This document provides an overview of queuing theory. It discusses key concepts like arrival processes, queue structures, service systems, queuing models and their assumptions. Examples are given of single and multiple channel queuing models. Metrics like average time in the system and average wait time are defined. Applications of queuing theory include telecommunications, healthcare, manufacturing and more.

1. “
“QUEUING THEORY”
QUEUING THEORY”
Presented By--
Presented By-- Anil Kumar
Anil Kumar
Avtar Singh
Avtar Singh

2. Queuing Theory
Queuing Theory

Queuing theory is the mathematics of waiting lines.
Queuing theory is the mathematics of waiting lines.
 It is extremely useful in predicting and evaluating
It is extremely useful in predicting and evaluating
system performance.
system performance.
 Queuing theory has been used for operations
Queuing theory has been used for operations
research, manufacturing and systems analysis.
research, manufacturing and systems analysis.
Traditional queuing theory problems refer to
Traditional queuing theory problems refer to
customers visiting a store, analogous to requests
customers visiting a store, analogous to requests
arriving at a device.
arriving at a device.

3. Applications of Queuing Theory
Applications of Queuing Theory
 Telecommunications
Telecommunications
 Traffic control
Traffic control
 Determining the sequence of computer
Determining the sequence of computer
operations
operations
 Predicting computer performance
Predicting computer performance
 Health services (e.g.. control of hospital bed
Health services (e.g.. control of hospital bed
assignments)
assignments)
 Airport traffic, airline ticket sales
Airport traffic, airline ticket sales
 Layout of manufacturing systems.
Layout of manufacturing systems.

4. Queuing System
Queuing System
 Model processes in which customers arrive.
Model processes in which customers arrive.
 Wait their turn for service.
Wait their turn for service.
 Are serviced and then leave.
Are serviced and then leave.
input
Server
Queue
output

5. Characteristics of Queuing
Systems
 Key elements
Key elements of queuing systems
of queuing systems
•
• Customer:--
Customer:-- refers to anything that arrives at a facility
refers to anything that arrives at a facility
and requires service, e.g., people, machines, trucks,
and requires service, e.g., people, machines, trucks,
emails.
emails.
•
• Server:--
Server:-- refers to any resource that provides the
refers to any resource that provides the
requested service, eg. repairpersons, retrieval machines,
requested service, eg. repairpersons, retrieval machines,
runways at airport.
runways at airport.

6. System Customers Server
Reception desk People Receptionist
Hospital Patients Nurses
Airport Airplanes Runway
Road network Cars Traffic light
Grocery Shoppers Checkout
station
Computer Jobs CPU, disk, CD
Queuing examples
Queuing examples

7. Components of a Queuing System
Components of a Queuing System
Arrival Process
Servers
Queue or
Waiting Line
Service Process
Exit

8. Parts of a Waiting Line
Parts of a Waiting Line
Dave’s
Dave’s
Car Wash
Car Wash
enter
enter exit
exit
Population of
Population of
dirty cars
dirty cars
Arrivals
Arrivals
from the
from the
general
general
population …
population …
Queue
Queue
(waiting line)
(waiting line)
Service
Service
facility
facility
Exit the system
Exit the system
Exit the system
Exit the system
Arrivals to the system
Arrivals to the system In the system
In the system
Arrival Characteristics
Arrival Characteristics
•Size of the population
Size of the population
•Behavior of arrivals
Behavior of arrivals
•Statistical distribution
Statistical distribution
of arrivals
of arrivals
Waiting Line
Waiting Line
Characteristics
Characteristics
•Limited vs. unlimited
Limited vs. unlimited
•Queue discipline
Queue discipline
Service
Service
Characteristics
Characteristics
•Service design
Service design
•Statistical
Statistical
distribution of
distribution of
service
service

9. 1.
1. Arrival Process
Arrival Process
 According to source
According to source
 According to numbers
According to numbers
 According to time
According to time
2. Queue Structure
• First-come-first-served (FCFS)
• Last-come-first-serve (LCFS)
• Service-in-random-order (SIRO)
• Priority service

10. 3.
3. Service system
Service system
1.
1. A single service system.
A single service system.
Queue
Arrivals
Arrivals
Service
facility
Departures
Departures
after service
after service
e.g- Your family dentist’s office, Library counter
e.g- Your family dentist’s office, Library counter

11. 2. Multiple, parallel server, single
2. Multiple, parallel server, single
queue model
queue model
Queue
Service
facility
Channel 1
Service
facility
Channel 2
Service
facility
Channel 3
Arrivals
Arrivals
Departures
Departures
after service
after service
e.g- Booking at a service station
e.g- Booking at a service station

12. 3. Multiple, parallel facilities with
3. Multiple, parallel facilities with
multiple queues Model
multiple queues Model
Service station Customers
leave
Queues
Arrivals
e.g.- Different cash counters in electricity office

13. 4. Service facilities in a series
4. Service facilities in a series
Arrivals
Queues
Service station 1 Service station 2
Queues
Customers
leave
Phase 1 Phase 2
e.g.- Cutting, turning, knurling, drilling, grinding,
packaging operation of steel

