**Basics of Queuing Theory**

**Introduction**

The queuing theory is a mathematical model used to analyze where there is a queue or lines of queues people arrive at random or in a particular probability distribution and there is an average rate of arrivals and the queue or queues are served by service centers in one line or in several lines. The service rate is also has an average rate and it has a probability distribution. It is also assumed the system after some long time goes in to a steady state and the queues performances can be calculated mathematically.

However, there are certain assumptions made in the queuing theory models. The assumptions relates to the characteristics of the queuing characteristics in terms of finite or infinite queue, the finite or infinite nature of the population, the finite and infinite number of people in the queue and its probability distribution of arrival rate normally assumed to be a Poisson distribution.

As well, it assumes the service rate is negatively exponential in its probability distribution and there exist an average rate of arrival and average rate of service. In addition, the structure or levels of and phase of service centers determine the complexity of the queuing models formula for calculating the performance characteristics of the queuing system.

**Definition of the queuing system parameters**

Queue is a single waiting line

Waiting line system consists of arrivals in line queues, servers and waiting line structures.

Calling population means source of customers or messages, the nature of customer numbers whether it is infinite or finite, which defines a large number of customers or a limited number of customers.

Arrival rate means the number of customers arriving at an average and the probability distribution of arrivals normally assumed to be a Poisson distribution.

Service time is the time taken to serve a customer and it is normally assumed to be a probability distribution that follows negative exponential distribution and the service rate has an average rate.

In addition, the arrival rate is normally less than service rate. This is because of the fact the system never clears out.

**Components of the queuing system**

Source of customers

Arrivals and waiting line in queues

Server system

Served customers

In the queuing system it must have a queue discipline and must also define the length of the queue whether it is infinite or finite depending on the limitation of the physical structure. Normally queue length is infinite however in some circumstances they are finite.

Waiting line structures of servers can have one or more than one channels parallel to each queue and can have different phases in one channel.

**Operating Characteristics of the queuing system**

The mathematical model of the queuing system does not provide optimal solution. It calculates the operating characteristics of the system performance.

The steady state gives the average performance characteristics of the system that the system will reach after a long time.

The queuing model calculates the following operating characteristics as follows:

- Average number of customers in the system waiting and being served
- Average number of customers waiting in the line
- Average time a customer spends in the system waiting and being served
- Average time a customer spends waiting in the waiting line or queue.
- Probability no customers in the system
- Probability n customers in the system
- Utilization rate: The proportion of time the system is in use.

**Basic Single-Server queuing Model**

Assumptions of the model are:

Arrival rate has a Poisson probability distribution

Exponential service time

First come first serve queue discipline

Queue length is infinite

Calling population is infinite

Say a= mean arrival rate

Say b = mean service rate

**Formulas for Single service Model**

Probability that no customers are in the system Po = (1-a/b)

Probability of exactly n customers in the system Pi = (a/b)n* Po

Average number of customers in the system L = a/(b-a)

Average number of customers in the queue L q = a2/a*(b-a)

Average time customer spends in the system = 1/(a-b)=L/a

Average time customet in the queue = W q = a/b*(b-a)

Probability the server is busy or utilization factor u = a/b

Probability the server is idle & customer can be served I = 1-u = (1-a/b) = Po