1. “QUEUING THEORY”

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

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

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

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

6. Queuing examples 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

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

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

9. 1. Arrival Process 2. Queue Structure According to source
According to numbers
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. Service system 1. A single service system.
Queue
Service facility
Departures
after service
Arrivals
e.g- Your family dentist’s office, Library counter

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

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

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

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

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

16. 2.Probabilistic queuing model Probability that n customers will arrive in the system in time interval T is

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

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

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

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

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

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

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

24. Where the Time Goes person will spend :
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
FOUR YEARS Doing housework
FIVE YEARS Waiting in line
SIX YEARS Eating

25. ANY QUESTIONS PLEASE ??