DECISION MODELING WITH DECISION MODELING WITH MICROSOFT EXCELMICROSOFT EXCEL

Chapter 15Chapter 15Chapter 15Chapter 15

Copyright 2001Copyright 2001Prentice Hall Publishers and Prentice Hall Publishers and

Ardith E. BakerArdith E. Baker

Part 1Part 1

QUEUINGQUEUING

INTRODUCTIONINTRODUCTION________models are everywhere. For example, ________models are everywhere. For example, airplanes “queue up” in holding patterns, waiting for airplanes “queue up” in holding patterns, waiting for a runway so they can land. Then, they line up again a runway so they can land. Then, they line up again to take off.to take off.

People line up for_________, to buy groceries, etc. People line up for_________, to buy groceries, etc. Jobs line up for machines, orders line up to be filled, Jobs line up for machines, orders line up to be filled, and so on.and so on.

A. K. Erlang (a Danish engineer) is credited with A. K. Erlang (a Danish engineer) is credited with founding queuing _________by studying telephone founding queuing _________by studying telephone switchboards in Copenhagen for the Danish switchboards in Copenhagen for the Danish Telephone Company.Telephone Company.

Many of the queuing results used today were Many of the queuing results used today were developed by____________.developed by____________.

A _______________is one in which you have a A _______________is one in which you have a sequence of times (such as people) _________at a sequence of times (such as people) _________at a facility for service, as shown below:facility for service, as shown below:

Consider St. Luke’s Hospital in Philadelphia and the Consider St. Luke’s Hospital in Philadelphia and the following three queuing models.following three queuing models.

Model 1: St. Luke’s Hematology LabModel 1: St. Luke’s Hematology Lab St. Luke’s St. Luke’s treats a large number of patients on an outpatient treats a large number of patients on an outpatient basis (i.e., not admitted to the hospital).basis (i.e., not admitted to the hospital).

_____________plus those admitted to the 600-bed _____________plus those admitted to the 600-bed hospital produce a large flow of new patients each hospital produce a large flow of new patients each day.day.

ArrivalsArrivals

0000000000 Service FacilityService Facility

Most new patients must visit the ___________ Most new patients must visit the ___________ laboratory as part of the diagnostic process. Each laboratory as part of the diagnostic process. Each such patient has to be seen by a_____________.such patient has to be seen by a_____________.

After seeing a doctor, the _________arrives at the After seeing a doctor, the _________arrives at the laboratory and checks in with a clerk. laboratory and checks in with a clerk.

Patients are assigned on a___________, first-served Patients are assigned on a___________, first-served basis to test rooms as they become available.basis to test rooms as they become available.

The technician assigned to that room performs the The technician assigned to that room performs the tests ordered by the_________. When the testing is tests ordered by the_________. When the testing is complete, the patient goes on to the next step in the complete, the patient goes on to the next step in the ________and the technician sees a new patient.________and the technician sees a new patient.

We must decide how many _______________to hire.We must decide how many _______________to hire.

WATS (Wide Area Telephone Service) is an acronym WATS (Wide Area Telephone Service) is an acronym for a special flat-rate, long distance service offered for a special flat-rate, long distance service offered by some phone companies.by some phone companies.

Model 2: Buying WATS LinesModel 2: Buying WATS Lines As part of its As part of its ____________process, St. Luke’s is designing a new ____________process, St. Luke’s is designing a new communications system which will include ______ communications system which will include ______ lines. lines.

We must decide how many WATS lines the hospital We must decide how many WATS lines the hospital should buy so that a ____________of busy signals should buy so that a ____________of busy signals will be encountered.will be encountered.

When all the phone lines allocated to WATS are in When all the phone lines allocated to WATS are in ____, the person dialing out will get a ______signal, ____, the person dialing out will get a ______signal, indicating that the call can’t be completed. indicating that the call can’t be completed.

The equipment includes __________devices such asThe equipment includes __________devices such as

Model 3: Hiring RepairpeopleModel 3: Hiring Repairpeople St. Luke’s hires St. Luke’s hires repairpeople to ____________repairpeople to ____________2020 individual pieces of individual pieces of electronic equipment. electronic equipment.

If a piece of equipment _____and all the repairpeople If a piece of equipment _____and all the repairpeople are occupied, it must wait to be repaired.are occupied, it must wait to be repaired.

