Review for Exam IIReview for Exam II

Tues. July 29, 2003Tues. July 29, 2003

(50% multiple choice, 50% (50% multiple choice, 50% problems)problems)

BringBring

an orange scantron sheetan orange scantron sheet pencilpencil calculatorcalculator nothing else (exam is closed book)nothing else (exam is closed book)

Exam will coverExam will cover

Chapter 6Chapter 6 Chapter 9, M/M/1, M/M/K, M/G/1Chapter 9, M/M/1, M/M/K, M/G/1

Chapter 6Chapter 6

DMUU—Decision Making Under DMUU—Decision Making Under Uncertainty Uncertainty

DMUR—Decision Making Under DMUR—Decision Making Under RiskRisk

UT—Utility TheoryUT—Utility Theory GT—Game TheoryGT—Game Theory

DMUU & DMUR DMUU & DMUR ComponentsComponents

A set of Future States of Nature, SA set of Future States of Nature, S• Mutually exclusiveMutually exclusive• collective exhaustivecollective exhaustive

A set of alternatives, AA set of alternatives, A• Mutually exclusiveMutually exclusive

A set of payoffs defined on A x SA set of payoffs defined on A x S• payoffs for each alternative/state pairpayoffs for each alternative/state pair

A Decision CriterionA Decision Criterion

DMUU CriteriaDMUU Criteria

PessimistPessimist• what is the payoff for minimizing losses or what is the payoff for minimizing losses or

costs???costs??? OptimistOptimist

• What is the payoff for minimizing losses or What is the payoff for minimizing losses or costs??costs??

InbetweenistInbetweenist RegrettistRegrettist Insufficient Reason Insufficient Reason

DMUR CriteriaDMUR Criteria

EREV -- expected valueEREV -- expected value Expected RegretExpected Regret ERPI -- expected return with perfect ERPI -- expected return with perfect

informationinformation EVPI -- expected value of perfect informationEVPI -- expected value of perfect information ERSI -- expected return with sample ERSI -- expected return with sample

informationinformation EVSI -- expected value of sample informationEVSI -- expected value of sample information

Relationships betweenRelationships between

EVSI, ERSI, EVPI, ERPIEVSI, ERSI, EVPI, ERPI EREV & Expected RegretEREV & Expected Regret Minimal Expected Regret and EVPIMinimal Expected Regret and EVPI Expected Regret + EREV, for any Expected Regret + EREV, for any

alternative = alternative =

EVPI = Expected Value of EVPI = Expected Value of Perfect InformationPerfect Information

= ERPI – EREV for the optimal = ERPI – EREV for the optimal alternativealternative

= minimum expected regret= minimum expected regret

EVPI and EVSIEVPI and EVSI

EVPI = ERPI - EREV for the optimal EVPI = ERPI - EREV for the optimal alternativealternative

EVSI = ERSI - EREV for the optimal EVSI = ERSI - EREV for the optimal alternativealternative

Bayesian RevisionBayesian Revision

When is it needed?When is it needed? Starts with what?Starts with what? Ends with what what?Ends with what what?

Combines prior and sample Combines prior and sample information as inputsinformation as inputs

Outputs posterior (revised) Outputs posterior (revised) informationinformation

Two ways to do Bayesian Two ways to do Bayesian RevisionRevision

TablesTables Probability treesProbability trees

Probability TreesProbability Trees

Construct backward-looking tree Construct backward-looking tree • Find joint probabilities at the end nodes by Find joint probabilities at the end nodes by

taking the product of all probabilities leading taking the product of all probabilities leading out to the end nodeout to the end node

Construct forward-looking treeConstruct forward-looking tree• Move joint probabilities to their appropriate Move joint probabilities to their appropriate

end nodesend nodes• Calculate marginal probabilities of indicator Calculate marginal probabilities of indicator

statesstates• Calculate posterior conditional probabilities Calculate posterior conditional probabilities

The revised probabilitiesThe revised probabilities

Are posterior probabilitiesAre posterior probabilities Replace the prior probabilities Replace the prior probabilities

whenwhen• A particular predictive state is known A particular predictive state is known

(like (like successsuccess or or failurefailure))

Decision TreesDecision Trees

For multi-stage decisionsFor multi-stage decisions Two node types, basically--decision Two node types, basically--decision

nodes and chance nodesnodes and chance nodes Know how to solve the decision Know how to solve the decision

treetree

Utility TheoryUtility Theory

What is it for?What is it for? What kinds of decision makers are What kinds of decision makers are

there?there?

