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QUEUING THEORY Body of knowledge about waiting lines Helps managers to better understand systems in manufacturing, service, and maintenance Provides competitive advantage and cost saving A QUEUE REPRESENTS ITEMS OR PEOPLE AWAITING SERVICE Applied Management Science for Decision Making, 1e © 2012 Pearson Prentice-Hall, Inc. Philip A. Vaccaro, PhD

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Queue Characteristics Average number of customers in a line Average number of customers in a line Average number of customers in a service facility Average number of customers in a service facility Probability a customer must wait Probability a customer must wait Average time a customer spends in a waiting line. Average time a customer spends in a waiting line. Average time a customer spends in a service facility Average time a customer spends in a service facility Percentage of time a service facility is busy Percentage of time a service facility is busy

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Queuing System Examples SystemCustomersServers Grocery Store Shoppers Checkout Clerks Phone System Phone Calls Switching Equipment Toll Highway VehiclesTollgate Restaurant Parties of Diners Tables & Waitstaff FactoryProductsWorkers

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The Father of Queuing Theory Danish engineer, who, in 1909 experimented with fluctuating demand in telephone traffic in Copenhagen. In 1917, he published a report addressing the delays in auto- matic telephone dialing equip- ment. At the end of World War II, his work was extended to more general problems, including waiting lines in business. AGNER K. ERLANG

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Lack of Managerial Intuition Surrounding Waiting Lines Queuing theory is not a matter of common sense. It is one of those applications where diligent, intelligent managers will arrive at drastically wrong solutions if they fail to thoroughly appreciate and understand the mathematics involved.

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Cost MinimumTotalCost Low Level Of Service High Level Of Service Optimal Service Level Cost of Waiting Time ( time x value of time ) Cost of Providing Service ( salaries + benefits ) Total Cost THE QUEUING COST TRADE-OFF

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Aspects of a Queuing Process SYSTEM ARRIVALS SYSTEM ARRIVALS THE QUEUE ITSELF THE QUEUE ITSELF THE SERVICE FACILITY THE SERVICE FACILITY

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The Calling Population The source of all system arrivals arrivals It is usually of infinite size It is usually of infinite size Theoretically, any person Theoretically, any person or object can enter the or object can enter the service facility during service facility during operating hours operating hours

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Poisson Arrival Distribution Poisson Probability Distribution for λ = 2 (estimated mean arrival rate) X ( the number of arrivals ) P ( probability )

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Poisson Arrival Distribution Poisson Probability Distribution for λ = 4 (estimated mean arrival rate) X ( the number of arrivals ) P ( probability )

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Establishing A Discrete Poisson Arrival Distribution Given any average arrival rate ( λ ) in seconds, minutes, hours, days: P ( X ) = ε λ X! ( FOR X = 0,1,2,3,4,5, etc. ) Where : P ( X ) = probability of X arrivals X = number of arrivals per time unit λ = the average arrival rate ε = ( base of the natural logarithm ) - λx

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EXAMPLE If the average arrival rate per hour is two people ( λ = 2 ), what is the probability of three ( 3 ) arrivals per hour?

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Solution Given λ = 2 : P ( 3 ) = ! 3 = [ 1 / ] x 8 (3)(2)(1) =.1353 x 8 = % P ( X ) = - λ ελ X X !

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The Remaining Probabilities GIVEN THAT λ = 2 P ( 0 arrivals ) = 14% P ( 1 arrival ) = 28% P ( 2 arrivals ) = 28% P ( 3 arrivals ) = 18% P ( 4 arrivals ) = 9% P ( 5 arrivals ) = 4% P ( 6 arrivals ) = 2% P ( 7 arrivals ) = 1% P ( 8 arrivals ) =.8% P ( 9 arrivals ) =.6% P ( => 10 ) = 0%

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Poisson Probability Table For a given value of λ, entry indicates the probability of obtaining a specified value of X X λ = 1.8 λ = 1.9 λ = 2.0 EXAMPLE