We must decide how many __________to hire and We must decide how many __________to hire and balance their cost against the cost of having ______ balance their cost against the cost of having ______ equipment.equipment.

electrocardiogram machines electrocardiogram machines

small dedicated computerssmall dedicated computers

CAT scannerCAT scanner

other equipmentother equipment

All three of these models fit the general description All three of these models fit the general description of a queuing model as described below: of a queuing model as described below:

PROBLEMPROBLEM ARRIVALS ARRIVALS SERVICE FACILITYSERVICE FACILITY

11 Patients Patients Technicians Technicians22 Telephone Calls Telephone Calls Switchboard Switchboard33 Broken Equipment Repairpeople Broken Equipment Repairpeople

These models will be resolved by using a These models will be resolved by using a combination of __________and simulation models.combination of __________and simulation models.

To begin, let’s start with the _______queuing model. To begin, let’s start with the _______queuing model.

THE BASIC MODELTHE BASIC MODELConsider the Xerox machine located in the fourth-Consider the Xerox machine located in the fourth-floor secretarial service suite. Assume that users floor secretarial service suite. Assume that users _______at the machine and form a ______________. _______at the machine and form a ______________.

Each arrival in turn uses the machine to perform a Each arrival in turn uses the machine to perform a specific ______which varies from obtaining a copy specific ______which varies from obtaining a copy of a of a 11-page letter to producing -page letter to producing 100100 copies of a copies of a 2525--page report. page report.

This system is called a ________________(or single-This system is called a ________________(or single-__________) queue. __________) queue.

2.2. The number of people in the ________(waiting The number of people in the ________(waiting for service).for service).

3.3. The ____________in the system (the interval The ____________in the system (the interval between when an individual enters the system between when an individual enters the system and when he or she leaves the system).and when he or she leaves the system).

4.4. The waiting time in the queue (the time The waiting time in the queue (the time between _________the system and the between _________the system and the beginning of service).beginning of service).

Questions about this or any other queuing system Questions about this or any other queuing system center on four _____________:center on four _____________:

1.1. The number of people in the _________(those The number of people in the _________(those being served and waiting in line).being served and waiting in line).

ASSUMPTIONS OF THE BASIC MODELASSUMPTIONS OF THE BASIC MODEL

1.1. ________ ________Process.Process. Each arrival will be called a Each arrival will be called a “job.” The __________“job.” The __________timetime (the time between (the time between arrivals) is not known. arrivals) is not known.

Therefore, the ___________Therefore, the ___________probabilityprobability distributiondistribution (or (or negative exponential negative exponential distributiondistribution) will be used to describe the ) will be used to describe the interarrival times for the basic model.interarrival times for the basic model.

The exponential distribution is completely The exponential distribution is completely specified by one______________, specified by one______________, , the mean , the mean arrival rate (i.e., how many jobs arrive on the arrival rate (i.e., how many jobs arrive on the average during a specified _______period).average during a specified _______period).

________interarrival time is the average time ________interarrival time is the average time between two________. Thus, for the between two________. Thus, for the exponential distributionexponential distribution

Avg. time between jobs = mean interarrival time = Avg. time between jobs = mean interarrival time = 11

Thus, if Thus, if = 0.05 = 0.05

mean interarrival time = = = 20mean interarrival time = = = 2011

11 0.050.05

2.2. _________ _________Process.Process. In the basic model, the In the basic model, the time that it takes to ____________a job (the time that it takes to ____________a job (the service timeservice time) is also treated with the ) is also treated with the _____________distribution. _____________distribution.

The _________for this exponential distribution The _________for this exponential distribution is called is called (the (the mean service ratemean service rate in jobs per in jobs per minute).minute).

TT is the number of ____that would be served is the number of ____that would be served (on the________) during a period of (on the________) during a period of TT minutes minutes if the machine were ______during that time.if the machine were ______during that time.

The The meanmean or or averageaverage, _____________(the , _____________(the average time to complete a job) isaverage time to complete a job) is

Avg. service time =Avg. service time = 11

Thus, if Thus, if = 0.10 = 0.10

mean service time = = = 10mean service time = = = 1011

11 0.100.10

3.3. _______ _______Size.Size. There is no limit on the number There is no limit on the number of jobs that can ______in the queue (an of jobs that can ______in the queue (an _________queue length)._________queue length).

