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UNIT I
INTRODUCTION TO MODELING ANDSIMULATION
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A System is defined as an aggregation or assemblage ofobjects joined in some regular interaction orinterdependence. While this definition is broad enough toinclude static systems, the principal interest will be indynamic systems where the interactions cause changes overtime.
Desire
Heading
Gyroscope Control
Surface
Airframe
Actual Heading
An aircraft under autopilot control
SYSTEM
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SIMULATION EXAMPLE
A FACTORY SYSTEM
PRODUCTION
CONTROL DEPT.
PURCHASING
DEPTFABRICATION
DEPT
ASSEMBLING
DEPT
SHIPPING
DEPT
CUSTOMER
ORDERRAW
MATERIALS
PRODUCT
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SYSTEM ENVIRONMENT
System is affected by changes occurring outside the system.
Such changes occurring Outside the systemare said to occur in system environment.
ENDOGENEOUS
Used to describe activitiesoccurring within the system
EXOGENEOUS
Used to describe activities
In the environment thataffect the system
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ACTIVITIES
DITERMINISTIC
Outcome of activity can bedescribe
completely in terms of input
STOCHASTIC
Effect of activity varyrandomly Over
various possible outcome.
SYSTEM SYSTEM
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CONTINUOUS ANDDISCRETE SYSTEM
In continuoussystem, changesare predominantly
smooth.
Example: Aircraft.
In discrete system,changes arepredominantly
discontinuous.
Example: Factory.
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SYSTEM MODELING
Model is defined as the body ofinformation about a system gathered forthe purpose of studying the system.
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TYPES OF MODEL
PHYSICAL
It is based on analogy
between such systemas mechanical andelectrical. In this,system attributes arerepresented by
measurements such asvoltage or position ofshaft
MATHEMATICAL
It uses symbolic
notation andmathematical equationto represent system.
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Physical Model Mathematical Model
Static Dynamic Static Dynamic
Numerical Analytical Numerical
SystemSimulation
MODEL
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STATIC PHYSICALMODEL
An example of a static physical model is a stick model of
a water molecule, with two small hydrogen "balls" stuck
with short sticks on either side of the oxygen "ball." This
model does not change with time. Another physical
model is that of a tank of water with sand, which shows
the effect of the wind and the movement of water.
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MATHEMATICAL MODEL
A mathematical model is a description of a system
using mathematical language. The process of
developing a mathematical model is termed
mathematical modeling (also written modeling).
Mathematical models are used not only in the natural
science (such as physics, biology, earth science,
meteorology) and engineering disciplines, but also in
the social science (such as economics, psychology,
sociology and political science); economists use
mathematical models most extensively.
http://en.wikipedia.org/wiki/Physicshttp://en.wikipedia.org/wiki/Biologyhttp://en.wikipedia.org/wiki/Earth_sciencehttp://en.wikipedia.org/wiki/Meteorologyhttp://en.wikipedia.org/wiki/Economicshttp://en.wikipedia.org/wiki/Psychologyhttp://en.wikipedia.org/wiki/Sociologyhttp://en.wikipedia.org/wiki/Political_sciencehttp://en.wikipedia.org/wiki/Economisthttp://en.wikipedia.org/wiki/Economisthttp://en.wikipedia.org/wiki/Political_sciencehttp://en.wikipedia.org/wiki/Sociologyhttp://en.wikipedia.org/wiki/Psychologyhttp://en.wikipedia.org/wiki/Economicshttp://en.wikipedia.org/wiki/Meteorologyhttp://en.wikipedia.org/wiki/Earth_sciencehttp://en.wikipedia.org/wiki/Biologyhttp://en.wikipedia.org/wiki/Physics8/8/2019 Simulation 4 Unit
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EXAMPLES OFMATHEMATICAL MODEL
Population Growth.
Model of a particle in a potential-field.
Model of rational behavior for a
consumer.
