irwin/mcgraw-hill chapter 11 queuing models 11.1

Irwin/McGraw-Hill

Chapter 11

Queuing Models

11.1

11-2

A Basic Queueing System

CustomersQueue

Served Customers

Queueing System

Service facility

SSSS

CCCC

C C C C C C C

Served Customers

11-3

Where is There Waiting?

• Service Facility– Fast-food restaurants

– Post office

– Grocery store

– Bank

• Disneyland

• Highway traffic

• Manufacturing

• Equipment awaiting repair

• Phone or computer network

• Product orders

11-4

Examples of Commercial Service SystemsThat Are Queueing Systems

Type of System Customers Server(s)

Barber shop People Barber

Bank teller services People Teller

ATM machine service People ATM machine

Checkout at a store People Checkout clerk

Plumbing services Clogged pipes Plumber

Ticket window at a movie theater People Cashier

Check-in counter at an airport People Airline agent

Brokerage service People Stock broker

Gas station Cars Pump

Call center for ordering goods People Telephone agent

Call center for technical assistance People Technical representative

Travel agency People Travel agent

Automobile repair shop Car owners Mechanic

Vending services People Vending machine

Dental services People Dentist

Roofing Services Roofs Roofer

11-5

Examples of Internal Service SystemsThat Are Queueing Systems


Secretarial services Employees Secretary

Copying services Employees Copy machine

Computer programming services Employees Programmer

Mainframe computer Employees Computer

First-aid center Employees Nurse

Faxing services Employees Fax machine

Materials-handling system Loads Materials-handling unit

Maintenance system Machines Repair crew

Inspection station Items Inspector

Production system Jobs Machine

Semiautomatic machines Machines Operator

Tool crib Machine operators Clerk

11-6

Examples of Transportation Service SystemsThat Are Queueing Systems


Highway tollbooth Cars Cashier

Truck loading dock Trucks Loading crew

Port unloading area Ships Unloading crew

Airplanes waiting to take off Airplanes Runway

Airplanes waiting to land Airplanes Runway

Airline service People Airplane

Taxicab service People Taxicab

Elevator service People Elevator

Fire department Fires Fire truck

Parking lot Cars Parking space

Ambulance service People Ambulance

11-7

System Characteristics

• Number of servers

• Arrival and service pattern– rate of arrivals and service

– distribution of arrivals and service

• Maximum size of the queue

• Queue disciplince– FCFS?

– Priority system?

• Population size– Infinite or finite?

11-8

Measures of System Performance

• Average number of customers waiting– in the system

– in the queue

• Average time customers wait– in the system

– in the queue

• Which measure is the most important?

11-9

Population Source

• Most queueing models assume an infinite population source.

• If the number of potential customers is small, a finite source model can be used.

– Number in system affects arrival rate (fewer potential arrivals when more in system)

– Okay to assume infinite if N > 20.

. . .

Customer Source System

. . .

Customer Source System

11-10

Maximum Size of Queue

• Most queueing models assume an infinite queue length is possible.

• If the queue length is limited, a finite queue model can be used.

. . .

11-11

The Queue

• The number of customers in the queue (or queue size) is the number of customers waiting for service to begin.

• The number of customers in the system is the number in the queue plus the number currently being served.

• The queue capacity is the maximum number of customers that can be held in the queue.

• An infinite queue is one in which, for all practical purposes, an unlimited number of customers can be held there.

• When the capacity is small enough that it needs to be taken into account, then the queue is called a finite queue.

• The queue discipline refers to the order in which members of the queue are selected to begin service.

– The most common is first-come, first-served (FCFS).– Other possibilities include random selection, some priority procedure, or even last-

come, first-served.

11-12

Number of Servers

• Single Server

• Multiple Servers

. . .

Customers ServiceCenter

. . .

Customers

ServiceCenters

. . .

. . .

. . .

Customers ServiceCenters

11-13

Arrivals

• The time between consecutive arrivals to a queueing system are called the interarrival times.

• The expected number of arrivals per unit time is referred to as the mean arrival rate.

• The symbol used for the mean arrival rate is

= Mean arrival rate for customers coming to the queueing system

where is the Greek letter lambda.

• The mean of the probability distribution of interarrival times is

1 / = Expected interarrival time

• Most queueing models assume that the form of the probability distribution of interarrival times is an exponential distribution.

11-14

Arrival Pattern

• A Poisson distribution is usually assumed.

• A good approximation of random arrivals.

• Lack-of-memory property: Probability of an arrival in the next instant is constant, regardless of the past.

0 1 2 3 4 5 6 7 8 9 10 11 12 13

Customers per time unit

RelativeFrequency

.02

.04

.06

.08

.10

.12

.14

.16

.18

11-15

Service

• When a customer enters service, the elapsed time from the beginning to the end of the service is referred to as the service time.

• Basic queueing models assume that the service time has a particular probability distribution.

• The symbol used for the mean of the service time distribution is

1 / = Expected service time

where is the Greek letter mu.

• The interpretation of itself is the mean service rate.

= Expected service completions per unit time for a single busy server

11-16

Some Service-Time Distributions

• Exponential Distribution– The most popular choice.

– Much easier to analyze than any other.

– Although it provides a good fit for interarrival times, this is much less true for service times.

– Provides a better fit when the service provided is random than if it involves a fixed set of tasks.

– Standard deviation: = Mean

• Constant Service Times– A better fit for systems that involve a fixed set of tasks.

– Standard deviation: = 0.

11-17

Service Pattern

• Either an exponential distribution is assumed,– Implies that the service is usually short, but occasionally long

– If service time is exponential then service rate is Poisson

– Lack-of-memory property: The probability that a service ends in the next instant is constant (regardless of how long its already gone).

– Decent approximation if the jobs to be done are random.

– Not a good approximation if the jobs to be done are always the same.

Service Time

RelativeFrequency (%)

• Or any distribution– Only single-server model is easily solved.

11-18

Properties of the Exponential Distribution

• There is a high likelihood of small interarrival times, but a small chance of a very large interarrival time. This is characteristic of interarrival times in practice.

• For most queueing systems, the servers have no control over when customers will arrive. Customers generally arrive randomly.

• Having random arrivals means that interarrival times are completely unpredictable, in the sense that the chance of an arrival in the next minute is always just the same.

• The only probability distribution with this property of random arrivals is the exponential distribution.

• The fact that the probability of an arrival in the next minute is completely uninfluenced by when the last arrival occurred is called the lack-of-memory property.

11-19

Notation

• Parameters:

= customer arrival rate = service rate (1/ = average service time)s = number of servers

• Performance Measures

Lq = average number of customers in the queueL = average number of customers in the systemWq = average waiting time in the queueW = average waiting time (including service)Pn = probability of having n customers in the system = system utilization

11-20

Application of Queueing Models

• We can use the results from queueing models to make the following types of decisions:

– How many servers to employ.

– How large should the waiting space be.

– Whether to use a single fast server or a number of slower servers.

– Whether to have a general purpose server or faster specific servers.

11-21

Herr Cutter’s Barber Shop

• Herr Cutter is a German barber who runs a one-man barber shop.

• Herr Cutter opens his shop at 8:00 A.M.

• The table shows his queueing system in action over a typical morning.

CustomerTime ofArrival

HaicutBegins

Durationof Haircut

HaircutEnds

1 8:03 8:03 17 minutes 8:20

2 8:15 8:20 21 minutes 8:41

3 8:25 8:41 19 minutes 9:00

4 8:30 9:00 15 minutes 9:15

5 9:05 9:15 20 minutes 9:35

6 9:43 — — —

11-22

Evolution of the Number of Customers

20 40 60 80

1

2

3

4

Number of Customers in the System

0

Time (in minutes)100

11-23

Labels for Queueing Models

To identify which probability distribution is being assumed for service times (and for interarrival times), a queueing model conventionally is labeled as follows:

Distribution of service times

— / — / — Number of Servers

Distribution of interarrival times

The symbols used for the possible distributions areM = Exponential distribution (Markovian)D = Degenerate distribution (constant times)GI = General independent interarrival-time distribution (any distribution)G = General service-time distribution (any arbitrary distribution)

11-24

Summary of Usual Model Assumptions

1. Interarrival times are independent and identically distributed according to a specified probability distribution.

2. All arriving customers enter the queueing system and remain there until service has been completed.

3. The queueing system has a single infinite queue, so that the queue will hold an unlimited number of customers (for all practical purposes).

