustered with a higher standard deviation between events than the more evenly distributed Erlang distribution. This makes the Poisson the more prefered model to use in waiting lines where the arrival of customers tends to be clustered.

Calculating Estimated Customers in Line

If the arrival rate and the processing rate are known, then the number of customers in a single waiting line by one workstation can be estimated using the following formula:

Note: A complete list of formulas can be found on page 12 of Waiting Line Management.

Wait Times

The average waiting time for a customer in line with one workstation can be estimated by knowing the arrival rate and the processing time according to the following formula: