Queuing (Waiting Line) Models

7/28/2019 Queuing (Waiting Line) Models

1/34

Queuing (Waiting Line) Models


2/34

Examples of a Queue

Patients waiting at the doctors clinic

Customers waiting at booking windows.

Letters to be typed at a typists desk. Ships to be loaded or unloaded.

T.V. sets to be repaired at the repairers shop.

Phone calls arriving at the operators board.


3/34

Waiting Line Examples

Situation Arrivals Servers Service Process

Bank Customer Teller Deposit, etc.

Doctors Clinic Patient Doctor Treatment

Traffic

Intersection

Cars Light Controlled

Passage

Assembly Line Parts Workers Assembly

Tool crib Workers Clerks Check out / in

tools


4/34

FACT !!

Thus, queues not only

comprise of people butalso of goods.


5/34

Components of a Queuing System

P

o

p

ul

a

t

i

o

n

Arrival

Process

Balk

Renege

Jockey

Queue

Discipline

Balking: Customer decides not to enter the waiting line.

Reneging: Customer enters the line but decides to exit before being served.

Jockeying: Customer enters one line and then switches to a different line in an effort to reduce

the waiting time.


6/34

Queuing System:

Performance Measures1. Time related questions for the customers

a) What is the average (or expected) time an arriving customer has to wait in a queue beforebeing served.

b) What is the average (or expected) time an arriving customer spends in the system, includingwaiting and service.

2. Quantitative questions related tot the no. of customersa) The expected no. of customers who are in queue (queue length) for service.

b) The expected no. of customers who are in the system either waiting in the queue or beingserviced.

3. Questions involving value of time both for customers and serversa) What is the probability that an arriving customer has to wait before being served?

b) What is the probability that a server is busy at any particular point in time?

c) What is the probability of n customers being in the queuing system when it is in steadystate condition?

d) What is the probability of service denial when an arriving customer cannot enter the systembecause the queue is full?

4. Cost related questionsa) What is the average cost needed to operate the system per unit of time?

b) How many servers (service centres) are needed to achieve cost effectiveness?


7/34

Cost Relationship in Waiting Line

Analysis

Expected

costs

Level of service

Total cost

Service cost

Waiting Costs


8/34

Suggestions for Managing Queues

1. Determine an acceptable waiting time for your customers.

2. Try to divert your customers attention when waiting.

3. Inform your customers of what to expect.

4. Keep employees not serving the customers out of sight.

5. Segment customers.6. Train your servers to be friendly.

7. Encourage customers to come during the slack periods, will alsohelp in slackening the load.

8. Take a long-term perspective toward getting rid of the queues.Develop plans for alternative ways to serve your customers.Where appropriate, develop plans for automating or speeding upthe process in some manner. This is not to say you want toeliminate personal attention; some customers expect this.


9/34

Terminologies used

Service Station

Unit being serviced

Units in QUnits in

the system

Arrivals


10/34

Waiting Line Terminology

Queue: Waiting line

Arrival: A person, machine, part, etc. that

arrives and demands service

Queue discipline: Rules for determining the

order according to which that arrivals receive

service

Channel: Number of servers

Phase: Number of steps in service


11/34

Input

source Service

facility

Waiting

line

Service system

1995 Corel Corp.

Line istoo long!

Balking


12/34

Reneging

Input

source Service

facility

Waiting

line

Service system

1995 Corel Corp.

I give up!


13/34

Behavior of the arrivalsjoin the queue, and wait

until served

No balking; refusal to

join the line No reneging; leaving the

line

Service

FacilityWaiting Line

Pattern of arrivalsrandomscheduled

Arrival Characteristics of aWaiting Line System


14/34

Line Characteristics of a

Waiting Line System - continued

Length of thequeue limitedunlimited

Service priority FIFO LIFO

SIROPriority based

Pre-emptive &non pre-emptive

etc.

Service

FacilityWaiting Line

SIRO = Service in Random Order


15/34

Arrival Rate

Constant or Periodic

Ex. Intermediate Output of a production process.

Variable or Random

Customers arriving at a railway ticket counter.

Assumption:

the number of arrivals per time unit is Poisson

distributed, i.e., it follows an exponential pattern.


16/34

Where is the mean no. of arrivals per timeperiod.

Probability of n arrivals during time period isgiven by:


17/34

Example: if the mean arrival rate of units into a

system is three per minute ( = 3) and we want to

find theprobability that exactly five units will arrive

within a one-minute period(n = 5, T = 1), we have

That is, there is a 10.1 percent chance that there will

be five arrivals in any one-minute interval.


18/34

Waiting Line Models


19/34

Properties of some specific

Waiting Line models


20/34

Deterministic Waiting Line Models

Arrival rate = , customers arriving per unit time

Service rate = , customers per unit time

If > ,1. Queue formation.2. Indefinite Lengthening of queue.