4. The queue discipline is first-come, first-served.

5. The queueing system has a specified number of servers, where each server is capable of serving any of the customers.

6. Each customer is served individually by any one of the servers.

7. Service times are independent and identically distributed according to a specified probability distribution.

11-25

Choosing a Measure of Performance

• Managers who oversee queueing systems are mainly concerned with two measures of performance:

– How many customers typically are waiting in the queueing system?

– How long do these customers typically have to wait?

• When customers are internal to the organization, the first measure tends to be more important.

– Having such customers wait causes lost productivity.

• Commercial service systems tend to place greater importance on the second measure.

– Outside customers are typically more concerned with how long they have to wait than with how many customers are there.

11-26

Defining the Measures of Performance

L = Expected number of customers in the system, including those being served (the symbol L comes from Line Length).

Lq = Expected number of customers in the queue, which excludes customers being served.

W = Expected waiting time in the system (including service time) for an individual customer (the symbol W comes from Waiting time).

Wq = Expected waiting time in the queue (excludes service time) for an individual customer.

These definitions assume that the queueing system is in a steady-state condition.

11-27

Relationship between L, W, Lq, and Wq

• Since 1/ is the expected service time

W = Wq + 1/

• Little’s formula states that

L = W

and

Lq = Wq

• Combining the above relationships leads to

L = Lq +

11-28

The Dupit Corp. Problem

• The Dupit Corporation is a longtime leader in the office photocopier marketplace.

• Dupit’s service division is responsible for providing support to the customers by promptly repairing the machines when needed. This is done by the company’s service technical representatives, or tech reps.

• Current policy: Each tech rep’s territory is assigned enough machines so that the tech rep will be active repairing machines (or traveling to the site) 75% of the time.

– A repair call averages 2 hours, so this corresponds to 3 repair calls per day.

– Machines average 50 workdays between repairs, so assign 150 machines per rep.

• Proposed New Service Standard: The average waiting time before a tech rep begins the trip to the customer site should not exceed two hours.

11-29

Alternative Approaches to the Problem

• Approach Suggested by John Phixitt: Modify the current policy by decreasing the percentage of time that tech reps are expected to be repairing machines.

• Approach Suggested by the Vice President for Engineering: Provide new equipment to tech reps that would reduce the time required for repairs.

• Approach Suggested by the Chief Financial Officer: Replace the current one-person tech rep territories by larger territories served by multiple tech reps.

• Approach Suggested by the Vice President for Marketing: Give owners of the new printer-copier priority for receiving repairs over the company’s other customers.

11-30

The Queueing System for Each Tech Rep

• The customers: The machines needing repair.

• Customer arrivals: The calls to the tech rep requesting repairs.

• The queue: The machines waiting for repair to begin at their sites.

• The server: The tech rep.

• Service time: The total time the tech rep is tied up with a machine, either traveling to the machine site or repairing the machine. (Thus, a machine is viewed as leaving the queue and entering service when the tech rep begins the trip to the machine site.)

11-31

Notation for Single-Server Queueing Models

• = Mean arrival rate for customers= Expected number of arrivals per unit time

1/ = expected interarrival time

• = Mean service rate (for a continuously busy server)= Expected number of service completions per unit time

= expected service time

• = the utilization factor= the average fraction of time that a server is busy serving customers=

11-32

The M/M/1 Model

• Assumptions1. Interarrival times have an exponential distribution with a mean of 1/.