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STATIC V/S DYNAMICMODEL
Static vs. dynamic: A static model does
not account for the element of time, while a
dynamic model does. Dynamic models
typically are represented with difference
equation or differential equations.
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PRINCIPLES USED INMODELING
Block Building
Description of system should be
organized in series of blocks.
Relevance
Model should only include those aspectsof the system that are relevant to thestudy objectives.
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UNIT II
SYSTEM SIMULATION AND CONTINUOUSSYSTEM SIMULATION
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TECHNIQUE OF SIMULATION
ANALYTICAL NUMERICAL
It produces directlythe general
solution
It produces solutionin
steps
Dynamic Problems Static Problems
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Monte Carlo Simulation
Select numbers randomly from aprobability distribution
Use these values to observe how amodel performs over time
Random numbers each have an equallikelihood of being selected at random
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MONTE-CARLOSIMULATION
.
RANDOM
NUMBER
DISTRIBUTION
RANDOM
VARIABLE
SIMULATION
OUTPUT
REAL
SYSTEM
MODEL
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TYPES OF SYSTEM
CONTINUOUSSYSTEM
In continuoussystem, changesare predominantlysmooth.
Example: Aircraft.
DISCRETE SYSTEM
In discrete system,changes arepredominantlydiscontinuous.
Example: Factory.
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DISTRIBUTED LAGMODEL
Model that have the properties of changingonly at fixed interval of time are calleddistributed lag model.
These are used in economic studies wherethe uniform steps corresponds to a timeinterval, such as month or a year.
These model consist of linear, algebraicequations.
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COBWEB MODEL
The cobweb model or cobweb theory is aneconomic model that explains why prices might
Be subject to periodic fluctuations in certain types
of markets. It describes cyclical supply anddemand in a market where the amount producedmust be chosen before prices are observed.Producers' expectations about prices are assumed
to be based on observations of previous prices.
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Two other possibilities are:
Fluctuations may also remain of constant magnitude,so a plot of the equilibria would produce a simplerectangle, if the supply and demand curves have
exactly the same slope. If the supply curve is less steep than the demand
curve near the point where the two curves cross, butmore steep when we move sufficiently far away, thenprices and quantities will spiral away from theequilibrium price but will not diverge indefinitely;instead, they may converge to a limit cycle.
COBWEB MODEL
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Continuous SystemModels Continuous system models were the first widely
employed models and are traditionally described byordinary and partial differential equations.
Such models originated in such areas as physics andchemistry, electrical circuits, mechanics, andaeronautics.
They have been extended to many new areas such asbio-informatics, homeland security, and social
systems.
Continuous differential equation models remain anessential component in multi-formalism compositions.
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Analog computer
Analog computer measures andanswer the questions by the methodof HOW MUCH. The input data is not
a number infect a physical quantitylike tem, pressure, speed, velocity.
Signals are continuous of (0 to 10 V)
Accuracy 1% Approximately High speed
Output is continuous
Time is wasted in transmission time
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Digital Computers
Digital computer counts and answer thequestions by the method of HOW Many.The input data is represented by a number.
These are used for the logical andarithmetic operations.
Signals are two level of (0 V or 5 V)
Accuracy unlimited
low speed sequential as well as parallelprocessing
Output is continuous but obtain when
computation is completed.
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Hybrid Computer
The combination of features of analogand digital computer is called Digitalcomputer. The main example are
central national defense andpassenger flight radar system. Theyare also used to control robots.
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UNIT III
SYSTEM DYNAMICS & PROBABILITYCONCEPT IN SIMULATION
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EXPONENTIAL GROWTHMODEL
Exponential growth (including exponential decal occurswhen the growth rate of a mathematical function ispropotional to the function's current value.
Human Population, if the number of births and deaths perperson per year were to remain at current levels
Heat Transfer experiments yield results whose best fit line areexponential growth curves.
Compound Interest at a constant interest rate providesexponential growth of the capital.
