Queuing Theory

What is queuing? 

When waiting for a service or event, a group of people or goods are organized and managed in a particular order, usually in a line or sequence, which is known as queuing. In many industries, including retail, transportation, healthcare, and customer service, waiting in line is a prevalent practice. The formation of queue (line) or waiting lines occurs whenever the present demand for a service exceeds the present capacity. It is difficult to predict about an arrival and the type of service required. If we provide too much service, it would involve unnecessary excessive costs. On the other hand, if we do not provide enough service capacity, this will result in a long waiting line, which proves costly. So we should have an economic balance between the cost of service and the cost associated with waiting for the service. 

Application of queuing: 

1. Industries: Supply of raw materials, dispatch of finished products, assembly lines. 
2. Transportation: Numbers of terminals, runway, and platforms. 
3. Communication: Boots/lines. 
4.  Public service organization: Hospitals, Banks. 
5. Others: Schedule of meeting, human coordination logistics, finance, emergency services, computing. 

Queuing system:

As shown in figure, the basic queuing system consists of two major components 

1. Input unit: Customers arriving at a queuing system, wait in queue to get some service. 
2. Service unit: If the system is idle or empty, the arriving customer may be serviced immediately. Once the service is over the customer leaves the system. 

Some common terms used in queuing are: 

1. Customer: The arriving unit that requires some service to be provided is called the customer. The customer may represent people, machines, etc. 
2. Server: A server is one who provides the arriving customer the necessary service. It may be persons in the counter or machines, etc. 
3. Service Channel: The type of service provided. It can be only one service unit- single channel/server model, More than one server: multichannel service. 
4. Arrival Rate: Rate at which customers arrive to be serviced. It is treated as a random variable for which a certain probability distribution is to be assumed. In queuing theory arrival rate is randomly distributed according to the Poisson distribution. The mean value of the arrival rate is denoted by λ; the unit is usually customers/time period. 
5. Service Rate: Rate at which the service is offered to the customers. The service rate refers to service offered by a single service channel. It is a random variable as the service to one customer may be different from the other. It is also assumed to follow in general, the Poisson distribution. The average service rate is μ; the unit is customers/time period. 
6. Infinite Queue: If the customers who arrive and form the queue are from a large population (more than 30) then the queue is referred to as infinite queuing model. 
7. Finite Queue: If the customers arrive from a small number of population say less than 30, then this is treated as a finite queue. 
8. Priority: Refers to the method of deciding which customer will be served next. The most common assumption is first come, first served (FCFS). 
9. Expected Number in Queue: The average or mean number of customers waiting to be serviced. This is denoted by Lq. 
10. Expected number in system: The average number of customer either waiting in the line or/and serviced. 
11. Expected time in queue: The expected or mean time a customer waiting in a line and is denoted by Wq. 
12. System Utilization or Traffic Intensity: This is the ratio between arrival and service rate denoted by ρ given by λ / μ. 

Essential features of queuing system 

1. Input source (or calling population) 
2. Queuing process 
3. Queue discipline 
4. Service process (or mechanism) 
5. Departure pattern 

Input source 

The input sources are the finite or infinite sources of potential customers commonly known as populations that will use the service. Input source has the following three characteristics: 

• Size of the population 
• Behavior of the arrival 
• Pattern of arrival of customers 

Queuing process 

The queuing process refers to the number of queues and their respective lengths. It depends upon the layout of the service system and the type of the service to be rendered. The number of queues may be single or multiple and may be of finite length or infinite length. 

Queuing discipline 

1. FCFS: First come ,first served 
2. LCFS: Last come, first served 
3. SIRO: Service in random order 
4. Priority Service 
5. Pre-emptive priority and Non-pre-emptive priority 

Service process 

Service process is characterized by: 

• The arrangement and capacity of service facility: Single queue single server; single queue multiple server. 
• The distribution of service time: Service time distribution can be described in Exponential distribution 

Departure pattern 

• Generally, It is ignored but it may influence the service and/or arrival pattern.
• In the multistage channel facilities, the departure pattern may influence the service system.

Models 

Single-channel infinite population model: 

Assumptions: •

• The arrivals or services are random and independent of all other conditions. The distribution of arrival rate can be shown to be Poisson. 
• Arrivals are served on the first come first served (FCFS).
• The mean is independent of time and not affected by a number of customers in waiting line, previously serviced etc. 
• The service time distribution is either Poisson or exponential. 
• Service time also varies from one customer to next but their service rate is known. 

Note: (Masters in Engineering Management, Subject: Operation Research)