2. Service times have an exponential distribution with a mean of 1/.

3. The queueing system has one server.

• The expected number of customers in the system is

L = 1 – = –

• The expected waiting time in the system is

W = (1 / )L = 1 / ( – )

• The expected waiting time in the queue is

Wq = W – 1/ = / [( – )]

• The expected number of customers in the queue is

Lq = Wq = 2 / [( – )] = 2 / (1 – )

11-33

The M/M/1 Model

• The probability of having exactly n customers in the system is

Pn = (1 – )n

Thus,P0 = 1 – P1 = (1 – )P2 = (1 – )2

::

• The probability that the waiting time in the system exceeds t is

P(W > t) = e–(1–)t for t ≥ 0

• The probability that the waiting time in the queue exceeds t is

P(Wq > t) = e–(1–)t for t ≥ 0

11-34

M/M/1 Queueing Model for the Dupit’s Current Policy

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B C D E G HData Results

3 (mean arrival rate) L = 3 4 (mean service rate) Lq = 2.25s = 1 (# servers)

W = 1Pr(W > t) = 0.368 Wq = 0.75

when t = 1 0.75

Prob(Wq > t) = 0.276when t = 1 n Pn

0 0.251 0.18752 0.14063 0.10554 0.07915 0.05936 0.04457 0.03348 0.02509 0.0188

10 0.0141

11-35

John Phixitt’s Approach (Reduce Machines/Rep)

• The proposed new service standard is that the average waiting time before service begins be two hours (i.e., Wq ≤ 1/4 day).

• John Phixitt’s suggested approach is to lower the tech rep’s utilization factor sufficiently to meet the new service requirement.

Lower = / , until Wq ≤ 1/4 day,where

= (Number of machines assigned to tech rep) / 50.

11-36

M/M/1 Model for John Phixitt’s Suggested Approach(Reduce Machines/Rep)

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2 (mean arrival rate) L = 1 4 (mean service rate) Lq = 0.5s = 1 (# servers)

W = 0.5Pr(W > t) = 0.135 Wq = 0.25

when t = 1 0.5


0 0.51 0.252 0.12503 0.06254 0.03135 0.01566 0.00787 0.00398 0.00209 0.0010

10 0.0005

11-37

The M/G/1 Model


2. Service times can have any probability distribution. You only need the mean (1/) and standard deviation ().

3. The queueing system has one server.

• The probability of zero customers in the system isP0 = 1 –

• The expected number of customers in the queue isLq = 22 + 2] / [2(1 – )]

• The expected number of customers in the system isL = Lq +

• The expected waiting time in the queue isWq = Lq /

• The expected waiting time in the system isW = Wq + 1/

11-38

The Values of and Lq for the M/G/1 Modelwith Various Service-Time Distributions

Distribution Mean Model Lq

Exponential 1/ 1/ M/M/1 2 / (1 – )

Degenerate (constant) 1/ 0 M/D/1 (1/2) [2 / (1 – )]

Erlang, with shape parameter k 1/ (1/k) (1/) M/Ek/1 (k+1)/(2k) [2 / (1 – )]

11-39

VP for Engineering Approach (New Equipment)


• The Vice President for Engineering has suggested providing tech reps with new state-of-the-art equipment that would reduce the time required for the longer repairs.

• After gathering more information, they estimate the new equipment would have the following effect on the service-time distribution:

– Decrease the mean from 1/4 day to 1/5 day.

– Decrease the standard deviation from 1/4 day to 1/10 day.

11-40

M/G/1 Model for the VP of Engineering Approach(New Equipment)

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B C D E F GData Results

3 (mean arrival rate) L = 1.163 0.2 (expected service time) Lq = 0.563

0.1 (standard deviation)s = 1 (# servers) W = 0.388

Wq = 0.188

0.6

P0 = 0.4

11-41

The M/M/s Model


2. Service times have an exponential distribution with a mean of 1/3. Any number of servers (denoted by s).

• With multiple servers, the formula for the utilization factor becomes

= / s

but still represents that average fraction of time that individual servers are busy.

11-42

Values of L for the M/M/s Model for Various Values of s

s = 1

s = 2

s = 3s = 4s = 5s = 7s = 10

s = 15s = 20s = 25

100

10

0.5

0.1

0.2

0 0.1 0.3 0.5 0.7 0.9 1.0Utilization factor

s

Steady-state expected number of customers in the queueing system

11-43

CFO Suggested Approach (Combine Into Teams)


• The Chief Financial Officer has suggested combining the current one-person tech rep territories into larger territories that would be served jointly by multiple tech reps.