4.4. QueueQueue______________________.. Jobs are served on a Jobs are served on a first-come, _________basis (i.e., in the same first-come, _________basis (i.e., in the same order as they arrive at the queue).order as they arrive at the queue).

5.5. TimeTime__________________.. The system operates as The system operates as described _______________over an infinite described _______________over an infinite horizon.horizon.

6.6. SourceSource______________________.. There is an infinite There is an infinite population available to__________.population available to__________.

CHARACTERISTICS OF THE BASIC MODELCHARACTERISTICS OF THE BASIC MODELThe values of the ___________The values of the ___________ and and (together with (together with the____________) are all that is needed to calculate the____________) are all that is needed to calculate several important ________________________of the several important ________________________of the basic model.basic model.

NOTE: these formulas hold only if NOTE: these formulas hold only if < < . .

CHARACTERISTICCHARACTERISTIC SYMBOLSYMBOL FORMULA FORMULA

UtilizationUtilization -- -- //

Exp. No. in SystemExp. No. in System LL /(/(– – ))

Exp. No. in QueueExp. No. in Queue LLqq 22//((– – ))

Exp. Waiting TimeExp. Waiting Time WW 11/(/(– – ))

Exp. Time in QueueExp. Time in Queue WWqq //((– – ))

Prob. System is EmptyProb. System is Empty PP00 1 1 – (– (//))

Consider a specific example where Consider a specific example where = 0.25= 0.25 and and = 0.10= 0.10. Thus, on the average, a job ________every . Thus, on the average, a job ________every 1/1/ = 1 / 0.25 = 4 = 1 / 0.25 = 4 minutes. minutes.

Similarly, the time it takes to complete a job, on Similarly, the time it takes to complete a job, on __________, is __________, is 1/1/ = 1 / 0.10 = 10 = 1 / 0.10 = 10 minutes. minutes.

In this case, since______, the service operation will In this case, since______, the service operation will get further ________(the queue will grow longer) as get further ________(the queue will grow longer) as time goes by.time goes by.

Now, return to the Xerox model, in which Now, return to the Xerox model, in which << . . Spreadsheets are ideal for crunching the numerical Spreadsheets are ideal for crunching the numerical results from such formulas. results from such formulas.

The following Excel spreadsheet was originally The following Excel spreadsheet was originally developed by Professor David Ashley. It already has developed by Professor David Ashley. It already has the formulas entered. Here is the introductory page.the formulas entered. Here is the introductory page.

Plugging the __________values from the Xerox Plugging the __________values from the Xerox model, model, = 0.05= 0.05 and and = 0.10= 0.10 into the appropriate into the appropriate cells of the _____worksheet yields the following cells of the _____worksheet yields the following results:results:

Steady-State ResultSteady-State Result Let’s interpret the results. Let’s interpret the results. LL is is the expected _________of people in the system after the expected _________of people in the system after the queue has reached the queue has reached steady-statesteady-state..

______________________________means that the means that the probability that probability that

you will observe a you will observe a certain number of certain number of

people in the people in the system does not system does not

depend on the depend on the _____at which you _____at which you

count them.count them.

Thus, in a steady-state, Thus, in a steady-state, 1.1. The system is ________with probability The system is ________with probability ½½

2.2. On average, there is _________in the queue On average, there is _________in the queue

3.3. On average, an ________must wait On average, an ________must wait 1010 min. min. before starting to use the machine. before starting to use the machine.

4.4. On average, an arrival will spend On average, an arrival will spend 2020 minutes minutes in the_________.in the_________.

Using the ResultsUsing the Results These results hold for the ______ These results hold for the ______ model and the particular values for the ___________ model and the particular values for the ___________ (( = 0.05= 0.05 and and = 0.10= 0.10 ). ).Suppose, for example, _______________makes the Suppose, for example, _______________makes the following calculations:following calculations:

Since Since = 0.05= 0.05, on the average , on the average 55//100100 of a job arrives of a job arrives

each minute.each minute.During each During each 88-hour day there are -hour day there are 8 x 60 = 480 8 x 60 = 480 ____________________. Thus, during each day there is on the . Thus, during each day there is on the average a total of average a total of (0.05)(480) = 24(0.05)(480) = 24 arrivals. arrivals.

