## What is Queuing Theory?

The mathematical study of a queue or waiting lines is called the queuing theory. In queuing theory, a mathematical model is constructed to predict the queue length and waiting time for the service.

**Waiting Lines**

We observe waiting for lines every day at various instances. For example, waiting at the cash counter of the grocery store, at some ticket booking counter of a movie theatre, at the bank to check the account balance, in a car repair shop, patients waiting in the queue at the doctor's office, calling the customer care number of a company, at diners, etc. Queues could be formed by items like trucks in lines that need to be unloaded, machines waiting to be repaired, airplanes lined up on the runway for take-off, etc. At any given time, when there are more customer service requests than the service which can be provided, a waiting line occurs.

## Major Components of a Waiting Line (Queuing System)

Wait time in a line is impacted by the design of the waiting line system (or queuing system).

- Customer or Arrivals or Input to the System: The arriving unit requires some service. The customer may be a person, machine, vehicle, item, etc. These have characteristics such as customer population size, behavior, etc.
- Queue (Waiting Line): The number of customers waiting to be serviced. The queue does not include the customer being serviced.
- Service Channel: The process or facility which is providing service to the customer. It may be single or multiple.

We now explore each of the three parts in detail.

### Arrival Characteristics

The input source of a queuing model has important features, which are as follows.

I. Size of Arrival (or customer) Population: The customer population can be finite or infinite. When the number of people or objects in the waiting line is a very small portion of all potential arrivals at any point in time, then population size is considered infinite. Real-life examples of infinite arrival population would be the number of vehicles at a highway toll booth, people at the cash counter of a grocery shopping, etc. A class of 20 students waiting in line to meet their career counselor will be considered a finite population. Most waiting line models assume an infinite arrival population. In the case of finite arrival population, queuing model becomes much more complex.

ii. Behavior of the Arrivals: Most queuing model assumes that an arriving customer is a patient customer and will wait in the queue until they are served. But in real life, people ** balk**,

**, and**

*renege***. Balking means a customer refuses to join the line because it is too long. Reneging refers to customers who join the line but grows impatient and leave the line without being served. Jockeying occurs when a customer switches lines for quicker service time.**

*jockey*iii. Arrival Pattern: Customer may arrive at a service facility randomly or on a scheduled time like an airplane lands every 5 minutes at an airport. The probability of arrival can be established by the discrete Poisson distribution formula.

$P(x)=\frac{{e}^{-\lambda}{\lambda}^{x}}{x!}\text{for}x=1,2,3,4,\mathrm{...}$where,

$P(x)=$ Probability of $x$ arrivals.

$x=$ Number of arriving customers per unit of time

$\lambda =$ Average or mean arrival rate

$e=2.7183$ (base of natural logarithm).

### Waiting-Line Characteristic

** **The second component of a queuing model is the waiting line itself. It has two main features as follows:

i. Length of the Queue: The length of a waiting line can be finite or infinite. The waiting line at a hair salon would be of finite length due to the limited number of waiting-chair present. While in the case of highway toll booths, the number of vehicles waiting in line can be treated as infinite length. Most queuing models assume the length of the queue to be infinite.

ii. Queue Discipline: Queue discipline refers to the priority based on which a customer is served. Most queuing systems use the **first-in, first-out (FIFO)** rule. In this discipline, the first customer in line receives the service first. This may not be the case always. Like in the hospital emergency room, priority changes are based on the injury of the patient.

### Service Channel Characteristic

The third component of a queuing model is the service facility, also called the service channel. Its important features are service infrastructure which includes the number of waiting queues, the number of servers, placement of the servers, and service pattern.

Based on the number of servers and number of services stops that must be made (phases) before leaving the system, service systems are classified into four types- i. Single-Channel, Single-Phase System: Example would be a queue at the ATM machine for cash withdrawal or checking account balance.

ii. Single-Channel, Multiphase System: Example, a canteen where the customer first needs to collect the food coupon and then go to other servers for collecting the food.

iii. Multiple-Channel, Single Phase System: Example, a hair salon where one can receive service from the first available barber.

iv. Multiple-Channel, Multi-Phase System: Example, a driver's license agency where one has to wait for submitting an application, then go to another station for a driving test, and then pay the application fee on another counter. For each phase, there are multiple service channels available.