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LIMITATIONS
Exponential growth models of physical phenomenaonly apply within limited regions, as unboundedgrowth is not physically realistic. Although growthmay initially be exponential, the modeledphenomena will eventually enter a region in whichpreviously ignored Negative feedback factors
become significant (leading to a Logistic growthmodel) or other underlying assumptions of theexponential growth model, such as continuity orinstantaneous feedback, break down.
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EXPONENTIAL DECAY
MODEL
A quantity is said to be subject to exponential
decay if it decreases at a rate proportional to itsvalue. Symbolically, this process can be modeledby the following differential equation, where Nisthe quantity and (lambda) is a positive number
called the decay constant:
EXPONENTIAL DECAY
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Exponential decay occurs in a wide variety ofsituations. Most of these fall into the domain of thenatural sciences. Any application of mathematics tothe
Social science or humanities is risky and uncertain,because of the extraordinary complexity of humanbehavior. However, a few roughly exponentialphenomena have been identified there as well.Many decay processes that are often treated as
exponential, are really only exponential so long asthe sample is large and the law of large numbersholds. For small samples, a more general analysis isnecessary, accounting for a Poission process.
EXPONENTIAL DECAYMODEL
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APPLICATIONS
Ecology
Neural Network
Statistics In medicine: modeling of growth of
tumors
S t d i
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System dynamicsDiagram
System dynamics is an approach tounderstanding the behaviour ofcomplexsystems over time. It deals with internalfeedback loops and time delays that affect
the behaviour of the entire system.[1]What makes using system dynamicsdifferent from other approaches tostudying complex systems is the use offeedback loops and stocks and flows.
These elements help describe how evenseemingly simple systems display bafflingnonlinearity.
http://en.wikipedia.org/wiki/Complex_systemhttp://en.wikipedia.org/wiki/Complex_systemhttp://en.wikipedia.org/wiki/Feedbackhttp://en.wikipedia.org/wiki/Stock_and_flowhttp://en.wikipedia.org/wiki/Nonlinearityhttp://en.wikipedia.org/wiki/Nonlinearityhttp://en.wikipedia.org/wiki/Stock_and_flowhttp://en.wikipedia.org/wiki/Feedbackhttp://en.wikipedia.org/wiki/Complex_systemhttp://en.wikipedia.org/wiki/Complex_system8/8/2019 Simulation 4 Unit
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Causal loop diagrams
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Stock and flow diagrams
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MULTI SEGMENT MODEL A multi-segment model is used to investigate optimal
compliant-surface jumping strategies and is applied tospringboard standing jumps. The human model hasfour segments representing the feet, shanks, thighs,and trunkheadarms. A rigid bar with a rotationalspring on one end and a point mass on the other end(the tip) models the springboard. Board tip mass,length, and stiffness are functions of the fulcrumsetting. Body segments and board tip are connectedby frictionless hinge joints and are driven by joint
torque actuators at the ankle, knee, and hip.
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RANDOM NUMBER
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RANDOM NUMBERGENERATION
A random number generator (oftenabbreviated as RNG) is acomputational or physical device
designed to generate a sequence ofnumbers or symbols that lack anypattern, i.e. appear random.
P actical applications
http://en.wikipedia.org/wiki/Computerhttp://en.wikipedia.org/wiki/Numberhttp://en.wikipedia.org/wiki/Randomhttp://en.wikipedia.org/wiki/Randomhttp://en.wikipedia.org/wiki/Numberhttp://en.wikipedia.org/wiki/Computer8/8/2019 Simulation 4 Unit
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Practical applicationsand uses
Gambling
Statistical sampling
Computer Simulation Cryptography
Completely randomized design
SIMULATION OF QUEUING SYSTEM
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UNIT IV
SIMULATION OF QUEUING SYSTEMAND DISCRETE SYSTEM SIMULATION
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El t f W iti
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Elements of WaitingLines
Queue is another name for a waitingline.