(24 arrivals/day)(20 minutes/arrival) = 480 minutes(24 arrivals/day)(20 minutes/arrival) = 480 minutesoror 8 hours 8 hours

We know that on the average each person spends We know that on the average each person spends 220 0 minutes in the _________(minutes in the _________(WW = 20 = 20). Thus, the total ). Thus, the total time spent at the _____________is:time spent at the _____________is:

If management feels that If management feels that 88 hours is too ________to hours is too ________to spend at the facility, then the following steps might spend at the facility, then the following steps might be taken:be taken:

1.1. A new machine might be _____________with a A new machine might be _____________with a smaller mean service time.smaller mean service time.

2.2. Another machine might be purchased and Another machine might be purchased and both machines used to satisfy the__________. both machines used to satisfy the__________. This would change the system to a _________ This would change the system to a _________ queue.queue.

3.3. Some __________might be sent to a different Some __________might be sent to a different and less busy copying facility.and less busy copying facility.

This would change the __________process.This would change the __________process.

In any case, management must _________the cost of In any case, management must _________the cost of providing service against the cost of waiting.providing service against the cost of waiting.

A TAXONOMY OF A TAXONOMY OF QUEUING MODELSQUEUING MODELS

There are many possible ______________models. There are many possible ______________models. Therefore, to facilitate communication among those Therefore, to facilitate communication among those working on queuing models, D. G. Kendall proposed working on queuing models, D. G. Kendall proposed a _________based on the following notation:a _________based on the following notation:

A/B/sA/B/s

A A = arrival distribution= arrival distributionB B = service distribution= service distributions s = number of servers= number of servers

wherewhere

Different _________are used to designate certain Different _________are used to designate certain _______________. Placed in the _______________. Placed in the AA or the or the BB position, position, they indicate the arrival or the service distribution, they indicate the arrival or the service distribution, respectively.respectively.

M M == ______________ ______________distributiondistribution

D D = deterministic____________= deterministic____________

G G = any (a general) distribution of _______times= any (a general) distribution of _______times

GIGI = any (a general) distribution of ______times = any (a general) distribution of ______times

The following conventions are in general use:The following conventions are in general use:

For example, the Xerox model is an For example, the Xerox model is an M/M/M/M/11 model model (i.e., a _____________queue with exponential inter-(i.e., a _____________queue with exponential inter-arrival and __________times).arrival and __________times).

LITTLE’S FLOW EQUATION LITTLE’S FLOW EQUATION AND RELATED RESULTSAND RELATED RESULTS

It can be proven that in a ________________queuing It can be proven that in a ________________queuing processprocess

LL = = WW

In the Xerox model, In the Xerox model, LL = 0.05 x 20 = 1.0 = 0.05 x 20 = 1.0

Consider the diagram below:Consider the diagram below:

Scene 1:Scene 1:

Our Hero ArrivesOur Hero Arrives

In scene 1 our hero arrives and joins the queue.In scene 1 our hero arrives and joins the queue.

This equation is often called ________flow equation.This equation is often called ________flow equation.

Arrived during the time our hero Arrived during the time our hero waited and was servedwaited and was served

Scene 2:Scene 2:

HeHecompletescompletes

serviceservice

In scene 2 he has just completed service.In scene 2 he has just completed service.

Assume the _______is in steady-state. Since in this Assume the _______is in steady-state. Since in this case the average number of people in the system is case the average number of people in the system is _______________of time, let’s measure this quantity _______________of time, let’s measure this quantity when our hero completes being served.when our hero completes being served.

At this time, the number of people in the ________is At this time, the number of people in the ________is precisely the total number who ________after he did.precisely the total number who ________after he did.

Therefore, if Therefore, if WW is his ____________time and people is his ____________time and people arrive at a rate of arrive at a rate of , we would expect , we would expect LL to equal____. to equal____.

Note that Little’s flow equation applies to any Note that Little’s flow equation applies to any ___________queuing process and is thus applicable ___________queuing process and is thus applicable to a wide variety of models.to a wide variety of models.

The proof used to establish Little’s flow equation The proof used to establish Little’s flow equation also shows that also shows that

LLqq = = WWqq

In the Xerox model, In the Xerox model, LLqq = 0.05 x 10 = 0.5 = 0.05 x 10 = 0.5

It is essential that It is essential that represent the _____at which represent the _____at which arrivals arrivals join join the queue.the queue.

