QUEUING THEORY

Quantitative Techniques for Decision MakingM.P. Gupta & R.B. Khanna© Prentice Hall India

QUEUING THEORY 17CHAPTER


Learning Objectives

• Characteristics of a queue.

• Single Channel Single Server Queuing Model

• Utilisation Factor

• Economic Aspects of Queuing.


Queuing

• Whenever any person or any thing has to wait for a service, there is economic loss due to the waiting time.

• This can be remedied by increasing the service facilities. This in turn add to the costs.

• A balance must be struck between loss due to waiting time and the cost of providing extra service facilities.

• Queuing Models deal with such problems.

• Queuing models are descriptive and not prescriptive.


Characteristics of a Queue

• The Calling Population– Size – Finite or infinite– Arrival characteristics

• Poisson Distribution• Other distributions

– Behaviour of the Calling Population• Reneges queue• Baulks queue• Patient caller



• The Service Facility – Physical Layout

Service Facility Type I

Service Facility Type 1


Single Channel, Single Server

Single Channel, Multi Server






Multi Channel Single Server








Multi Channel, Multi Server



• The Service Facility – Queue Discipline – First Come First Served or First In First Out

(FCFS or FIFO)– Last In First Out (LIFO)– Priority (PRI)

• Pre-emptive Priority• Non pre-emptive

– Service in Random Order (SIRO)



• The Service Facility – Service Time – Exponentially distributed– Other distribution

• The Queue – Size– Finite– Infinite



Increased Service

Costs

Waiting Costs

Cost of Facilities

Total costs

The aim is to reduce total cost

Increased Service

Costs

Waiting Costs

Cost of Facilities

Total costs


Single Channel Single Server Model M/M/1

• Arrivals follows a Poisson distribution (M)• Service times follow an exponential distribution (M)• Single Channel Single Server (1)• The queue discipline is FCFS – first come, first

served (FCFS)• The calling population is large enough to be

considered infinite (∞)• The length of the queue is also infinite (∞) • Kendall - Lee’s notation : M/M/1: FCFS/∞/∞.


Single Channel Single Server Model M/M/1

Waiting Time in System = sW1 1

or ( )S A

• If arrival rate is A (λ) and service rate is S (μ), then

Waiting time in queue or ( ) (time units)( ) ( )q

AW

S S A

Length in service or ( ) (numbers)s

AL

S A

Length in Queue

2 2

or ( )( ) ( )q

AL

S S A

(numbers)

(time units)


M/M/1 - Example

• Interval between aircraft arrivals is 20 minutes i.e. 3 per hour• Unloading time is 15 minutes per aircraft i.e. 4 aircraft per hour

2

3

4

1 11

4 33

454 4 3

33

4 3

3 32 25

4 4 3

hour

minutes( ) ( )

aircraft

. aircraft( ) ( )

s

q

s

q

A

S

WS A

AW

S S A

AL

S A

AL

S S A


• Aircraft are spending 1 hour on the ground instead of 15 minutes as planned

• If two unloading crews are used and the service rate doubled to 8 aircraft an hour, we get

2

3

8

1 112

8 33

4 58 8 3

30 6

8 3

3 30 225

8 8 3

minutes

. minutes( ) ( )

. aircraft

. aircraft( ) ( )

s

q

s

q

A

S

WS A

AW

S S A

AL

S A

AL

S S A

The aircraft will now be spending only 12 minutes on the ground and the planned tonnage can be delivered.


Utilisation Factor

• The ratio is called the utilisation factor. • It is also the probability that the system is

busy. • Probability that the system is busy

• Probability that the system is idle

or A

S

A

S

1 1 1A

S


Utilisation Factor

• The length of the queue increases sharply when the utilisation factor is more than 0.7.

• For practical purposes, a queue system should be so designed that its utilisation factor is around 0.7.

0

10

20

30

40

50

60

0 0.2 0.4 0.6 0.8 1

Utilisation Factor

Len

gth

of

Qu

eue


Economic Aspect of Queuing

•A computer maintenance contract is to be signed by your company office.•At an average three computers per month go off road due to various defects.•The cost of a computer being unavailable is Rs 8000 per month.•Alfa Computers have quoted at Rs 3000 per month, and can repair 5 computers per month•Beta Bytes has quoted at Rs 5000 per month for the contract and can repair 6 computers per month at an average•Who should get the contract?


M/M/1 - Example

Numbers in system

sA

LS A

31.5

5 3

3

16 3

1.5 8000 12000 1 8000 8000

Alfa Computers Beta Bytes

(a) Arrival rate of computers for repairs (A)

3 per month 3 per month

(b) Service Rate (S) 5 per month 6 per month

(c)

(d) Cost of off road computers

(e) Cost of Contract 3000 5000

(f) Total cost 15000 13000

QUEUING THEORY

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