ProcedureProcedure

The highest payoff is assigned a The highest payoff is assigned a value of ___value of ___

The lowest payoff is assigned a The lowest payoff is assigned a value ofvalue of

Intermediate payoffs are assigned Intermediate payoffs are assigned value by asking the DM an value by asking the DM an indifference questionindifference question

Most DM’sMost DM’s

Are Are risk-takingrisk-taking for small dollar for small dollar amounts and amounts and risk-averserisk-averse when the when the dollar amounts are large, as dollar amounts are large, as compared to their net worthcompared to their net worth

Most large corporationsMost large corporations

Are risk-neutral for small-to-Are risk-neutral for small-to-medium-sized risks, relative to the medium-sized risks, relative to the market capitalization of the firmmarket capitalization of the firm

In such situations payoffs work well In such situations payoffs work well in place of utilesin place of utiles

GamesGames

Where do such constructs make Where do such constructs make sense in business?sense in business?

What is a fair game?What is a fair game? For what kinds of games can pure For what kinds of games can pure

strategies be employed?strategies be employed? How are mixed strategies How are mixed strategies

obtained?obtained?

ExamplesExamples

Airlines competing on a particular Airlines competing on a particular routeroute

Retail grocers competing in a Retail grocers competing in a particular market, like Gold Beach particular market, like Gold Beach in the bookin the book

Television networks competing for Television networks competing for market share in prime timemarket share in prime time

History and RationaleHistory and Rationale

First formalized and explained by von First formalized and explained by von Neumann and Morgenstern in 1947 who Neumann and Morgenstern in 1947 who also made important contributions to also made important contributions to utility theoryutility theory

Resembles a decision theory problem in Resembles a decision theory problem in which the states of nature are managed which the states of nature are managed by a malevolent opponent who actively by a malevolent opponent who actively chooses his states or strategies so as to chooses his states or strategies so as to minimize the decision maker’s expected minimize the decision maker’s expected payoff or utility.payoff or utility.

Classification of GamesClassification of Games• Number of PlayersNumber of Players

– Two players - ChessTwo players - Chess– Multi-player - More than two competitors (Poker)Multi-player - More than two competitors (Poker)

• Total returnTotal return– Zero Sum - The amount won and amount lost by Zero Sum - The amount won and amount lost by

all competitors are equal (Poker among friends)all competitors are equal (Poker among friends)– Nonzero Sum -The amount won and the amount Nonzero Sum -The amount won and the amount

lost by all competitors are not equal (Poker In A lost by all competitors are not equal (Poker In A Casino)Casino)

• Sequence of MovesSequence of Moves– Sequential - Each player gets a play in a given Sequential - Each player gets a play in a given

sequence.sequence.– Simultaneous - All players play simultaneously.Simultaneous - All players play simultaneously.

Classification on the Classification on the basis of pure vs. mixed basis of pure vs. mixed strategiesstrategies

Games which possess a saddle point are said Games which possess a saddle point are said to have pure strategiesto have pure strategies

Example:Example:

This game has a Saddle Point--because there This game has a Saddle Point--because there is convergence upon the same cell of the is convergence upon the same cell of the tabletable

Player 2OP1 OP2 OP3 Row Min

S1 15 0 -2 -2Player 1 S2 0 -15 -1 -15

S3 1 2 0 0 <-----Col Max 15 2 0

Mixed Strategies Mixed Strategies

Most games do not possess saddle points Most games do not possess saddle points and hence cannot use pure strategiesand hence cannot use pure strategies

For such situations, it becomes necessary For such situations, it becomes necessary for both players to use mixed (probabilistic) for both players to use mixed (probabilistic) strategiesstrategies

The probabilities with which the players The probabilities with which the players execute each of their strategies can be execute each of their strategies can be determined as the solution to two linear determined as the solution to two linear programming problems--one for each programming problems--one for each playerplayer

Queuing TheoryQueuing Theory

9.2 Elements of the 9.2 Elements of the Queuing ProcessQueuing Process A queuing system consists of three basic A queuing system consists of three basic

components:components:• Arrivals: Customers arrive according to some Arrivals: Customers arrive according to some

arrival pattern.arrival pattern.

• Waiting in a queue: Arriving customers may Waiting in a queue: Arriving customers may have to wait in one or more queues for service.have to wait in one or more queues for service.

• Service: Customers receive service and leave Service: Customers receive service and leave the system.the system.