Consider, for example, a _______with an upper limit Consider, for example, a _______with an upper limit on the number of items that can wait in the queue on the number of items that can wait in the queue (called a______________).(called a______________).

In such a system, a person can call, find the system In such a system, a person can call, find the system ____(receive a busy signal) and be sent away (_____ ____(receive a busy signal) and be sent away (_____ up). up).

For example, a modern phone system holds only a For example, a modern phone system holds only a certain number of calls (say, certain number of calls (say, 1010) in a queue until they ) in a queue until they are answered by the next ____________service are answered by the next ____________service representative.representative.

This person does not join the queue. This is called a This person does not join the queue. This is called a _________._________.

Similarly, a customer may tire of waiting in line (or Similarly, a customer may tire of waiting in line (or being on hold) and ________without being served.being on hold) and ________without being served.

Here, the person joined the queue, became impatient Here, the person joined the queue, became impatient and left without completing the transaction. This is and left without completing the transaction. This is called______________called______________..

Another important general result depends on the Another important general result depends on the _______________that _______________that

expected waiting time = expected waiting time = expected waiting time in queue + expected service timeexpected waiting time in queue + expected service time

For the basic model, we found thatFor the basic model, we found that

Expected service time =Expected service time = 11

Putting the general result in __________yieldsPutting the general result in __________yields

WW = = WWqq + +11

For the Xerox model:For the Xerox model:

WW = 10 + = 20 = 10 + = 20 11 0.100.10

This general result holds for any queuing model in This general result holds for any queuing model in which a ________________occurs.which a ________________occurs.

Now we can easily compute the four _________ Now we can easily compute the four _________ characteristics characteristics LL, , LLqq, , WW, , WWqq::

First calculate First calculate LL: : --

LL = =

From __________flow equation we know that From __________flow equation we know that

LL = = WW

Thus, after computing Thus, after computing LL, we can now compute , we can now compute WW: :

WW = = LL//Knowing Knowing WW, we can now compute , we can now compute WWqq: :

WWqq = = WW - -11

LLqq = = WWqq

And knowing And knowing WWqq, we can now compute , we can now compute LLqq: :

THE THE MM//GG/1 QUEUE/1 QUEUEThe ________________distribution may not fit the The ________________distribution may not fit the service process very well.service process very well.

Fortunately, there is a generalization of the ______ Fortunately, there is a generalization of the ______ model that permits the _____________of the service model that permits the _____________of the service time to be____________.time to be____________.

It is not necessary to know the __________ It is not necessary to know the __________ distribution, only the mean service time, distribution, only the mean service time, 1/1/, and , and its___________, its___________, 22..

The operating characteristics for this model are:The operating characteristics for this model are:

CHARACTERISTICCHARACTERISTIC SYMBOLSYMBOL FORMULA FORMULA

UtilizationUtilization -- -- //

Exp. No. in SystemExp. No. in System LL LLqq + + //

Exp. No. in QueueExp. No. in Queue LLqq 2 2 2 2 + + ((//))22

2(1 – 2(1 – //))

Exp. Waiting TimeExp. Waiting Time WW W Wqq + 1 + 1//

Exp. Time in QueueExp. Time in Queue WWqq LLqq//

Prob. System is EmptyProb. System is Empty PP00 1 1 – (– (//))As As 22 increases, increases, LL, , LLqq, , WW, , WWqq all increase. This all increase. This

means that the ____________of a server may be as means that the ____________of a server may be as important as the speed of the server.important as the speed of the server.

For example, suppose you must hire a secretary and For example, suppose you must hire a secretary and you have to select one of two candidates.you have to select one of two candidates.

Secretary 1 is very____________, typing any Secretary 1 is very____________, typing any document in exactly document in exactly 1515 minutes. minutes.

Secretary 2 is somewhat faster, with an average of Secretary 2 is somewhat faster, with an average of 1414 minutes per___________, but with times varying minutes per___________, but with times varying according to the exponential distribution.according to the exponential distribution.

With the “With the “MG1MG1” and “” and “MMsMMs” worksheets, we can ” worksheets, we can easily determine which secretary will give shorter easily determine which secretary will give shorter average _____________times on documents.average _____________times on documents.

Since Secretary 1 types every document in exactly Since Secretary 1 types every document in exactly 1515 minutes, minutes, 22 = 0= 0. In addition, . In addition, = 3= 3 per hour (or per hour (or 0.050.05 per minute) and per minute) and = = 11//1515 per minute. per minute.

