Queuing (Waiting Line) Models
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Transcript of Queuing (Waiting Line) Models
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Queuing (Waiting Line) Models
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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.
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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
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FACT !!
Thus, queues not only
comprise of people butalso of goods.
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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.
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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?
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Cost Relationship in Waiting Line
Analysis
Expected
costs
Level of service
Total cost
Service cost
Waiting Costs
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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.
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Terminologies used
Service Station
Unit being serviced
Units in QUnits in
the system
Arrivals
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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
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Input
source Service
facility
Waiting
line
Service system
1995 Corel Corp.
Line istoo long!
Balking
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Reneging
Input
source Service
facility
Waiting
line
Service system
1995 Corel Corp.
I give up!
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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
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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
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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.
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Where is the mean no. of arrivals per timeperiod.
Probability of n arrivals during time period isgiven by:
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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.
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Waiting Line Models
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Properties of some specific
Waiting Line models
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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.
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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=
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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( - )
=
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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?
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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
( - )= =
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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
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Example Problem 1: Solution
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Example Problem 1: Solution
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.
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Multiple Server Formulas
form.eventuallywilllinelonginfinitlyan
case,not theisthisIfstability.systemfor:Note
nutilizatiosystemaverage
serversidenticalparallel,ofnumber
serverforrateservicemeanmu
ratearrivalmeanlambda
s
s
s
one
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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
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Multiple Server Formulas (continued)
systemincustomersofnumberaverage
serviceincludingsystemintimeaverage1
lineinwaitingspenttimeaverage
lineincustomersofnumberaverage1!
/
2
0
WL
WW
LW
s
P
L
q
qq
s
q
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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
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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)
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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
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Thank you!!