The Arrival ProcessThe Arrival Process

There are two possible types of There are two possible types of arrival processesarrival processes

• Deterministic arrival process.Deterministic arrival process.

• Random arrival process.Random arrival process. The random process is more The random process is more

common in businesses.common in businesses.

Under three conditions the arrivals can be Under three conditions the arrivals can be modeled as a Poisson processmodeled as a Poisson process• OrderlinessOrderliness : one customer, at most, will : one customer, at most, will

arrive during any time interval.arrive during any time interval.

• StationarityStationarity : for a given time frame, the : for a given time frame, the

probability of arrivals within a certain time probability of arrivals within a certain time

interval is the same for all time intervals of interval is the same for all time intervals of

equal length.equal length.

• Independence Independence : the arrival of one customer : the arrival of one customer

has no influence on the arrival of another.has no influence on the arrival of another.

The Arrival ProcessThe Arrival Process

P(X = k) =

Where = mean arrival rate per time unit.

t = the length of the interval.

e = 2.7182818 (the base of the natural logarithm).k! = k (k -1) (k -2) (k -3) … (3) (2) (1).

tke- t

k!

The Poisson Arrival The Poisson Arrival ProcessProcess

HANK’s HARDWARE – HANK’s HARDWARE – Arrival ProcessArrival Process

Customers arrive at Hank’s Hardware Customers arrive at Hank’s Hardware according to a Poisson distribution.according to a Poisson distribution.

Between 8:00 and 9:00 Between 8:00 and 9:00 A.MA.M.. an average of 6 an average of 6

customers arrive at the store.customers arrive at the store.

What is the probability that k customers will What is the probability that k customers will

arrive between 8:00 and 8:30 in the arrive between 8:00 and 8:30 in the

morning (k = 0, 1, 2,…)?morning (k = 0, 1, 2,…)?

k

Input to the Poisson Input to the Poisson distribution distribution

= 6 customers per = 6 customers per hour.hour.t t = 0.5 hour.= 0.5 hour.t t = (6)(0.5) = 3.= (6)(0.5) = 3.

t e- t

k !

0

0.049787

0

1!

1

0.1493612

2!0.224042

3

3! 0.224042

1 2 3 4 5 6 7

P(X = k )=

8

123

HANK’s HARDWARE – HANK’s HARDWARE – An illustration of the An illustration of the Poisson distribution. Poisson distribution.

HANK’s HARDWARE – HANK’s HARDWARE – Using Excel for the Using Excel for the Poisson probabilitiesPoisson probabilities

SolutionSolution• We can use the POISSON function in Excel We can use the POISSON function in Excel

to determine Poisson probabilities.to determine Poisson probabilities.• Point probability: P(X = k) = ?Point probability: P(X = k) = ?

– Use Poisson(k, Use Poisson(k, t, FALSE)t, FALSE)– Example: P(X = 0; Example: P(X = 0; t = 3) = POISSON(0, 1.5, t = 3) = POISSON(0, 1.5,

FALSEFALSE))

• Cumulative probability: P(XCumulative probability: P(Xk) = ?k) = ?– Example: P(XExample: P(X3; 3; t = 3) = Poisson(3, 1.5, t = 3) = Poisson(3, 1.5,

TRUETRUE))

HANK’s HARDWARE – HANK’s HARDWARE – Excel PoissonExcel Poisson

Factors that influence the Factors that influence the modeling of queuesmodeling of queues

– Line configurationLine configuration

– JockeyingJockeying

– BalkingBalking

The Waiting Line The Waiting Line CharacteristicsCharacteristics

– Priority

– Tandem Queues

– Homogeneity

A single service queue.A single service queue. Multiple service queue with single Multiple service queue with single

waiting line.waiting line. Multiple service queue with Multiple service queue with

multiple waiting lines.multiple waiting lines. Tandem queue (multistage service Tandem queue (multistage service

system).system).

Line ConfigurationLine Configuration

Jockeying occurs when customers Jockeying occurs when customers switch lines once they perceived switch lines once they perceived that another line is moving faster.that another line is moving faster.

Balking occurs if customers avoid Balking occurs if customers avoid joining the line when they perceive joining the line when they perceive the line to be too long.the line to be too long.

Jockeying and BalkingJockeying and Balking

These rules select the next customer These rules select the next customer for service.for service.

There are several commonly used There are several commonly used rules:rules:• First come first served (FCFS).First come first served (FCFS).• Last come first served (LCFS).Last come first served (LCFS).• Estimated service time.Estimated service time.• Random selection of customers for Random selection of customers for

service.service.