3. Service facility would always be busy.

4. Service system eventually fails.

If < ,1. No waiting time2. Proportion of time the service facility would be idle = ( - )/ = 1 - /

The ratio, / = is called the average utilisation, or traffic intensity, or clearingratio

1. If > 1, the system would ultimately fail

2. If 1, the system works and is the proportion of time it is busy.


21/34

Average utilisation of the server =

Average length of the system, LS =

LS

= Arrival rate / time difference of ( - )

Average no. in the waiting line, LQ=LQ= * LS

=

( - )LS =

2

( - )LQ=


22/34

Average Waiting time in the system, WS = LS/

Average Waiting time in the Queue, WQ= LQ/

= * WS the probability that n customers are in the service system

at a given time = Pn = (1 - )n

Note: Before using the above formulas, make sure that > , ie., Service rate is greater than Arrival rate.

1( - )

=


23/34

Example Problem 1

Western National Bank is considering opening a drive-through window forcustomer service. Management estimates that customers will arrive at therate of 15 per hour. The teller who will staff the window can servicecustomers at the rate of one every three minutes.

Part 1 Assuming Poisson arrivals and exponential service, find

1. Utilization of the teller.

2. Average number in the waiting line.

3. Average number in the system.

4. Average waiting time in line.

5. Average waiting time in the system, including service.

Part 2

Because of limited space availability and a desire to provide an acceptablelevel of service, the bank manager would like to ensure, with 95 percentconfidence, that no more than three cars will be in the system at any time.What is the present level of service for the three-car limit? What level of telleruse must be attained and what must be the service rate of the teller to ensurethe 95 percent level of service?


24/34

Average utilisation of the server = = / =15/20 = 75%

Average length of the system, LS =

LS = Arrival rate / time difference of ( - ) LS = / ( - ) = 3 customers

Average no. in the waiting line, LQ=

LQ= * Ls

LQ= 2 = 2.25 customers

( - )

Arrival rate

time difference of ( - )

= * LS =

=

= * WS =

1

( - )= =


25/34

Example Problem 1: Solution

Average Waiting time in the system, WS = LS/

= 0.2 hr = 12 minutes

Average Waiting time in the Queue, WQ= LQ/

= * WS = 0.15 hr = 9 minutes

The probability that n customers are in the

service system at a given time = Pn = (1 - )n

1

( - )=

= Pn = (1 - )n


26/34



27/34


Rate of service required to attain this 95 percentservice level = ???? solve the equation: /=0.47,

where =number of arrivals per hour = 15.

This gives; =32 per hour.

Conclusion: That is, the teller must serve approximately 32 people per

hour (a 60 percent increase over the original 20-per-hourcapability) for 95 percent confidence that not more thanthree cars will be in the system.

Note that the teller will be idle 53 percent of the time.


28/34

Multiple Server Formulas

form.eventuallywilllinelonginfinitlyan

case,not theisthisIfstability.systemfor:Note

nutilizatiosystemaverage

serversidenticalparallel,ofnumber

serverforrateservicemeanmu

ratearrivalmeanlambda

s

s

s

one


29/34

Multiple Server Formulas (continued)

in timepointgivenaatsystemin the

customersofyprobabilitfor

!

/

for

!

/

in timepointgivenaatsystemin thecustomers

zeroofyprobabilit

1

1

!

/

!

/

0

0

11

0

0

nsnP

ss

snP

nP

sn

P

sn

n

n

n

s

n

sn


30/34

Multiple Server Formulas (continued)

systemincustomersofnumberaverage

serviceincludingsystemintimeaverage1

lineinwaitingspenttimeaverage

lineincustomersofnumberaverage1!

/

2

0

WL

WW

LW

s

P

L

q

qq

s

q


31/34

Example: Multiple Server

Computer Lab Help Desk

Now 45 students/hour need help.

3 servers, each with service rate of 18

students/hour

Based on this, we know:

= 18 students/hour

s = 3 servers = 45 students/hour


32/34

Changing System Performance

Customer Arrival Rates

Try to smooth demand through non-peak discounts or pricepromotions

Number and type of service facilities

Increase or decrease number of servers, or dedicate specific serversfor certain tasks (e.g., express line for under 10 items)

Change Number of Phases

Can use multi-phase system instead of single phase. This spreads theworkload among more servers and may result in better flow (e.g., fastfood restaurants having an order phase, pay phase, and pick-up phaseduring busy hours)


33/34

Changing System Performance

Server efficiency Add resources to each phase (e.g., bagger helping a

checker at the grocery store)

Use technology (e.g. price scanners) to improve efficiency

Change priority rules Example: implement a reservation protocol

Change the number of lines Reduce multiple lines to single queue to avoid jockeying

Dedicate specific servers to specific transactions


34/34

Thank you!!

Queuing (Waiting Line) Models

Documents

Transcript of Queuing (Waiting Line) Models