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Precise Terminology THIS DISTRIBUTION MAY OR MAY NOT BE POISSON DISTRIBUTED. DISTRIBUTED. The discrete arrival probability distribution, based on the average arrival rate ( λ ) which was computed from the actual system observations Theoretical Distribution The actual discrete arrival probability distribution that was constructed from the actual system observations Observed Distribution

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THIS CAN BE ESTABLISHED BY A GOODNESS – OF - FIT HYPOTHESIS TEST The theoretical poisson arrival probability distribution must be statistically identical to the observed arrival probability distribution If the two probability distributions are not found to be statistically identical, we are forced to study and solve the problem via simulation modeling

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Service Times PROBABILITYPROBABILITY seconds Service times normally follow a negative exponential probability distribution THE PROBABILITY A CUSTOMER WILL REQUIRE THAT SERVICE TIME

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Queue Discipline Balking, reneging, and jockeying are not permitted in the service system. permitted in the service system. Jockeying is the switching from one waiting line to another. line to another. JOCKEYING CAN BE DISCOURAGED BY PLACING BARRICADES SUCH AS MAGAZINE RACKS AND IMPULSE ITEM DISPLAYS BETWEEN WAITING LINES

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Queuing Theory Variables Lambda ( λ ) is the average arrival rate of people or items into the service system. It can be expressed in seconds, minutes, hours, or days. From the Greek small letter L.

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Queuing Theory Variables Mu ( μ ) is the average service rate of the service system. It can be expressed as the number of people or items processed per second, minute, hour, or day. From the Greek small letter M.

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Queuing Theory Variables Rho ( ρ ) is the % of time that the service facility is busy on the average. It is also known as the utilization rate. From the Greek small letter R. BUSY IS DEFINED AS AT LEAST ONE PERSON OR ITEM IN THE SYSTEM

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Queuing Theory Variables Mu ( M ) is a channel or service point in the ser- vice system. Examples are gasoline pumps, checkout coun- ters, vending machines, bank teller windows. From the Greek large letter M.

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Queuing Theory Variables Phases are the number of service points that must be negotiated by a customer or item before leaving the service system. They have no symbol. A CARWASH TAKES A VEHICLE THROUGH SEVERAL PHASES: PRE-WASH, WASH, WAX, AND DRY BEFORE IT IS ALLOWED TO LEAVE THE FACILITY.

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Queuing Theory Variables Po or ( 1 – ρ ) is the percentage of time that the service facility is idle. L is the average number of people or items in the service system both waiting to be served and currently being served. Lq is the average number of people or items in the waiting line ( queue ) only !

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Queuing Theory Variables W is the average time a customer or item spends in the service system, both waiting and receiving service. Wq is the average time a customer or item spends in the waiting line ( queue ) only. P w is the probability that a customer or item must wait to be served.

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The average number of customers or items processed by the entire service system Queuing Theory Variables Mμ is the effective service rate.* * [ NUMBER OF SERVERS ] x [ AVERAGE SERVICE RATE PER SERVER ] It can be expressed in seconds, minutes, hours, or days

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IMPORTANT CONSIDERATION The average service rate must always exceed the average arrival rate. Otherwise, the queue will grow to infinity. μ > λ THERE WOULD BE NO SOLUTION !

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Single-Channel / Single-Phase System ONE WAITING LINE or QUEUE ONE SERVICE POINT or CHANNEL ONE SERVICE POINT or CHANNEL EXIT

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Dual-Channel / Single-Phase System ONE OR TWO WAITING LINES TWO DUPLICATE SERVICE POINTS TWO DUPLICATE SERVICE POINTS EXIT No Jockeying Permitted Between Lines

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Dual-Channel / Triple-Phase System TWO IDENTICAL SERVICE CHANNELS. EACH CHANNEL HAS 3 DISTINCT SERVICE POINTS ( A-B-C ) EXIT AA BB CC ENTER Jockeying Is Permitted Between Lines !