Priority RulesPriority Rules

Tandem QueuesTandem Queues

These are multi-server systems. These are multi-server systems. A customer needs to visit several A customer needs to visit several

service stations (usually in a distinct service stations (usually in a distinct order) to complete the service order) to complete the service process.process.

ExamplesExamples• Patients in an emergency room.Patients in an emergency room.• Passengers prepare for the next flight.Passengers prepare for the next flight.

A homogeneous customer population A homogeneous customer population is one in which customers require is one in which customers require essentially the same type of service.essentially the same type of service.

A non-homogeneous customer A non-homogeneous customer population is one in which customers population is one in which customers can be categorized according to: can be categorized according to: • Different arrival patternsDifferent arrival patterns• Different service treatments. Different service treatments.

HomogeneityHomogeneity

In most business situations, service In most business situations, service time varies widely among time varies widely among customers.customers.

When service time varies, it is When service time varies, it is treated as a random variable.treated as a random variable.

The exponential probability The exponential probability distribution is used sometimes to distribution is used sometimes to model customer service time. model customer service time.

The Service ProcessThe Service Process

f(t) = e-t

= the average number of customers who can be served per time period.Therefore, 1/ = the mean service time.

The probability that the service time X is less than some “t.”

P(X t) = 1 - e-t

The Exponential Service Time The Exponential Service Time DistributionDistribution

Schematic illustration of Schematic illustration of the exponential the exponential distributiondistribution

The probability that service is completed within t time units

P(X t) = 1 - e-t

X = t

HANK’s HARDWARE – HANK’s HARDWARE – Service timeService time

Hank’s estimates the average Hank’s estimates the average service time to be 1/service time to be 1/ = 4 minutes = 4 minutes per customer.per customer.

Service time follows an Service time follows an exponential distribution.exponential distribution.

What is the probability that it will What is the probability that it will take less than 3 minutes to serve take less than 3 minutes to serve the next customer?the next customer?

We can use the EXPDIST function in We can use the EXPDIST function in Excel to determine exponential Excel to determine exponential probabilities.probabilities.

Probability density: f(t) = ?Probability density: f(t) = ?• Use EXPONDIST(t, Use EXPONDIST(t, , , FALSEFALSE))

Cumulative probability: P(XCumulative probability: P(Xk) = ?k) = ?• Use EXPONDIST(t, Use EXPONDIST(t, , , TRUETRUE))

Using Excel for the Using Excel for the Exponential ProbabilitiesExponential Probabilities

The mean number of customers The mean number of customers served per minute is ¼ = ¼(60) = served per minute is ¼ = ¼(60) = 15 customers per hour. 15 customers per hour.

P(X < .05 hours) = 1 – eP(X < .05 hours) = 1 – e-(15)(.05) -(15)(.05) = ?= ? From Excel we have:From Excel we have:

• EXPONDIST(.05,15,EXPONDIST(.05,15,TRUETRUE) = .5276) = .5276

HANK’s HARDWARE – HANK’s HARDWARE – Using Excel for the Using Excel for the Exponential ProbabilitiesExponential Probabilities

3 minutes = .05 hours

HANK’s HARDWARE – HANK’s HARDWARE – Using Excel for the Using Excel for the Exponential ProbabilitiesExponential Probabilities

=EXPONDIST(B4,B3,TRUE)

Exponential Distribution for Mu = 15

0.0002.0004.0006.0008.000

10.00012.00014.00016.000

0.000 0.075 0.150 0.225 0.300 0.375

t

f(t)

=EXPONDIST(A10,$B$3,FALSE)Drag to B11:B26

• The memoryless property.The memoryless property.– No additional information about the time left for No additional information about the time left for

the completion of a service, is gained by recording the completion of a service, is gained by recording the time elapsed since the service started.the time elapsed since the service started.

– For Hank’s, the probability of completing a service For Hank’s, the probability of completing a service within the next 3 minutes is (0.52763) independent within the next 3 minutes is (0.52763) independent of how long the customer has been served already.of how long the customer has been served already.

• The Exponential and the Poisson distributions The Exponential and the Poisson distributions are related to one another.are related to one another.– If customer arrivals follow a If customer arrivals follow a Poisson distribution Poisson distribution

with mean rate with mean rate ,, their interarrival times are their interarrival times are exponentially distributed with mean time 1exponentially distributed with mean time 1//

The Exponential The Exponential Distribution -Distribution - CharacteristicsCharacteristics

9.3 Performance 9.3 Performance Measures of Queuing Measures of Queuing System System Performance can be measured by Performance can be measured by

focusing on:focusing on:• Customers in queue.Customers in queue.