14. Queuing Models
Queuing Models
1.
1. Deterministic queuing model
Deterministic queuing model
2.
2. Probabilistic queuing model
Probabilistic queuing model
• Deterministic queuing model
Deterministic queuing model :--
:--
λ
λ =
= Mean number of arrivals per time
Mean number of arrivals per time
period
period
µ
µ =
= Mean number of units served per
Mean number of units served per
time period
time period

15. Assumptions
Assumptions
• If
If λ
λ > µ, then waiting line shall be formed and
> µ, then waiting line shall be formed and
increased indefinitely and service system would fail
increased indefinitely and service system would fail
ultimately
ultimately
2. If
2. If λ
λ µ, there shall be no waiting line
µ, there shall be no waiting line
≤
≤

16. 2.Probabilistic queuing model
2.Probabilistic queuing model
Probability that n customers will arrive in the
Probability that n customers will arrive in the
system in time interval T is
system in time interval T is
( ) ( )
!
n
e
t
n
P
t
n
t
λ
λ −
=

17. Single Channel Model
Single Channel Model
λ
λ =
= Mean number of arrivals per time
Mean number of arrivals per time
period
period
µ
µ =
= Mean number of units served per
Mean number of units served per
time period
time period
L
Ls
s =
= Average number of units
Average number of units
(customers) in the system (waiting and being
(customers) in the system (waiting and being
served)
served)
=
=
W
Ws
s =
= Average time a unit spends in the
Average time a unit spends in the
system (waiting time plus service time)
system (waiting time plus service time)
=
=
λ
λ
µ –
µ – λ
λ
1
1
µ –
µ – λ
λ

18. L
Lq
q =
= Average number of units waiting
Average number of units waiting
in the queue
in the queue
=
=
W
Wq
q =
= Average
Average time a unit spends
time a unit spends
waiting in the queue
waiting in the queue
=
=
p
p =
= Utilization factor for the system
Utilization factor for the system
=
=
λ
λ2
2
µ(µ –
µ(µ – λ
λ)
)
λ
λ
µ(µ –
µ(µ – λ
λ)
)
λ
λ
µ
µ

19. P
P0
0 =
= Probability of
Probability of 0
0 units in the
units in the
system (that is, the service unit is idle)
system (that is, the service unit is idle)
=
= 1 –
1 –
P
Pn > k
n > k =
= Probability of more than k units in the
Probability of more than k units in the
system, where n is the number of units in
system, where n is the number of units in
the system
the system
=
=
λ
λ
µ
µ
λ
λ
µ
µ
k
k + 1
+ 1

20. Single Channel Model Example
Single Channel Model Example
λ
λ =
= 2
2 cars arriving/hour
cars arriving/hour
µ
µ = 3
= 3 cars serviced/hour
cars serviced/hour
L
Ls
s = = = 2
= = = 2 cars
cars
in the system on average
in the system on average
W
Ws
s =
= = = 1
= = 1
hour average waiting time in
hour average waiting time in
the system
the system
L
Lq
q =
= = =
= =
1.33
1.33 cars waiting in line
cars waiting in line
λ
λ2
2
µ(µ –
µ(µ – λ
λ)
)
λ
λ
µ –
µ – λ
λ
1
1
µ –
µ – λ
λ
2
2
3 - 2
3 - 2
1
1
3 - 2
3 - 2
2
22
2
3(3 - 2)
3(3 - 2)

21. Cont…
Cont…
λ
λ =
= 2
2 cars arriving/hour,
cars arriving/hour, µ
µ = 3
= 3 cars
cars
serviced/hour
serviced/hour
W
Wq
q = =
= =
= 40
= 40 minute
minute
average waiting time
average waiting time
p
p =
= λ
λ/µ = 2/3 =
/µ = 2/3 =
66.6%
66.6% of time mechanic
of time mechanic
is busy
is busy
λ
λ
µ(µ –
µ(µ – λ
λ)
)
2
2
3(3 - 2)
3(3 - 2)
λ
λ
µ
µ
P
P0
0 = 1 - = .33
= 1 - = .33 probability
probability
there are
there are 0
0 cars in the system
cars in the system

22. Suggestions for Managing Queues
Suggestions for Managing Queues
1.
1. Determine an acceptable waiting time for
Determine an acceptable waiting time for
your customers
your customers
2.
2. Try to divert your customer’s attention when
Try to divert your customer’s attention when
waiting
waiting
3.
3. Inform your customers of what to expect
Inform your customers of what to expect
4.
4. Keep employees not serving the customers
Keep employees not serving the customers
out of sight
out of sight
5.
5. Segment customers
Segment customers

23. 1.
1. Train your servers to be friendly
Train your servers to be friendly
2.
2. Encourage customers to come during the
Encourage customers to come during the
slack periods
slack periods
3.
3. Take a long-term perspective toward getting
Take a long-term perspective toward getting
rid of the queues
rid of the queues

24. Where the Time Goes
Where the Time Goes
In a life time, the average
person will spend :
SIX MONTHS Waiting at stoplights
EIGHT MONTHS Opening junk mail
ONE YEAR Looking for misplaced 0bjects
TWO YEARS Reading E-mail
FOUR YEARS Doing housework
FIVE YEARS Waiting in line
SIX YEARS Eating

25. ANY QUESTIONS
ANY QUESTIONS
PLEASE ??
PLEASE ??