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The Service System COMPRISED OF TWO GROUPS SUPERMARKET SHOPPERS ARE NOT IN THE SERVICE SYSTEM UNTIL THEY MOVE TO THE CHECKOUT AREA RESTAURANT PATRONS ENTER THE SERVICE SYSTEM AS SOON AS THEY ARRIVE SYSTEM AS SOON AS THEY ARRIVE Customers or items waiting to be served or processed Customers or items currently being served or processed

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Queuing Theory Limitations BJs WHOLESALE CLUB HAS FOURTEEN (14) CHECKOUTS. HOWEVER,THEY COULD BE DIVIDED INTO CONTRACTOR, EXPRESS, CASH-ONLY, AND CREDIT-CARD-ONLY SUBSYSTEMS. Formulae only accommodate eight ( 8 ) channels and / or eight ( 8 ) phases If service systems exceed the above, it may be possible to divide them into sub-systems for separate analyses.

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Stealth Queuing Systems NORMAL CHARACTERISTICS MISSING VISITING NURSES, PLUMBERS, ELECTRICIANS BROKEN MACHINES WAITING FOR A MECHANIC, OR SEATED PATIENTS IN A DENTISTS OFFICE, OR WORK-IN-PROCESS INVENTORY WAITING FOR PROCESSING. Fixed channels may be replaced by mobile servers who carry portable equipment and make housecalls. Moving waiting lines may be replaced by sitting customers or stockpiled items.

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Behavioral Considerations Customer willingness to wait depends on what is perceived as reasonable. Waiting lines that are always moving are perceived as less painful. Customer willingness to wait is higher if they know that others are also waiting their turn. Customers should be permitted to perform the services that they can easily provide for themselves. QUEUING THEORY

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Behavioral Considerations Well projected waiting times allow customers to adjust their expectations and therefore their aggravation. If customers are kept If customers are kept busy, their waiting time busy, their waiting time may not be construed as wasted time. may not be construed as wasted time. Customers should be rewarded with price discounts or gifts if they must wait beyond a certain period of time. QUEUING THEORY FILLING OUT SURVEYS AND FORMS, BEING ENTERTAINED IT SHOWS THAT THE FIRM VALUES THEIR TIME AND IS WILLING TO PAY THEM FOR IT IF THE WAIT IS TOO LONG

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Single-Channel / Single-Phase Model The Average Number of Customers in the System L = λ μ - λ

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Single-Channel / Single-Phase Model The Average Number Just Waiting in Line Lq = λ μ ( μ - λ ) 2

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Single-Channel / Single-Phase Model Average Customer Time Spent in the System W = ( μ - λ ) 1

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Single-Channel / Single-Phase Model Percentage of Time the System is Busy ρ = μ λ

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Therefore: μ = 30 λ = 20 M = 1 Single-Channel / Single-Phase Model A clerk can serve thirty customers per hour on average. Twenty customers arrive each hour on average. APPLICATION

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Single-Channel / Single-Phase Model APPLICATION The Average Number of Customers in the System L = 20 ( ) = 2

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Single-Channel / Single-Phase Model APPLICATION The Average Number Just Waiting in Line Lq = ( 20 ) 30 ( ) =

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Single-Channel / Single-Phase Model APPLICATION The Average Customer Time Spent in the System W = 1 ( ) =.10 hrs ( 6 minutes )

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Single-Channel / Single-Phase Model APPLICATION The Percentage of Time the System is Busy ρ = = 67%

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QM for Windows QUEUING APPLICATIONS Single-Channel Single-Phase Model

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WE SCROLL TO WAITING LINES

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One ( 1 ) Clerk On-Duty Twenty ( 20 ) Arrivals per Hour Thirty ( 30 ) Customers Can Be Served per Hour

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The probability of exactly seven (7) persons in the system is 2% ( = k ) The probability of seven or fewer persons in the system is 96% ( <= k ) The probability of more than seven (7) persons in the system is 4% ( > k )

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Queuing Theory Modeling with Single-Channel Single-Phase Model