A waiting line system consists of two
components: The customer population (people or objects
to be processed)
The process or service system
Whenever demand exceeds availablecapacity, a waiting line or queue forms There is a tradeoff between cost and
service level.
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Customer PopulationCharacteristics
Finite versus Infinite populations: Is the number of potential new customers materially
affected by the number of customers already in queue?
Balking When an arriving customer chooses not to enter a queue
because its already too long.
Reneging When a customer already in queue gives up and exits
without being serviced. Jockeying
When a customer switches between alternate queues inan effort to reduce waiting time.
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Service System
The service system is defined by:
The number of waiting lines
The number of servers The arrangement of servers
The arrival and service patterns
The service priority rules
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Number of Lines
Waiting lines systems can havesingle or multiple queues.
Single queues avoid jockeying behaviorand perceived fairness is usually high.
Multiple queues are often used whenarriving customers have differingcharacteristics (e.g. paying with cash,less than 10 items, etc.) and can bereadily segmented.
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Servers
Single servers or multiple, parallelservers providing multiple channels
Arrangement of servers (phases) Multiple phase systems require customers
to visit more than one server
Example of a multi-phase, multi-server
system:
C C C CC DepartArrivals
1
2
3 6
5
4
Phase 1 Phase 2
Example Queuing
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Example QueuingSystems
A i l & S i
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Arrival & ServicePatterns
Arrival rate:
The average number of customers arriving
per time period Modeled using the Poisson distribution
Arrival rate usually denoted by lambda ()
Example: =50 customers/hour; 1/=0.02hours between customer arrivals (1.2 minutesbetween customers)
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Arrival & Service Patterns
Service rate: The average number of customers that can be
served during the period of time
Service times are usually modeled using theexponential distribution
Service rate usually denoted by mu ()
Example: =70 customers/hour; 1/=0.014
hours per customer (0.857 minutes percustomer).
Even if the service rate is larger than thearrival rate, waiting lines form!
Reason is the variation in specific customer
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Example Priority Rules First come, first served
Best customers first (reward loyalty)
Highest profit customers first
Quickest service requirements first
Largest service requirements first
Earliest reservation first Emergencies first
Etc.
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Waiting Line PerformanceMeasures
Lq = The average number of customerswaiting in queue
L = The average number of customersin the system
Wq = The average waiting time inqueue
W= The average time in the system
p = The system utilization rate (% oftime servers are busy)
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Single-Server Waiting Line Assumptions
Customers are patient (no balking, reneging, orjockeying)
Arrivals follow a Poisson distribution with a meanarrival rate of. This means that the timebetween successive customer arrivals follows anexponential distribution with an average of 1/
The service rate is described by a Poissondistribution with a mean service rate of . Thismeans that the service time for one customer
follows an exponential distribution with anaverage of 1/
The waiting line priority rule is first-come, first-served
Infinite population
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Formulas: Single-ServerCase
= lambda= mean arrival rate
=mu= mean service rate
p=
= average system utilization
Note:> for system stability. If this is not the case,
an infinitl lon line will eventuall form.
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Formulas: Single-ServerCase (continued)
L=
= average number of customers in system
Lq =pL=average number of customers in line
W=1
= average time in system including service
Wq =pW=average time spent waiting
Pn= 1 p pn= probability ofn customers in the system
at a given point in time
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Example
A help desk in the computer lab servesstudents on a first-come, first servedbasis. On average, 15 students need
help every hour. The help desk canserve an average of 20 students perhour.
