1. QUEUING THEORY
Made by Abdulsalam shek salim

2. Introduction Body of knowledge about waiting lines
Helps managers to better understand systems in manufacturing, service, and maintenance
Provides competitive
advantage and cost saving
A QUEUE REPRESENTS ITEMS
OR PEOPLE AWAITING SERVICE

3. Queue Characteristics
1) Average number of customers in a line
2) Average number of customers in a service facility
3) Probability a customer must wait
4) Average time a customer spends in a waiting line.
5) Average time a customer spends in a service facility
6) Percentage of time a service facility is busy

4. Queuing System Examples

5. The Father of Queuing Theory
Danish engineer, who, in 1909 experimented with fluctuating demand in telephone traffic in Copenhagen.
In 1917, he published a report addressing the delays in auto-matic telephone dialing equip-ment.
At the end of World War II, his work was extended to more general problems, including waiting lines in business.
AGNER K. ERLANG

6. Lack of Managerial Intuition Surrounding Waiting Lines
Queuing theory is not a matter of common sense. It is one of those applications where diligent, intelligent managers will arrive at drastically wrong solutions if they fail to thoroughly appreciate and understand the mathematics involved.

7. THE QUEUING COST TRADE-OFF
Total Cost
Cost of Providing
Service
( salaries + benefits )
Minimum
Total
Cost
Cost of Waiting
Time
( time x value of time )
Low Level
Of Service
Optimal Service
Level
High Level
Of Service

8. Aspects of a Queuing Process
SYSTEM ARRIVALS
THE QUEUE ITSELF
THE SERVICE FACILITY

9. Poisson Arrival Distribution
( probability )
.25
.20
.15
.10
.05
.00
Poisson
Probability
Distribution
for
λ = 2
(estimated mean arrival rate)
X ( the number of arrivals )

10. Poisson Arrival Distribution
( probability )
.25
.20
.15
.10
.05
.00
Poisson
Probability
Distribution
for
λ = 4
(estimated mean arrival rate)
X ( the number of arrivals )

11. Establishing A Discrete Poisson Arrival Distribution
Given any average arrival rate ( λ ) in seconds, minutes, hours, days:
- λ x
P ( X ) = ε λ X!
( FOR X = 0,1,2,3,4,5, etc. )
Where : P ( X ) = probability of X arrivals
X = number of arrivals per time unit
λ = the average arrival rate
ε = ( base of the natural logarithm )

12. If the average arrival rate probability of three ( 3 )
EXAMPLE
If the average arrival rate
per hour is two people
( λ = 2 ) , what is the
probability of three ( 3 )
arrivals per hour?

13. Solution Working on the same formula which we explain it befor ..
- λ X ε λ
P ( X ) =
X !
- 2 3
Given λ = 2 :
P ( 3 ) =
3 !
= [ 1 / ] x 8
(3)(2)(1)
= x 8 = ≈
18% 6

14. Theoretical Distribution Observed Distribution
Precise Terminology
Theoretical Distribution
Observed Distribution
The discrete arrival probability distribution, based on the average arrival rate ( λ ) which was computed from the actual system observations.
The actual discrete arrival probability distribution that was constructed from the actual system observations.
THIS DISTRIBUTION MAY
OR MAY NOT BE POISSON
DISTRIBUTED.

15. Service Times Service times normally follow a negative exponential
probability
distribution
.25
.20
.15
.10
.05
.00 P R O B A I L T Y
THE PROBABILITY
A CUSTOMER
WILL REQUIRE
THAT SERVICE
TIME
seconds

16. Some of Queuing Theory Variables
Lambda ( λ ) is the average arrival rate
of people or items into the service system.
It can be expressed in seconds, minutes, hours, or days.
From the Greek small letter “ L “.

17. Queuing Theory Variables
Mu ( μ ) is the average service rate of the service system.
It can be expressed as the number of people or items processed per second, minute, hour, or day.
From the Greek small letter “ M “.

18. Queuing Theory Variables
Rho ( ρ ) is the % of time that the service facility is busy on the average.
It is also known as the
utilization rate.
From the Greek small letter “ R “.

19. Queuing Theory Variables
Mu ( M ) is a channel or service point in the ser-vice system.
Examples are gasoline pumps, checkout coun-ters, vending machines, bank teller windows.
From the Greek large letter “ M “.

20. IMPORTANT CONSIDERATION
The average service rate must always exceed the average arrival rate.
Otherwise, the queue will grow to infinity.
μ > λ
THERE WOULD BE NO SOLUTION !

21. Dual-Channel / Single-Phase System
EXIT
ONE WAITING LINE or QUEUE
ONE SERVICE POINT or CHANNEL

22. Dual-Channel / Single-Phase System
EXIT
EXIT No
Jockeying
Permitted
Between Lines
ONE OR TWO WAITING LINES
TWO DUPLICATE SERVICE POINTS

23. Dual-Channel / Single-Phase System
EXIT
EXIT
Jockeying
Is Permitted
Between
Lines !
ENTER
ENTER
TWO IDENTICAL SERVICE CHANNELS.
EACH CHANNEL HAS 3 DISTINCT SERVICE POINTS ( A-B-C )

24. Thank you all for listening
We have too much things to speak about Queuing Theory but because of the we are we are competing with the time it will be enough until here .