• A territory with two tech reps:– Number of machines = 300 (versus 150 before)

– Mean arrival rate = = 6 (versus = 3 before)

– Mean service rate = = 4 (as before)

– Number of servers = s = 2 (versus s = 1 before)

– Utilization factor = = /s = 0.75 (as before)

11-44

M/M/s Model for the CFO’s Suggested Approach(Combine Into Teams of Two)

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6 (mean arrival rate) L = 3.4286 4 (mean service rate) Lq = 1.9286s = 2 (# servers)

W = 0.5714Pr(W > t) = 0.169 Wq = 0.3214

when t = 1 0.75


0 0.14291 0.21432 0.16073 0.12054 0.09045 0.06786 0.05097 0.03818 0.02869 0.0215

10 0.0161

11-45

M/M/s Model for the CFO’s Suggested Approach(Combine Into Teams of Three)

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W = 0.4393Pr(W > t) = 0.090 Wq = 0.1893

when t = 1 0.75


0 0.07481 0.16822 0.18933 0.14194 0.10655 0.07986 0.05997 0.04498 0.03379 0.0253

10 0.0189

11-46

Comparison of Wq with Territories of Different Sizes

Number ofTech Reps

Number ofMachines s Wq

1 150 3 4 1 0.75 0.75 workday (6 hours)

2 300 6 4 2 0.75 0.321 workday (2.57 hours)

3 450 9 4 3 0.75 0.189 workday (1.51 hours)

11-47

Values of L for the M/D/s Model for Various Values of s

100

0.1

1.0

10

0 0.1 0.3 0.5 0.9 1.0

s = 2

s = 3s = 4

s = 5s = 7

s = 10s = 15

s = 20

s = 25

s = 1

0.7Utilization factor

Steady-state expected number of customers in the queueing system

s

11-48

Priority Queueing Models

• General Assumptions:– There are two or more categories of customers. Each category is assigned to a

priority class. Customers in priority class 1 are given priority over customers in priority class 2. Priority class 2 has priority over priority class 3, etc.

– After deferring to higher priority customers, the customers within each priority class are served on a first-come-fist-served basis.

• Two types of priorities– Nonpreemptive priorities: Once a server has begun serving a customer, the

service must be completed (even if a higher priority customer arrives). However, once service is completed, priorities are applied to select the next one to begin service.

– Preemptive priorities: The lowest priority customer being served is preempted (ejected back into the queue) whenever a higher priority customer enters the queueing system.

11-49

Preemptive Priorities Queueing Model

• Additional Assumptions1. Preemptive priorities are used as previously described.

2. For priority class i (i = 1, 2, … , n), the interarrival times of the customers in that class have an exponential distribution with a mean of 1/i.

3. All service times have an exponential distribution with a mean of 1/, regardless of the priority class involved.

4. The queueing system has a single server.

• The utilization factor for the server is

= (1 + 2 + … + n) /

11-50

Nonpreemptive Priorities Queueing Model

• Additional Assumptions1. Nonpreemptive priorities are used as previously described.

2. For priority class i (i = 1, 2, … , n), the interarrival times of the customers in that class have an exponential distribution with a mean of 1/i.

3. All service times have an exponential distribution with a mean of 1/, regardless of the priority class involved.

4. The queueing system can have any number of servers.

• The utilization factor for the servers is

= (1 + 2 + … + n) / s

11-51

VP of Marketing Approach (Priority for New Copiers)


• The Vice President of Marketing has proposed giving the printer-copiers priority over other machines for receiving service. The rationale for this proposal is that the printer-copier performs so many vital functions that its owners cannot tolerate being without it as long as other machines.

• The mean arrival rates for the two classes of copiers are– 1 = 1 customer (printer-copier) per workday (now)

– 2 = 2 customers (other machines) per workday (now)

• The proportion of printer-copiers is expected to increase, so in a couple years– 1 = 1.5 customers (printer-copiers) per workday (later)

– 2 = 1.5 customers (other machines) per workday (later)

11-52

Nonpreemptive Priorities Model forVP of Marketing’s Approach (Current Arrival Rates)

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B C D E F GData

n = 2 (# of priority classes) 4 (mean service rate)s = 1 (# servers)

i L Lq W Wq

Priority Class 1 1 0.5 0.25 0.5 0.25Priority Class 2 2 2.5 2 1.25 1Priority Class 3 1 #DIV/0! #DIV/0! #DIV/0! #DIV/0!Priority Class 4 1 #DIV/0! #DIV/0! #DIV/0! #DIV/0!Priority Class 5 1 1.75 1.5 1.75 1.5