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Template and Sample Data

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Multi-Channel Single-Phase Systems This system is a single waiting line serviced by more than one server. It assumes: an infinite calling population a first-come, first-served queue discipline a poisson arrival rate negative exponential service times Additional Parameters M = number of servers or channels Mμ = mean effective service rate for the facility

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Multi-Channel Single-Phase Systems The probability that the service facility is idle: Po = 1 n = M-1 n M Σ 1/n! (λ / μ) + 1 (λ / μ). Mμ n=0 M! Mμ-λ

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Multi-Channel Single-Phase Systems The average number of customers in the system: λμ ( λ / μ ) L =. Po + λ / μ (M-1)! (Mμ-λ) M 2

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Multi-Channel Single-Phase Systems The average number of customers in the queue: Lq = L – ( λ / μ ) The average time a customer spends in the system: W = L / λ The average waiting time in the queue: Wq = W – ( 1 / μ ) or Lq / λ The probability that all the systems servers are currently busy: 1 λ. Mμ. Po Pw = M! μ Mμ-λ M

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Multi-Channel Single-Phase Systems Application Example Application Example A bank has three loan officers on duty, each of whom can serve four customers per hour. Every hour, ten loan applicants arrive at the loan department and join a common queue. What are the systems operating characteristics? (10 / 4 ) + 1 (10 / 4 ) + 1 ( 10 / 4 ) + 1. ( 10 / 4 ). 3(4) 0! 1! 2! 3! 3(4) Po = =.045 = 4.5%

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Multi-Channel Single-Phase Systems Application Example Continued L = ( 10 )( 4 )( 10 / 4 ). (.045 ) + [ 10 / 4 ] = ( 3 – 1 ) ! [ 3 ( 4 ) – 10 ] 3 2 L = ( 40 )( 2.5 ) 3 2 ! [ ] 2. (.045 ) = L = ( 40 )( ) ( 2 )( 1 ) [ 2 ] 2 x = L = [ ( 625 / 8 ) x.045 ] = L =

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Multi-Channel Single-Phase Systems Application Example Continued Lq = 6 – [10/4] = 3.5 W = 6/10 =.60 hours ( 36 minutes ) Wq = 3.5 / 10 =.35 hours ( 21 minutes ) (4). (.045) =.703 = 70.3% 3! 4 3(4)-10 3 Pw = L = 6

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QM for Windows QUEUING APPLICATIONS Multi-Channel Single-Phase Model

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Queuing Theory Modeling with Multi-Channel Single-Phase Model

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Template and Sample Data

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Finite Calling Population Model Application Application A shop has fifteen (15) machines which are repaired in the same order in which they fail. The machines fail according to a poisson distribution, and the service times are expo- nentially distributed. One (1) mechanic is on-duty. A machine fails on average, every forty (40) hours. The average repair takes 3.6 hours. N = 15 machines λ = 1/40 th of a machine per hour =.0250 machine per hour μ = 1/3.6 th of a machine per hour =.2778 machine per hour

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Finite Calling Population Model N = size of the finite calling population Probability that the system is empty: N! λ (N – n)! μ 1 Σ n = 0 N n Po =

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Finite Calling Population Model Average length of the queue Lq = N – λ + μ ( 1 – Po ) λ

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Finite Calling Population Model Average number of customers (items) in the system: L = Lq + ( 1 – Po ) Average waiting time in the queue: ( N – L ) λ Lq Average time in the system: W = Wq + ( 1 / μ ) Wq =

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Finite Calling Population Model Application 15!.0250 ( 15 – n )! Σ n = 0 1 n =.0616 = 6.16% Po = Lq = 15 – ( ) = 3.63 machines.0250

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Finite Calling Population Model APPLICATION L = ( ) = 4.57 machines Wq = = hours ( 15 – 4.57 ) (.0250 ) W = ( 1 /.2778 ) = hours 3.63