• Customers in the system.Customers in the system.

Performance is measured for a Performance is measured for a system in steady state. system in steady state.

Roughly, this is a transient period…

n

Time

9.3 Performance 9.3 Performance Measures of Queuing Measures of Queuing System System

The transient The transient period occurs at period occurs at the initial time the initial time of operation.of operation.

Initial transient Initial transient behavior is not behavior is not indicative of indicative of long run long run performance.performance.

This is a steady state period………..

n

Time

9.3 Performance 9.3 Performance Measures of Queuing Measures of Queuing System System • The steady state

period follows the transient period.

• Meaningful long run performance measures can be calculated for the system when in steady state.

Roughly, this is a transient period…

kEach with

service rate of

kEach with

service rate of

…

For k serverswith service rates

…

For k serverswith service rates

For one server

For one server

In order to achieve steady state, theIn order to achieve steady state, theeffective arrival rate must be less than effective arrival rate must be less than

the sum of the effective service rates .the sum of the effective service rates .

In order to achieve steady state, theIn order to achieve steady state, theeffective arrival rate must be less than effective arrival rate must be less than

the sum of the effective service rates .the sum of the effective service rates .

9.3 Performance 9.3 Performance Measures of Queuing Measures of Queuing System System

k servers

P0 = Probability that there are no customers in the system.P0 = Probability that there are no customers in the system.

Pn = Probability that there are “n” customers in the system.Pn = Probability that there are “n” customers in the system.

L = Average number of customers in the system.L = Average number of customers in the system.

Lq = Average number of customers in the queue.Lq = Average number of customers in the queue.

W = Average time a customer spends in the system.W = Average time a customer spends in the system.

Wq = Average time a customer spends in the queue.Wq = Average time a customer spends in the queue.

Pw = Probability that an arriving customer must wait for service.

Pw = Probability that an arriving customer must wait for service.

= Utilization rate for each server (the percentage of time that each server is busy).

= Utilization rate for each server (the percentage of time that each server is busy).

Steady State Performance Steady State Performance MeasuresMeasures

Little’s Formulas represent important Little’s Formulas represent important relationships between L, Lrelationships between L, Lqq, W, and W, W, and Wqq..

These formulas apply to systems that meet These formulas apply to systems that meet the following conditions:the following conditions:• Single queue systems,Single queue systems,• Customers arrive at a finite arrival rate Customers arrive at a finite arrival rate andand• The system operates under a steady state The system operates under a steady state

condition.condition.

L =L =W LW Lqq = = W Wqq L = LL = Lqq + +

Little’s FormulasLittle’s Formulas

For the case of an infinite population

Queuing system can be classified by:Queuing system can be classified by:• Arrival process.Arrival process.• Service process.Service process.• Number of servers.Number of servers.• System size (infinite/finite waiting line).System size (infinite/finite waiting line).• Population size.Population size.

NotationNotation• M (Markovian) = Poisson arrivals or exponential M (Markovian) = Poisson arrivals or exponential

service time.service time.• D (Deterministic) = Constant arrival rate or D (Deterministic) = Constant arrival rate or

service time.service time.• G (General) = General probability for arrivals or G (General) = General probability for arrivals or

service time.service time.

Example:

M / M / 6 / 10 / 20

Example:

M / M / 6 / 10 / 20

Classification of QueuesClassification of Queues

9.4 M9.4 MMM1 Queuing 1 Queuing System - System - AssumptionsAssumptions

• Poisson arrival process.Poisson arrival process.

• Exponential service time distribution.Exponential service time distribution.

• A single server.A single server.

• Potentially infinite queue. Potentially infinite queue.

• An infinite population.An infinite population.

The probability thata customer waits in the system more than “t” is P(X>t) = e-( - )t

The probability thata customer waits in the system more than “t” is P(X>t) = e-( - )t

PP00 = 1 – ( = 1 – ())

PPnn = [1 – ( = [1 – ()])](())nn

L = L = (( – – ))

LLq q = = 2 2 [[(( – – )])]

W = 1 W = 1 (( – – ))

WWqq = = [[(( – – )])]

PPw w = =

= =

M / M /1 Queue - M / M /1 Queue - Performance MeasuresPerformance Measures