Based on this description, we know: = 20 students/hour (average service time
is 3 minutes)
= 15 students/hour (average timebetween student arrivals is 4 minutes)
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Average Utilization
p=
=
15
20 = 0.75 or75
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h S
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Average Time in the System,and in Line
W=1
=
1
20
15= 0 .2 hours
or 12 minutes
Wq =pW=0 .75 0 .2 = 0 . 15 hours
or 9 minutes
P b bili f
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Probability ofnStudents in the Line
P0= 1 p p0= 1 0 . 75 1= 0.25
P1=
1
p p
1=
1
0. 75 0 . 75=
0.188P2= 1 p p
2= 1 0. 75 0 . 752= 0.141
P3= 1 p p3= 1 0 .75 0 . 75
3= 0.105
P 4= 1 p p4= 1 0 . 75 0 .754= 0.079
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Single Server: SpreadsheetApproach
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A B C
QueuingAnalysis: SingleServer
Inputs
Timeunit hour
Arrival Rate(lambda) 15 customers/hour
ServiceRate(mu) 20 customers/hour
IntermediateCalculations
Averagetimebetweenarrivals 0.066667 hour
Averageservicetime 0.05 hour
PerformanceMeasures
Rho(averageserver utilization) 0.75
P0(probabilitythesystemisempty) 0.25
L(averagenumberinthesystem) 3 customersLq(averagenumber waitinginthequeue) 2.25 customers
W(averagetimeinthesystem) 0.2 hourWq(averagetimeinthequeue) 0.15 hour
Probabilityof aspecificnumber of customersinthesystemNumber 2
Probability 0.140625
Key FormulasB9: =1/B5B10: =1/B6B13: =B5/B6B14: =1-B13B15: =B5/(B6-B5)B16: =B13*B15B17: =1/(B6-B5)B18: =B13*B17B22: =(1-B$13)*(B13^B21)
Use Data Table (trackingB22) to easily computethe probability ofncustomers in the
system.
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Multiple Server Case
Assumptions
Same as Single-Server, except here we
have multiple, parallel servers Single Line
When server finishes with customer, firstperson in line goes to the idle server
All servers are identical
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Multiple Server Formulas
= lambda= mean arrival rate
=mu= mean service rate for one server
s= number of parallel, identical servers
p=
s= average system utilization
Note:s> for system stability. If this is not the case,an infinitly long line will eventually form.
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l l l
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Multiple Server Formulas(continued)
Lq=P
0/
sp
s! 1 p 2=
average number of customers in line
Wq=L
q/=average time spent waiting in line
W=Wq
1
= average time in system including service
L=W= average number of customers in system
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Example: Multiple Server
Computer Lab Help Desk
Now 45 students/hour need help.
3 servers, each with service rate of18 students/hour
Based on this, we know: = 18 students/hour
s = 3 servers
= 45 students/hour
Flexible Spreadsheet Approach
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Flexible Spreadsheet Approach
Formulas are somewhat complex to set up initially, butyou only need to do it once!
For other multiple-server problems, can just change theinput values.
This approach also makes sensitivity analysis possible.
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A B C
QueuingAnalysis: MultipleServers
Inputs
Timeunit hour
Arrival Rate(lambda) 45 customers/hour
ServiceRateper Server (mu) 18 customers/hour
Number of Servers(s) 3 servers
IntermediateCalculations
Averagetimebetweenarrivals 0.022222 hour
Averageservicetimeper server 0.055556 hour
Combinedservicerate(s*mu) 54 customers/hour
PerformanceMeasures
Rho(averageserver utilization) 0.833333P0(probabilitythesystemisempty) 0.044944
L(averagenumberinthesystem) 6.011236 customers
Lq(averagenumber waitinginthequeue) 3.511236 customersW(averagetimeinthesystem) 0.133583 hour
Wq(averagetimeinthequeue) 0.078027 hour
Probabilityof aspecificnumber of customersinthesystemNumber 5
Probability 0.081279
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108
109
E F G H
WorkingCalculations, mainlyfor P0Calculation
lambda/mu 2.5
s! 6
n (/)n n! Sum0 1 1 1
1 2.5 1 3.5
2 6.25 2 6.625
3 15.625 6 9.229166667
4 39.0625 24 10.85677083
5 97.65625 120 11.67057292
6 244.14063 720 12.00965712
7 610.35156 5040 12.13075862
8 1525.8789 40320 12.16860284
9 3814.6973 362880 12.17911512
10 9536.7432 3628800 12.18174319
11 23841.858 39916800 12.18234048
12 59604.645 47900160012.18246492
13 149011.61 6.227E+0912.18248885
14 372529.03 8.718E+1012.18249312
15 931322.57 1.308E+1212.18249383