### Waiting-Line Performance Measures

** **Performance measures help in understanding a waiting-line system's performance. These measures include the following:

- average queue length;
- the average time that each arrival or customer spends in the queue;
- the average number of customers in the queueing system;
- average time each customer spends in the system, including waiting time plus service time;
- average or mean service time;
- utilization factor for the queueing system;
- probability of a specific number of customers in the queueing system; and
- the probability that the service facility will be idle.

### Single Channel Waiting-line Model

We now try to understand the various performance measures of a single channel waiting-line model. This involves a single line, single server, and single-phase system. This is the easiest waiting-line model. Certain assumptions as follows are made while modeling such scenarios.

- Arrivals or customer population size is infinite, and they are patient. Customers don't balk, renege, or jockey.
- The queue discipline follows a first-come, first-serve basis.
- The arrival rate for each customer varies, but the mean customer arrival rate(λ) is known and does not change over time and follows the Poisson distribution.
- The service rate for each customer varies, but the mean service rate (μ) is constant and follows the Poisson distribution.

- For the determination of arrival rate and service rate, the same time period is used. For example, if $\lambda $ is the average number of arrivals per hour, then μ is also the average number of customers served per hour.

After taking into consideration, these assumptions, operating characteristic or performance measures of a single-channel waiting line system is calculated using below formulas.

Let, $\lambda =$ mean or average arrival rate (average number of customers arriving per unit of time) and $\mu =$ mean service rate (average number of customers being served per unit of time), then the below formulas are used for the analysis of performance measures of a single-channel waiting-line system.

- The average number of customers $\left(L\right)$ in the queue system, which includes both customers waiting and customers being served, is given by:

- The average waiting time of customers $\left(W\right)$in the queue system, which includes both the time spent in the line plus time spent being served, is given by:

- The utilization factor $\left(\rho \right)$ of the system is given by:

- The average number of customers in the queue $\left({L}_{q}\right)$ is given by:

- The average time customers spend waiting in the queue$\left({W}_{q}\right)$ is given by:

- The probability that the queuing system is idle, that is, there are zero customers in the system, is given by:

- The probability that there are more than $k$ customers in the system is given by:

### Multi-Channel Waiting-Line Model

We now try to understand the various performance measures of a multi-channel waiting-line model. This involves a single line, more than one server, and a single-phase system. One such example would be a hair salon where multiple barbers are serving, and the customer can go to the first available barber to get serviced. This also takes into consideration all assumptions made for the single-channel waiting line model.

Let, $\lambda =$ mean or average arrival rate (average number of customers arriving per unit of time), $\mu =$ mean service rate (average number of customers being served per unit of time)and $m=$ number of active service channel,

The below formulas are used for the analysis of performance measures of a multi-channel waiting-line system.

- The probability that the queuing system is idle, that is, there are zero customers in the system, is given by:

- The average number of customers $\left(L\right)$ in the queue system, which includes both customers waiting and customers being served, is given by:

- The average waiting time of customers $\left(W\right)$in the queue system includes both the time spent in the line plus time spent being served is given by:

- The utilization factor $\left(\rho \right)$ of the system is given by:

- The average number of customers in the queue $\left({L}_{q}\right)$ is given by:

- The average time customers spend waiting in the queue$\left({W}_{q}\right)$ is given by:

### Application

Application of the queueing theory is in determining the waiting cost and service cost of the queuing system. Cost analysis or economic analysis helps in deciding the best among various waiting-line system designs for a business requirement.

Waiting cost is labeled in terms of customer dissatisfaction and loss of goodwill, while service cost is considered the cost for running a service channel.

