Table of Contents
Chapter 14 (Queueing Models)
Elements of a Queueing Model (Section 14.1)
Some Examples of Queueing Systems (Section 14.2)
Measures of Performance for Queueing Systems (Section 14.3)
A Case Study: The Dupit Corp. Problem (Section 14.4)
Some Single-Server Queueing Models (Section 14.5)
Some Multiple-Server Queueing Models (Section 14.6)
Priority Queueing Models (Section 14.7)
Some Insights about Designing Queueing Systems (Section 14.8)
Economic Analysis of the Number of Servers to Provide (Section 14.9)
14.2–14.13
14.14–14.16
14.17–14.20
14.21–14.23
14.24–14.33
14.34–14.42
14.43–14.50
14.51–14.53
14.54–14.57
Queueing Models (UW Lecture)
14.58–14.76
These slides are based upon a lecture from the MBA elective course “Modeling with Spreadsheets”
at the University of Washington (as taught by one of the authors).
Queueing Applications (UW Lecture)
14.77–14.90
These slides are based upon a lecture from the MBA elective course “Modeling with Spreadsheets”
at the University of Washington (as taught by one of the authors).
McGraw-Hill/Irwin
14.1
© The McGraw-Hill Companies, Inc., 2003
A Basic Queueing System
Served Customers
Queueing System
Queue
Customers
CCCCCCC
C
C
C
C
S
S
S
S
Service
facility
Served Customers
McGraw-Hill/Irwin
14.2
© The McGraw-Hill Companies, Inc., 2003
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.
Customer
Time of
Arrival
Haicut
Begins
Duration
of Haircut
Haircut
Ends
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
—
—
—
McGraw-Hill/Irwin
14.3
© The McGraw-Hill Companies, Inc., 2003
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
l = Mean arrival rate for customers coming to the queueing system
where l is the Greek letter lambda.
•
The mean of the probability distribution of interarrival times is
1 / l = Expected interarrival time
•
Most queueing models assume that the form of the probability distribution of
interarrival times is an exponential distribution.
McGraw-Hill/Irwin
14.4
© The McGraw-Hill Companies, Inc., 2003
Evolution of the Number of Customers
4
Number of
Customers
in the
System
3
2
1
0
McGraw-Hill/Irwin
20
40
60
Time (in minutes)
14.5
80
100
© The McGraw-Hill Companies, Inc., 2003
The Exponential Distribution for Interarrival Times
0
McGraw-Hill/Irwin
Mean
Time
14.6
© The McGraw-Hill Companies, Inc., 2003
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.
McGraw-Hill/Irwin
14.7
© The McGraw-Hill Companies, Inc., 2003
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 lastcome, first-served.
McGraw-Hill/Irwin
14.8
© The McGraw-Hill Companies, Inc., 2003
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 / m = Expected service time
where m is the Greek letter mu.
•
The interpretation of m itself is the mean service rate.
m = Expected service completions per unit time for a single busy server
McGraw-Hill/Irwin
14.9
© The McGraw-Hill Companies, Inc., 2003
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: s = Mean
•
Constant Service Times
– A better fit for systems that involve a fixed set of tasks.
– Standard deviation: s = 0.
•
Erlang Distribution
– Fills the middle ground between the exponential distribution and constant.
– Has a shape parameter, k that determines the standard deviation.
– In particular, s = mean / (k)
McGraw-Hill/Irwin
14.10
© The McGraw-Hill Companies, Inc., 2003
Standard Deviation and Mean for Distributions
McGraw-Hill/Irwin
Distribution
Standard Deviation
Exponential
mean
Degenerate (constant)
0
Erlang, any k
(1 / k) (Mean)
Erlang, k = 2
(1 / 2) (Mean)
Erlang, k = 4
(1 / 2) (Mean)
Erlang, k = 8
(1 / 22) (Mean)
Erlang, k = 16
(1 / 4) (Mean)
14.11
© The McGraw-Hill Companies, Inc., 2003
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 are
M = Exponential distribution (Markovian)
D = Degenerate distribution (constant times)
Ek = Erlang distribution (shape parameter = k)
GI = General independent interarrival-time distribution (any distribution)
G = General service-time distribution (any arbitrary distribution)
McGraw-Hill/Irwin
14.12
© The McGraw-Hill Companies, Inc., 2003
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.