16 2328306.4 2.092E+1312.18249394
17 5820766.1 3.557E+1412.18249396
18 14551915 6.402E+1512.18249396
99 2.489E+39 9.33E+15512.18249396
100 6.223E+39 9.33E+15712.18249396
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Key Formulas for Spreadsheet
F10: =F$5^E10 (copied down)
G10: =E10*G9 (copied down)
H10: =H9+(F10/G10) (copied down)
F5: =B5/B6
F6: =INDEX(G9:G109,B7+1) B10: =1/B5
B11: =1/B6
B12: =B7*B6
B15: =B5/B12
B16: = (INDEX(H9:H109,B7)+ (((F5^B7)/F6)*((1)/(1-B15))))^(-1)
B17: =B5*B19 B18: =(B16*(F5^B7)*B15)/(INDEX(G9:G109,B7+1)*(1-B15)^2)
B19: =B20+(1/B6)
B20: =B18/B5
B24: =IF(B23
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Probability ofn students in thesystem
Probability of Number in System
0.0000
0.0200
0.0400
0.0600
0.0800
0.1000
0.1200
0.1400
0.1600
0 2 4 6 810
12
14
16
18
20
22
24
26
28
30
Number in System
Probability
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Changing System Performance
Customer Arrival Rates Try to smooth demand through non-peak discounts
or price promotions
Number and type of service facilities Increase or decrease number of servers, or dedicate
specific servers for certain tasks (e.g., express linefor under 10 items)
Change Number of Phases
Can use multi-phase system instead of single phase.This spreads the workload among more servers andmay result in better flow (e.g., fast food restaurantshaving an order phase, pay phase, and pick-upphase during 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) toimprove 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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Supplement D Highlights
The elements of a waiting line system include the customerpopulation source, the patience of the customer, the servicesystem, arrival and service distributions, waiting line priorityrules, and system performance measures. Understanding theseelements is critical when analyzing waiting line systems.
Waiting line models allow us to estimate system performance bypredicting average system utilization, average number ofcustomers in the service system, average number of customerswaiting in line, average time a customer waits in line, and theprobability ofn customers in the service system.
The benefit of calculating operational characteristics is to
provide management with information as to whether systemchanges are needed. Management can change the operationalperformance of the waiting line system by altering any or all ofthe following: the customer arrival rates, the number of servicefacilities, the number of phases, server efficiency, the priorityrule, and the number of lines in the system. Based on proposedchanges, management can then evaluate the expectedperformance of the system.
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Changing System Performance
Customer Arrival Rates Try to smooth demand through non-peak discounts
or price promotions
Number and type of service facilities Increase or decrease number of servers, or dedicate
specific servers for certain tasks (e.g., express linefor under 10 items)
Change Number of Phases
Can use multi-phase system instead of single phase.This spreads the workload among more servers andmay result in better flow (e.g., fast food restaurantshaving an order phase, pay phase, and pick-upphase during 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) toimprove 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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Supplement D Highlights
The elements of a waiting line system include the customerpopulation source, the patience of the customer, the servicesystem, arrival and service distributions, waiting line priorityrules, and system performance measures. Understanding theseelements is critical when analyzing waiting line systems.
Waiting line models allow us to estimate system performance bypredicting average system utilization, average number ofcustomers in the service system, average number of customerswaiting in line, average time a customer waits in line, and theprobability ofn customers in the service system.
The benefit of calculating operational characteristics is to
provide management with information as to whether systemchanges are needed. Management can change the operationalperformance of the waiting line system by altering any or all ofthe following: the customer arrival rates, the number of servicefacilities, the number of phases, server efficiency, the priorityrule and the number of lines in the system Based on proposed