For Secretary 2, enter the ______________For Secretary 2, enter the ______________ = 14= 14, , = 0.05= 0.05, and , and = = 11//1414 per minute. per minute.

Even though Secretary 2 is “________,” her average Even though Secretary 2 is “________,” her average turnaround times are longer because of the high turnaround times are longer because of the high ____________of her service times.____________of her service times.

MODEL 1: AN MODEL 1: AN MM//MM//SS QUEUE QUEUE(HEMATOLOGY LAB)(HEMATOLOGY LAB)

Previously, the stated goal was to attack the three Previously, the stated goal was to attack the three St. Luke’s Hospital models with __________models.St. Luke’s Hospital models with __________models.

Consider the blood-testing model (Model 1):Consider the blood-testing model (Model 1):

000000

Server 1

Server 2

Server 3

Each patient joins a ________ Each patient joins a ________ queue and, on arriving at the queue and, on arriving at the head of the line, enters the first head of the line, enters the first ___________examining room.___________examining room.

This type of system must not be confused with a This type of system must not be confused with a system in which a queue forms _________of each system in which a queue forms _________of each server, as in a typical grocery store.server, as in a typical grocery store.

Now, assumeNow, assume

This implies that a new patient arrives every This implies that a new patient arrives every 55 minutes on average sinceminutes on average since

Mean interarrival = 1/Mean interarrival = 1/ = 1/0.20 = 5 = 1/0.20 = 5

The __________time is given by an exponential The __________time is given by an exponential distribution with ____________distribution with ____________ = 0.20 = 0.20 per min. per min.

Each server is______________Each server is______________

This implies that the _______service time is This implies that the _______service time is 88 minutes sinceminutes since

Mean service time for an individual server = Mean service time for an individual server = 1/1/ = 1/0.125 = 8 = 1/0.125 = 8

Each service time is given by an __________ Each service time is given by an __________ distribution with parameter distribution with parameter = 0.125 = 0.125 per min. per min.

Note that if there were only one server, the queue Note that if there were only one server, the queue would grow without _________because would grow without _________because > > ((0.20 >0.20 > 0.1250.125). ).

However, for a __________queue, a steady-state will However, for a __________queue, a steady-state will exist as long as exist as long as < < ss, where , where ss is the number of is the number of ___________. ___________.

For example, if there are 2 servers, a steady-state For example, if there are 2 servers, a steady-state will be __________because will be __________because 0.20 < (2*0.125)0.20 < (2*0.125)..

The Key EquationsThe Key Equations We want to find the values We want to find the values LL, , LLqq,,WW, and , and WWqq. However, since this is a . However, since this is a

__________ queue, we must use different_________.__________ queue, we must use different_________.

For this model, the probability that the system is For this model, the probability that the system is _________is:_________is:

PP0 0 = = 11

((//))nn nn!!

((//))ss ss!!

111 - (1 - (//ss))

++nn=0=0

ss-1-1

LLqq = = PP00

((//))ss+1+1

((ss-1)!(-1)!(ss- - //))22

The expected _________of people in the queue is:The expected _________of people in the queue is:

Assume now that we have decided to hire two Assume now that we have decided to hire two technicians. We can input the parameters technicians. We can input the parameters ss = 2= 2, , = 0.20= 0.20, and , and = 0.125 = 0.125 into the into the MMsMMs worksheet to worksheet to obtain the results.obtain the results.

Here are the results when a third technician is added Here are the results when a third technician is added ((ss = 3= 3). Note that the expected waiting time has been ). Note that the expected waiting time has been reduced from reduced from 22.2222.22 minutes to minutes to 9.569.56 minutes. minutes.

When a fourth technician is added (When a fourth technician is added (ss = 4= 4), the ), the expected waiting time has been reduced slightly expected waiting time has been reduced slightly more.more.

Adding additional servers may _________the waiting Adding additional servers may _________the waiting time, however, you have to consider not only how time, however, you have to consider not only how much extra you will ____for the additional servers much extra you will ____for the additional servers but also if those additional servers will be busy most but also if those additional servers will be busy most of the time.of the time.

The previous results show that as more servers are The previous results show that as more servers are added, the percentage of ________for the added, the percentage of ________for the technicians increases, which could lead to ________ technicians increases, which could lead to ________ and sloppy work.and sloppy work.