3 0.75

Results

11-53

Nonpreemptive Priorities Model forVP of Marketing’s Approach (Future Arrival Rates)

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B C D E F GData

n = 2 (# of priority classes) 4 (mean service rate)s = 1 (# servers)

i L Lq W Wq

Priority Class 1 1.5 0.825 0.45 0.55 0.3Priority Class 2 1.5 2.175 1.8 1.45 1.2Priority Class 3 1 #DIV/0! #DIV/0! #DIV/0! #DIV/0!Priority Class 4 1 #DIV/0! #DIV/0! #DIV/0! #DIV/0!Priority Class 5 1 1.75 1.5 1.75 1.5

3 0.75

Results

11-54

Some Insights About Designing Queueing Systems

1. When designing a single-server queueing system, beware that giving a relatively high utilization factor (workload) to the server provides surprisingly poor performance for the system.

2. Decreasing the variability of service times (without any change in the mean) improves the performance of a queueing system substantially.

3. Multiple-server queueing systems can perform satisfactorily with somewhat higher utilization factors than can single-server queueing systems. For example, pooling servers by combining separate single-server queueing systems into one multiple-server queueing system greatly improves the measures of performance.

4. Applying priorities when selecting customers to begin service can greatly improve the measures of performance for high-priority customers.

11-55

Total Cost

• The goal is to minimize total cost = cost of servers + cost of waiting

Cost of ServiceCapacity

Cost of customerswaiting

Total Cost

Optimum

Service Capacity

Cost

11-56

Effect of High-Utilization Factors (Insight 1)

345


0.5 (mean arrival rate) L = 1 1 (mean service rate) Lq = 0.5

9

10

1112

13

141516171819202122232425

A B C D EData Table Demonstrating the Effect ofIncreasing on Lq and L for M/M/1

Lq L

1 0.5 1

0 0.01 0.0001 0.01010 0.25 0.0833 0.33330 0.5 0.5 10 0.6 0.9 1.50 0.7 1.6333 2.33330 0.75 2.25 30 0.8 3.2 40 0.85 4.8167 5.66670 0.9 8.1 90 0.95 18.05 190 0.99 98.01 990 0.999 998.001 999

0

20

40

60

80

100

0 0.2 0.4 0.6 0.8 1

System Utilization (r)A

vera

ge

Lin

e L

en

gth

(L

)

11-57

Economic Analysis of the Number of Servers to Provide

• In many cases, the consequences of making customers wait can be expressed as a waiting cost.

• The manager is interested in minimizing the total cost.TC = Expected total cost per unit timeSC = Expected service cost per unit timeWC = Expected waiting cost per unit time

The objective is then to choose the number of servers so as toMinimize TC = SC + WC

• When each server costs the same (Cs = cost of server per unit time),SC = Cs s

• When the waiting cost is proportional to the amount of waiting (Cw = waiting cost per unit time for each customer),

WC = Cw L

11-58

Acme Machine Shop

• The Acme Machine Shop has a tool crib for storing tool required by shop mechanics.

• Two clerks run the tool crib.

• The estimates of the mean arrival rate and the mean service rate (per server) are

= 120 customers per hour = 80 customers per hour

• The total cost to the company of each tool crib clerk is $20/hour, so Cs = $20.

• While mechanics are busy, their value to Acme is $48/hour, so Cw = $48.

• Choose s so as to Minimize TC = $20s + $48L.

11-59

Excel Template for Choosing the Number of Servers

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B C D E F GData Results


W = 0.014473684Pr(W > t) = 0.02581732 Wq = 0.001973684

when t = 0.05 0.5

Prob(Wq > t) = 0.00058707when t = 0.05 n Pn

0 0.2105263161 0.315789474

Cs = $20.00 (cost / server / unit time) 2 0.236842105Cw = $48.00 (waiting cost / unit time) 3 0.118421053

4 0.059210526Cost of Service $60.00 5 0.029605263Cost of Waiting $83.37 6 0.014802632

Total Cost $143.37 7 0.007401316

Economic Analysis:

11-60

Model 1 (M/M/1)

Customers arrive to a small-town post office at an average rate of 10 per hour (Poisson distribution). There is only one postal employee on duty and he can serve customers in an average of 5 minutes (exponential distribution).