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Finite Calling Population Model Performance Summary Number of Mechanics Utilization Rate ( ρ ) Average Wait Line (Lq) Average Number In System (L) Average Wait (Wq) Average Time In System (W) Probability of Waiting (Pw) M = Machines 4.57 Machines Hours Hours91.14% M = Machines Machines Hours Hours39.49% M = Machines Machines.198 Hours Hours11.47% M = Machines Machines.0287 Hours Hours.0251

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Finite Calling Population Model Cost Summary Number of Mechanics Total Hourly Wage Average Number In System (L) Total Hourly Opportunity Cost Total Cost M = 1$ Machines $13,710.00$13, M = 2$ Machines 4,944.00$4, M = 3$ Machines $3,900.00$3, M = 4$ Machines $3,741.00$3, Assume Machine Hourly Opportunity Cost of $3,000.00

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QM for Windows QUEUING APPLICATIONS Single-Channel Single-Phase Finite Calling Population Model The Mechanic Problem operational analysis and cost analysis

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We assume that a mechanic earns $24.00 per hour on average. We also assume that the opportunity cost of an out-of-service machine is $3, per hour.

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Queuing Theory Modeling with Single-Channel Single-Phase Finite Calling Population Model The Mechanic Problem ( operational analysis only )

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Template and Sample Data

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Kendall-Lee Convention Widely accepted classification system for queuing models. Indicates the pattern of arrivals, the service time distribution, and the number of channels in a model. Often encountered in queuing software. Known also as the Kendall Notation.

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Basic Three - Symbol Notation Arrival Distribution Service Time Distribution Service Channels Open 1 st 2 nd 3 rd Where: M = poisson distribution D = constant (deterministic) rate D = constant (deterministic) rate G = general distribution with mean and variance known G = general distribution with mean and variance known m / s = number of channels or servers m / s = number of channels or servers The Template

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1st Example MM1 Poisson PoissonArrivalDistributionNegativeExponential Service Time DistributionSINGLE-CHANNELSINGLE-SERVERMODEL

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2nd Example MMm Poisson PoissonArrivalDistributionNegativeExponential Service Time DistributionMULTI-CHANNELMULTI-SERVERMODEL

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2nd Example MMs PoissonArrivalDistributionNegativeExponential Service Time DistributionMULTI-CHANNELMULTI-SERVERMODEL

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3rd Example MM1 PoissonArrivalDistributionNegativeExponential Service Time DistributionSINGLE-CHANNELSINGLE-SERVERMODEL WITH FINITE POPULATION WITH FINITE CALLING POPULATION ImposedLegend

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4th Example MD3 PoissonArrivalDistributionConstant Service Time DistributionTRIPLE-CHANNELTRIPLE-SERVERMODEL

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MD3 Example 3 stamping machines in a work center 5 second fixed stamp time per inserted disc blank steel discs follow a poisson arrival pattern into the center

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5th Example MG2 PoissonArrivalDistribution The Normal Curve Service Time DistributionDUAL-CHANNELDUAL-SERVERMODEL

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MG2 Example An office copy room has 2 copiers Copy job time is normally distributed The mean copy job time is 2 minutes The standard deviation is 30 seconds Employees follow a poisson arrival pattern into the copy room μ = 2.0 min σ = 30 secs M = 2

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What We Have Seen The Convenience Store Clerk The Loan Officers The Mechanic M / M / 1 M / M / m or M / M / s M / M / 1 with finite calling population PROBLEM MODEL

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QUEUING THEORY Applied Management Science for Decision Making, 1e © 2012 Pearson Prentice-Hall, Inc. Philip A. Vaccaro, PhD

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Solved Problems Queuing Theory Computer-Based Manual Applied Management Science for Decision Making, 1e © 2010 Pearson Prentice-Hall, Inc. Philip A. Vaccaro, PhD

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The Post Office Queuing Theory Problem 1 A post office has a single line for customers to use while waiting for the next available postal clerk. There are two postal clerks who work at the same rate. The arrival rate of customers follows a poisson dis- tribution, while the service time follows an exponential distribution. The average arrival rate is one customer every three ( 3 ) minutes and the average service rate is one customer every two ( 2 ) minutes for each of the two clerks. The facility is idle 50% of the time ( Po =.50 ).