ECONOMIC ANALYSIS OFECONOMIC ANALYSIS OFQUEUING SYSTEMSQUEUING SYSTEMS

The cost of hiring ____________technicians is fairly The cost of hiring ____________technicians is fairly clear. Now, let’s determine the cost of___________. clear. Now, let’s determine the cost of___________.

The cost to the patient is __________to the decision, The cost to the patient is __________to the decision, except as it affects the patient’s willingness to use except as it affects the patient’s willingness to use the hospital.the hospital.

Besides the possible effect on demand, the Besides the possible effect on demand, the hematology lab could _____the hospital money if it hematology lab could _____the hospital money if it reduced the output of the hospital. reduced the output of the hospital.

Cost ParametersCost Parameters If you are willing and able to If you are willing and able to _________certain costs, you can build expected cost _________certain costs, you can build expected cost models of _________systems.models of _________systems.

Consider, for example, the hematology lab model (in Consider, for example, the hematology lab model (in general terms any _________queue with exponential general terms any _________queue with exponential ___________and service times), and suppose the ___________and service times), and suppose the manager is willing to specify two costs:manager is willing to specify two costs:

CCss = cost per hour of having a server available = cost per hour of having a server available

CCWW = cost per hour of having a person wait in the = cost per hour of having a person wait in the

system (a very “fuzzy” or qualitative cost)system (a very “fuzzy” or qualitative cost)

Let’s start by calculating the ___________of hiring Let’s start by calculating the ___________of hiring 22 servers for an servers for an 88-hour day. -hour day.

There are two components:There are two components:

_________cost =_________cost = ((CCss)(2)(8))(2)(8)

where where CCss is the cost per hour for one server is the cost per hour for one server

22 is the number of servers is the number of servers

88 is the number of hours each server works is the number of hours each server works

_________cost =_________cost = ((CCWW)()(LL22)(8))(8)

where where LL22 is the number of people in the queue when is the number of people in the queue when

there are two servers. there are two servers.

To calculate the ___________of using To calculate the ___________of using 44 servers for a servers for a 66-hour day, take-hour day, take

((CCss)(4)(6) + ()(4)(6) + (CCWW)()(LL44)(6))(6)oror

[([(CCss)(4) + ()(4) + (CCWW)()(LL44)]6)]6

The Total Cost per HourThe Total Cost per Hour We now define We now define

TC(TC(ss) = total cost per hour of using ) = total cost per hour of using ss servers servers= (= (CCss)()(ss) + () + (CCWW)()(LLss))

The goal is to find the value of The goal is to find the value of ss that will _________ that will _________ the sum of these two costs.the sum of these two costs.

Note that as Note that as ss, the number of_______, increases, the , the number of_______, increases, the waiting cost will _________and the server cost will waiting cost will _________and the server cost will increase.increase.

Unfortunately, it is not possible to derive a formula Unfortunately, it is not possible to derive a formula that gives the ___________value of that gives the ___________value of ss..

However, consider the following worksheet…However, consider the following worksheet…

In this worksheet, specify In this worksheet, specify CCss = $50/server/hour= $50/server/hour and and

CCww = $100/customer/hour = $100/customer/hour. .

The results show that The results show that 33 servers minimizes the Total servers minimizes the Total Cost.Cost.

an 8- an 8- hour hour shift.shift.

Compare the cost overCompare the cost over

Next, create a data table to determine the sensitivity Next, create a data table to determine the sensitivity of this decision to the “fuzzy” cost, of this decision to the “fuzzy” cost, CCWW (($0$0 to to $180$180).).

Highlight the Highlight the block, click on block, click on

Data – TableData – Table..

Enter Enter B2B2 as as the the Column Column

InputInput and click and click OK.OK.

Excel automatically fills in the table as shown below.Excel automatically fills in the table as shown below.

These results can be ______to look for patterns and These results can be ______to look for patterns and ________. ________.

You can see that You can see that 22 servers is __________for servers is __________for CCWW = 0 = 0, ,

while while 33 servers is optimal from servers is optimal from CCWW = $20 = $20 up toup to $180 $180. .

For values of For values of CCWW >> $200 $200, , 44 servers is optimal. servers is optimal.

End of Part 1End of Part 1Please continue to Part 2Please continue to Part 2