Let, $m=$ number of channels,

${C}_{s}=$ Labor cost or service of each channel

${C}_{w}=$ Waiting cost of each channel

$\begin{array}{c}\text{Totalservicecost}=\left(\text{servicecostperchannel}\right)\left(\text{numberofchannel}\right)\\ =m{C}_{s}\end{array}$

$\begin{array}{c}\text{Totalwaitingcost}=\left(\text{Numberofarrivals}\right)\left(\text{Averagewaitingtimeperarrival}\right)\left(\text{waitingcost}\right)\\ =\lambda W{C}_{w}\end{array}$

$\begin{array}{c}\text{Totalcost}=\text{servicecost}+\text{waitingcost}\\ =m{C}_{s}+\lambda W{C}_{w}\end{array}$

## Practice Problem

Mr. A is running a hair salon and has employed Mr. D as a specialist. The average number of customers arriving per hour is three, and Mr. D is able to service four customers per hour. The waiting cost is $20 per hour, and the labor cost of Mr. D is $20 per hour. Should Mr. A hire another person, Mr. J, in place of Mr. D, who can service five customers per hour but demands more, $25 per hour?

**Solution:**

The scenario satisfies all assumptions of a single-channel queuing system.

Here,

$\begin{array}{l}\lambda =\text{meanarrivalrate}=3\text{perhour}\\ {\mu}_{1}=\text{meanservicerateofMr}\text{.D}=4\text{perhour}\\ {\mu}_{2}=\text{meanservicerateofMr}\text{.J}=5\text{perhour}\\ m=\text{numberofchannel}=1\\ \text{ForMr}\text{.D,}{C}_{s}=\$20\text{perhour,}{C}_{w}=\$20\text{perhour}\\ \text{ForMr}\text{.J,}{C}_{s}=\$25\text{perhour,}{C}_{w}=\$20\text{perhour}\end{array}$Performance measures | Mr. D | Mr. D |

$L=\frac{\lambda}{\mu -\lambda}$ | $L=\frac{3}{4-3}=3$ | $L=\frac{3}{5-3}=1.5$ |

$W=\frac{1}{\mu -\lambda}$ | $W=\frac{1}{4-3}=1$ | $W=\frac{1}{5-3}=0.5$ |

$\rho =\frac{\lambda}{\mu}$ | $\rho =\frac{3}{4}=0.75$ | $\rho =\frac{3}{5}=0.6$ |

${L}_{q}=\rho L=\frac{{\lambda}^{2}}{\mu \left(\mu -\lambda \right)}$ | ${L}_{q}=\rho L=0.75x3=2.25$ | ${L}_{q}=\rho L=0.6x1.5=0.90$ |

${W}_{q}=\rho W=\frac{\lambda}{\mu \left(\mu -\lambda \right)}$ | ${W}_{q}=\rho W=0.75x1=0.75$ | ${W}_{q}=\rho W=0.6x0.5=0.30$ |

${P}_{0}=1-\frac{\lambda}{\mu}$ | ${P}_{0}=1-\frac{3}{4}=0.25$ | ${P}_{0}=1-\frac{3}{5}=0.4$ |

$Total\mathrm{cos}t=serivice\mathrm{cos}t+waiting\mathrm{cos}t\phantom{\rule{0ex}{0ex}}=m{C}_{s}+\lambda W{C}_{w}$ | $Total\mathrm{cos}t=m{C}_{s}+\lambda W{C}_{w}\phantom{\rule{0ex}{0ex}}=1x\$20+3x1x\$20\phantom{\rule{0ex}{0ex}}=\$80$ | $Total\mathrm{cos}t=m{C}_{s}+\lambda W{C}_{s}\phantom{\rule{0ex}{0ex}}=1x\$25+3x0.5x\$20\phantom{\rule{0ex}{0ex}}=\$55$ |

Thus, we can see the total cost is lower in case Mr. A hires Mr. J as a specialist in place of Mr. D.

## Context and Application

This topic is significant in professional exams for both undergraduate and graduate courses, especially for B.Sc. and M.Sc. in Mathematics. It is also important for designing a queuing system to optimize waiting-line service, minimize service time, and advance research in operation management

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