# Queuing Theory and Queuing Models

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Queuing theory is the study of processes which involve waiting in line (queue) for a service. It consists of the development of mathematical models which mimic a real life situation and the application of these models to the real life situation. Queuing theory is can be considered a branch of operations management where it is used to make models of real situations and hence improve the efficiency of service for customers.

The advantages of queuing theory are that by modelling the queue we can predict how many customers might have to wait and depending on that we can increase or decrease the number of service personnel. This helps us to save costs because if waiting time is more some customers may leave and we can lose revenue. If we can calculate the waiting time in advance we can increase number of servers beforehand and prevent losses. This explains why queuing theory is so important.

The limitations or disadvantage of queuing theory are that the models it develops are only as good as the assumptions behind them. The assumptions are necessarily simplifications of complex real world situations and hence will never be completely accurate.

The queuing models which are developed involve three aspects-

1. Arrival Process- We first model how the customer arrives for service and what is the inter-arrival time between two customers. We generally use the Poisson distribution to model the number of arriving customers and the exponential distribution to model the inter-arrival time. Sometimes Erlang distribution is also used.
2. Queue discipline- This deals with how the customers are served when they arrive for service. First Come First Serve(FCFS) is a static queuing model where the customers who arrives first is served first. SIRO(Service In Random Order) is a dynamic queuing model where customers are served randomly.
3. Service Process- This deals with the time taken to serve a customer. Generally the service time distribution is modelled using exponential distribution.

Some examples where queuing models may be applied are in railway ticket lines, traffic stops which involve waiting, lines at banks and restaurants and at schools and public places.

Types of Queuing Models

The queueing models are classified on the basis of distributions they use, the number of servers, the number of customers in service and queue discipline.

We use the notation {(a/b/c):(d/e)} for Queuing models where:

a = arrivals distribution

b= service time distribution

c= number of service channels

d= maximum number of servers allowed in system

e= queue discipline

The different queueing distributions used are:

M= Markovian (exponential) inter-arrival time and service distribution

D= Deterministic (constant) inter-arrival time and service distribution

E= Erlang Distribution

Gl= General probability distribution- normal, uniform or any empirical distribution.

Commonly used queuing models:

Some commonly used queuing models are:

1. {(M/M/1):(∞/FCFS)} === Here we use exponential distribution to model inter-arrival time and service distribution assuming single server with no limit on customers and FCFS(First Come First Serve) queue discipline.
2. {(M/M/1):(N/FCFS)} === Here we use exponential distribution to model inter-arrival time and service distribution assuming single server but only fixed number of customers N and FCFS(First Come First Serve) queue discipline.
3. {(M/M/c):(∞/FCFS)} === Here we use exponential distribution to model inter-arrival time and service distribution assuming more than one server with no limit on customers and FCFS(First Come First Serve) queue discipline.
4. {(M/G/1):( ∞ /FCFS)} === As opposed to above three models here we use exponential distribution to model inter-arrival time and use general distribution for service time.

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