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The Post Office Queuing Theory REQUIREMENT: 1.What is the average length of the line? 2.How long does the average person spend waiting for a clerk to become available? 3.What proportion of the time are both clerks idle? Po =.50 BUT NOW SHOW ALL SUPPORTING CALCULATIONS

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The Post Office Queuing Theory Problem 1 Given: λ = 20 and μ = 30 ( per hour rates ) and M = 2 The average number of customers in the system ( L ) * : L = ( 20 x 30 ) (.67 ) ( )! (2 x 30 – 20) x (.50 ) L = 600 (.4489 ) ( 60 – 20 ) 2 X (.50 ) +.67 = * L needs to be calculated before Lq can be found. Po =.50

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The Post Office Queuing Theory Problem 1 Given: λ = 20 and μ = 30 ( per hour rates ) and M = 2 The average length of the line ( Lq ) : Lq = L – ( λ / μ ) Lq =.7541 – ( 20 / 30 ) =.0841

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The Post Office Queuing Theory REQUIREMENT: 1.What is the average length of the line? 2.How long does the average person spend waiting for a clerk to become available? 3.What proportion of the time are both clerks idle? Po =.50 BUT NOW SHOW ALL SUPPORTING CALCULATIONS

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The Post Office Queuing Theory W = L / λ = (.7541 / 20 ) =.0377 of an hour or 2.25 minutes Problem 1 The average time in the system ( W ) : The average time in the queue ( Wq ) : Wq = Lq / λ = (.0841 / 20 ) =.0042 of an hour or.25 minutes

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The Post Office Queuing Theory REQUIREMENT: 1.What is the average length of the line? 2.How long does the average person spend waiting for a clerk to become available? 3.What proportion of the time are both clerks idle? Po =.50 BUT NOW SHOW ALL SUPPORTING CALCULATIONS

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The Post Office Queuing Theory x 20 x 2(30) 0! 30 1! 30 (2)(1) 30 2(30) Po = [ ] +.50 (.4489 ) ( 1.5 ) 1 Po =.50 - THE PROBABILITY THAT THE POST OFFICE IS IDLE +

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The Post Office Revisited Queuing Theory Problem 2 A post office has a single line for customers to use while waiting for the next available postal clerk. There are three postal clerks who work at the same rate. The arrival rate of customers follows a poisson dis- tribution, while the service time follows an exponential distribution. The average arrival rate is one customer every three ( 3 ) minutes and the average service rate is one customer every two minutes for each of the three clerks. The facility is idle 51.22% of the time ( Po =.5122 ).

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The Post Office Revisited Queuing Theory REQUIREMENT: 1.What is the average length of the line? 2.How long does the average person spend waiting for a clerk to become available? 3.What proportion of the time are all three clerks idle? Po =.5122 BUT NOW SHOW ALL SUPPORTING CALCULATIONS

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The Post Office Revisted Queuing Theory Problem 2 Given: λ = 20 and μ = 30 ( per hour rates ) and M = 3 The average number of customers in the system ( L ) * : L = ( 20 x 30 ) (.67 ) ( )! (3 x 30 – 20) x (.5122 ) L = 600 ( ) 2! ( 90 – 20 ) 2 X (.5122 ) +.67 =.6794 * L needs to be calculated before Lq can be found. Po =.5122 Given

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The Post Office Revisited Queuing Theory Problem 2 Given: λ = 20 and μ = 30 ( per hour rates ) and M = 3 The average length of the line ( Lq ) : Lq = L – ( λ / μ ) Lq =.6794 – ( 20 / 30 ) =.0094

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The Post Office Revisted Queuing Theory REQUIREMENT: 1.What is the average length of the line? 2.How long does the average person spend waiting for a clerk to become available? 3.What proportion of the time are all three clerks idle? Po =.5122 BUT NOW SHOW ALL SUPPORTING CALCULATIONS

