Queueing theory studies waiting lines and delays. It can be used to analyze systems like emergency rooms with patients waiting for service. Models represent the input of customers, the queue where they wait, and the service process. In an M/M/1 model, arrivals and service times follow exponential distributions, there is one server, and customers are served in order of arrival. Key metrics include the expected length of the system, length of the queue, and waiting times in both the system and queue. Queueing theory provides insights to reduce delays by adjusting staffing levels or other factors.

1. QUEUING THEORY
Mr.P.D.Devan
Assistant Professor
Kumaraguru College of Technology
Coimbatore

2. INTRODUCTION
• Queueing theory is the study of waiting in all these various guises.
Ex: Consider assigning an extra doctor to the emergency room, which has one
doctor already.
How much can we reduce the average waiting time for patients if the extra
doctor is hired?

3. STRUCTURE OF QUEUEING MODELS

4. CONT…
Input Source:
• The size is the total number of customers. The size may be infinite (default one) or finite.
• When will each one arrive?
• Associate with a distribution—usually, Poisson distribution (the number of customers generated
until any specific time) or Exponential distribution (interarrival time).
• A customer may be balking, who refuses to enter the system and is lost if the queue is too long.

5. CONT…
Quee
• The queue is where customers wait before
being served.
• A queue is characterized by the maximum
permissible number of customers that it can
contain. Queue may be infinite (default one)
or finite.
Queue Discipline
• Refers to the order in which members of the
queue are selected for service.
• First-come-first-serve is normally used.

6. CONT…
Service Mechanism:
• Consists of one or more service facilities, each of which contains one or more parallel service
channels, called servers.
• At a given facility, the customer enters one of the parallel service channels and is served by that
server.
• Most elementary models assume one service facility with either one or a finite number of servers.
• Service time is usually defined by a probability distribution.

7. TYPES OF QUEUING MODELS

8. CONT…
Single Server:
• A single server serves customers one at a time from the front of the queue, according to a first-
come, first-served discipline.
• When the service is complete the customer leaves the queue and the number of customers in the
system reduces by one.
Ex: Doctor attending a patient

9. CONT…
Multiple Server:
• Multi-server systems include more that one server, and these provide service to the customers
arriving into the customer queue(s).
• The models of multi-server systems can be designed with several similar servers or with different
types of servers.
Ex: Railway reservation system

10. CONT…
Finite Queue Length:
• In finite queue length models we try to restrict the queue length to a certain limit after which we
say that if this threshold limit is reached, people who come into the system do not join the system.
Ex: No of persons on a vehicle.
• After certain limit, no person can be accommodate in the vehicle.
• After threshold limit, the person will leave the system without joining the line without getting
served.

11. CONT…
Infinite Queuing System:
• The infinite queue length model assumes, that every person who comes joins the line.
• There is no restriction on the number of people who are actually waiting or there is no restriction
on the length of the queue.
• The queue length can theoretically be infinite so it can go on and on.
Ex: People are waiting for bus (No limitation)

12. FINITE VS INFINITE POPULATION

13. CHARACTERISTICS OF QUEUEING SYSTEM
Arrival Pattern:
• Based on the collected data what is actually happening and it can fit in to corresponding
distribution.
• In most cases, arrival follows Poisson distribution, with arrival rate called λ/hour.
Service Pattern:
• It is also observed from practice, that service times are exponentially distributed, at the rate of
µ/hour.

14. RELATIONSHIP BETWEEN λ & µ
• For infinite queue length models ------
λ
µ
< 1 (
𝐴𝑟𝑟𝑖𝑣𝑎𝑙
𝑆𝑒𝑟𝑣𝑖𝑐𝑒
< 1)
• For finite queue length models ------
λ
µ
> 1 (
𝐴𝑟𝑟𝑖𝑣𝑎𝑙
𝑆𝑒𝑟𝑣𝑖𝑐𝑒
>1)

15. CUSTOMER BEHAVIOUR
Collusion: Some customers who do not want to wait make one customer as their representative
and he represents a group of customers. Now, only representatives wait in queue and not all
members of the group.
Balking: When a customer does not wait to join the queue at the correct place he wants because of
his arrival. He wants to jump the queue and move ahead of others to reduce their waiting time in the
queue.
Jockeying: This is the process of a customer leaving the queue which he had joined and goes and
joins another queue to get advantage of being served earlier because the new queue has lesser
customers ahead of him.
Reneging: Some customers either do not have time to wait in queue for a long time or they do not
have the patience to wait, they leave the queue without being served.

16. QUEUING DISCIPLINE
First in, First Served (FIFS): This is the most commonly used method and the customers are
served in the order of their arrival (First in, First out-FIFO).
Last in, First Served (LIFS): This is rarely used as it will create controversies and ego problems
amongst the customers. Anyone who comes first expects to be served first. It is used in store
management, where it is convenient to issue the store last received and is called (LIFO) i.e. Last in,
First Out.
Service in Priority (SIP): The priority in servicing is allotted based on the special requirement of
a customer like a doctor may attend to a serious patient out of turn, and this also maybe the case
with a vital machine which has broken down.

17. KENDALL’S NOTATION FOR QUEUES

18. VARIOUS QUEUING MODELS

19. MODEL I- M/M/1: FCFS/∞/∞
M/M/1: FCFS/∞/∞
Arrival
Service
Number
of servers
First Come First Serve
Queue length
Population
M/M/1: FCFS/∞/∞ - Single server model with Poisson arrival and exponential service
following the memory less property, the infinity, infinity represent infinite queue length and
infinite population model.

20. MAJOR FIND OUTS
From a customer point of view the customer is interested in four important parameters.
1. Length of the system (Ls) - Number of people who are actually in the system
2. Length of the queue (Lq) - Expected number of people who are waiting for service
3. Waiting time in system (Ws)- Persons interested in waiting time in the system
4. Waiting time in queue (Wq)- Person is interested in waiting time in the queue
5. P- Probability that there are n people in this system (i.e. 0 ≤ P ≤ ∞)