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A B C D E G H I

Template for the M/M/s Queueing Model

Data Results 10 (mean arrival rate) L = 5 12 (mean service rate) Lq = 4.166666667s = 1 (# servers) minutes

W = 0.5 30Pr(W > t) = 0.13533528 Wq = 0.416666667 25

when t = 1 0.833333333


0 0.1666666671 0.1388888892 0.1157407413 0.0964506174 0.0803755145 0.0669795956 0.0558163297 0.0465136088 0.038761349 0.032301117

10 0.026917597

11-61

Model 2 (M/G/1)

ABC Car Wash is an automated car wash. Each customer deposits four quarters in a coin slot, drives the car into the auto-washer, and waits while the car is automatically washed. Cars arrive at an average rate of 20 cars per hour (Poisson). The service time is exactly 2 minutes.

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A B C D E F G H

Template for the M/G/1 Queueing Model

Data Results 20 (mean arrival rate) L = 1.333

0.03333333 (expected service time) Lq = 0.667 0 (standard deviation) minutess = 1 (# servers) W = 0.067 4

Wq = 0.033 2

0.666666667

P0 = 0.333333333

11-62

Model 3 (M/M/s)

A grocery store has three registers open. Customers arrive to check out at an average of 1 per minute (Poisson). The service time averages 2 minutes (exponential).

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A B C D E G H

Template for the M/M/s Queueing Model

Data Results 1 (mean arrival rate) L = 2.888888889 0.5 (mean service rate) Lq = 0.888888889s = 3 (# servers)

W = 2.888888889Pr(W > t) = 0.74131525 Wq = 0.888888889

when t = 1 0.666666667


0 0.1111111111 0.2222222222 0.2222222223 0.1481481484 0.0987654325 0.0658436216 0.0438957487 0.0292638328 0.0195092219 0.013006147

10 0.008670765

11-63

One Fast Server or Many Slow Servers

• A McDonalds is considering changing the way that they serve customers.

• Customers arrive at an average rate of 50 per hour.

• Current System: For most of the day (all but their lunch hour), they have three registers open. Each cashier takes the customer’s order, collects the money, and then gets the burgers and pours the drinks. This takes an average of 3 minutes per customer (exponential distribution).

• Proposed System: They are considering having just one cash register. While one person takes the order and collects the money, another will pour the drinks, and another will get the burgers (like Wendys). The three together think they can serve a customer in an average of 1 minute.

Question: Should they switch to the proposed system?

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3 Slow Servers (McDonalds)

1 Fast Server (Wendys)

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10

B C D E G H IData Results

50 (mean arrival rate) L = 6.011235955 20 (mean service rate) Lq = 3.511235955s = 3 (# servers) minutes

W = 0.120224719 7.21Pr(W > t) = 0.46194225 Wq = 0.070224719 4.21

when t = 0.1 0.833333333

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10

B C D E G H IData Results

50 (mean arrival rate) L = 5 60 (mean service rate) Lq = 4.166666667s = 1 (# servers) minutes

W = 0.1 6Pr(W > t) = 4.54E-05 Wq = 0.083333333 5

when t = 1 0.833333333

11-65

LL Bean

• LL Bean’s mail order business– Mail order phone lines open 24 hours per day, 365 days per year

– 78,000 calls per week (average)

– Seasonal variations as well as variability during each day

• How LL Bean estimates the number of servers needed– Each of the week’s 168 hours in a week is modeled separately as a period to be

staffed

– Each hour modeled as an M/M/s queue

– Arrival rates and service rates estimated from historical data

– Service standard: no more than 15% of calls wait more than 20 seconds

– Full-time, part-time, and temporary workers scheduled to meet service standard

irwin/mcgraw-hill chapter 11 queuing models 11.1

Documents

queue size

basic queueing system

queue queue disciplince

maximum size of queue

infinite queue length

number of potential

finite queue model

queue average time customers