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The Post Office Revisited Queuing Theory W = L / λ = (.6794 / 20 ) =.0339 of an hour or minutes Problem 2 The average time in the system ( W ) : The average time in the queue ( Wq ) : Wq = Lq / λ = (.0094 / 20 ) = of an hour or.028 minutes

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The Post Office Revisited Queuing Theory REQUIREMENT: 1.What is the average length of the line? 2.How long does the average person spend waiting for a clerk to become available? 3.What proportion of the time are all three clerks idle? Po =.5122 BUT NOW SHOW ALL SUPPORTING CALCULATIONS

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The Post Office Revisited Queuing Theory x 20 x 3(30) 0! 30 1! 30 2! 30 (3)(2)(1) 30 3(30) Po = [ ] (.3007 ) ( ) 1 Po.5122 THE PROBABILITY THAT THE POST OFFICE IS IDLE + 2 +

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The Computer Technician Queuing Theory Problem 3 A technician monitors a group of five (5) computers that run an automated manufacturing facility. It takes an average of fifteen (15) minutes ( exponentially distributed ) to adjust a computer that developes a problem. The computers run for an average of eighty-five (85) minutes ( poisson distributed ) without requiring adjustments.

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The Computer Technician Queuing Theory Problem 3 REQUIREMENT : 1.What is the average number of computers waiting for adjustment? 2.What is the average number of computers not in working order? 3.What is the probability that the system is empty? 4.What is the average time in the queue? 5.What is the average time in the system? Note: Po =.344 or 34.4% ( no need to manually compute! )

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The Computer Technician Queuing Theory Problem 3 Given: λ = 60/85 =.706 computers ; μ = 4 computers ; N = 5 M = 1 ( technician ) ; Po =.344 Average number of computers waiting for adjustment : Lq = N - λ + μ (1 – Po ) λ Lq = 5 – (.66 ) = 5 – 4.4 =

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The Computer Technician Queuing Theory Problem 3 REQUIREMENT : 1.What is the average number of computers waiting for adjustment? 2.What is the average number of computers not in working order? 3.What is the probability that the system is empty? 4.What is the average time in the queue? 5.What is the average time in the system? Note: Po =.344 or 34.4% ( no need to manually compute! )

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The Computer Technician Queuing Theory Problem 3 The average number of computers not in working order: L = Lq + ( 1 – Po ) L = ( ) = 1.24

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The Computer Technician Queuing Theory Problem 3 REQUIREMENT : 1.What is the average number of computers waiting for adjustment? 2.What is the average number of computers not in working order? 3.What is the probability that the system is empty? 4.What is the average time in the queue? 5.What is the average time in the system? Note: Po =.344 or 34.4% ( no need to manually compute! )

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The Computer Technician Queuing Theory Problem 3 The probability that the system is empty : Po = ( as given )

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The Computer Technician Queuing Theory Problem 3 REQUIREMENT : 1.What is the average number of computers waiting for adjustment? 2.What is the average number of computers not in working order? 3.What is the probability that the system is empty? 4.What is the average time in the queue? 5.What is the average time in the system? Note: Po =.344 or 34.4% ( no need to manually compute! )

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The Computer Technician Queuing Theory Problem 3 The average time in the queue : Wq = Lq ( N – L ) λ Wq =.576 ( 5 – 1.24 )(.706 ) =.217 of an hour

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The Computer Technician Queuing Theory Problem 3 REQUIREMENT : 1.What is the average number of computers waiting for adjustment? 2.What is the average number of computers not in working order? 3.What is the probability that the system is empty? 4.What is the average time in the queue? 5.What is the average time in the system? Note: Po =.344 or 34.4% ( no need to manually compute! )

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The Computer Technician Queuing Theory Problem 3 The average time in the system : W = W q + ( 1 / μ ) W = ( 1 / 4 ) = =.467 of an hour

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Solved Problems Queuing Theory Computer-Based Manual Applied Management Science for Decision Making, 1e © 2010 Pearson Prentice-Hall, Inc. Philip A. Vaccaro, PhD

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