McGraw-Hill/Irwin
14.13
© The McGraw-Hill Companies, Inc., 2003
Examples of Commercial Service Systems
That 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
McGraw-Hill/Irwin
14.14
© The McGraw-Hill Companies, Inc., 2003
Examples of Internal Service Systems
That Are Queueing Systems
Type of System
Customers
Server(s)
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
McGraw-Hill/Irwin
14.15
© The McGraw-Hill Companies, Inc., 2003
Examples of Transportation Service Systems
That Are Queueing Systems
Type of System
Customers
Server(s)
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
McGraw-Hill/Irwin
14.16
© The McGraw-Hill Companies, Inc., 2003
Choosing a Measure of Performance
•
Managers who oversee queueing systems are mainly concerned with two
measures of performance:
–
–
•
When customers are internal to the organization, the first measure tends to be
more important.
–
•
How many customers typically are waiting in the queueing system?
How long do these customers typically have to wait?
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.
McGraw-Hill/Irwin
14.17
© The McGraw-Hill Companies, Inc., 2003
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.
McGraw-Hill/Irwin
14.18
© The McGraw-Hill Companies, Inc., 2003
Relationship between L, W, Lq, and Wq
•
Since 1/m is the expected service time
W = Wq + 1/m
•
Little’s formula states that
L = lW
and
Lq = lWq
•
Combining the above relationships leads to
L = Lq + l/m
McGraw-Hill/Irwin
14.19
© The McGraw-Hill Companies, Inc., 2003
Using Probabilities as Measures of Performance
•
In addition to knowing what happens on the average, we may also be
interested in worst-case scenarios.
– What will be the maximum number of customers in the system? (Exceeded no more
than, say, 5% of the time.)
– What will be the maximum waiting time of customers in the system? (Exceeded no
more than, say, 5% of the time.)
•
Statistics that are helpful to answer these types of questions are available for
some queueing systems:
– Pn = Steady-state probability of having exactly n customers in the system.
– P(W ≤ t) = Probability the time spent in the system will be no more than t.
– P(Wq ≤ t) = Probability the wait time will be no more than t.
•
Examples of common goals:
– No more than three customers 95% of the time: P0 + P1 + P2 + P3 ≥ 0.95
– No more than 5% of customers wait more than 2 hours: P(W ≤ 2 hours) ≥ 0.95
McGraw-Hill/Irwin
14.20
© The McGraw-Hill Companies, Inc., 2003
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.
McGraw-Hill/Irwin
14.21
© The McGraw-Hill Companies, Inc., 2003
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.
McGraw-Hill/Irwin
14.22
© The McGraw-Hill Companies, Inc., 2003
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.)
McGraw-Hill/Irwin
14.23
© The McGraw-Hill Companies, Inc., 2003
Notation for Single-Server Queueing Models
•
l
= Mean arrival rate for customers
= Expected number of arrivals per unit time
1/l = expected interarrival time
•
m
= Mean service rate (for a continuously busy server)
= Expected number of service completions per unit time
1/m = expected service time
•
r
= the utilization factor
= the average fraction of time that a server is busy serving customers
=l/m
McGraw-Hill/Irwin
14.24
© The McGraw-Hill Companies, Inc., 2003
The M/M/1 Model
•
Assumptions
1. Interarrival times have an exponential distribution with a mean of 1/l.
2. Service times have an exponential distribution with a mean of 1/m.
3. The queueing system has one server.
•
The expected number of customers in the system is
L = r / (1 – r) = l / (m – l)
•
The expected waiting time in the system is
W = (1 / l)L = 1 / (m – l)
•
The expected waiting time in the queue is
Wq = W – 1/m = l / [m(m – l)]
•
The expected number of customers in the queue is
Lq = lWq = l2 / [m(m – l)] = r2 / (1 – r)
McGraw-Hill/Irwin
14.25
© The McGraw-Hill Companies, Inc., 2003
The M/M/1 Model
•
The probability of having exactly n customers in the system is
Pn = (1 – r)rn
Thus,
•
P0 = 1 – r
P1 = (1 – r)r
P2 = (1 – r)r2
:
:
The probability that the waiting time in the system exceeds t is
P(W > t) = e–m(1–r)t for t ≥ 0
•
The probability that the waiting time in the queue exceeds t is
P(Wq > t) = re–m(1–r)t for t ≥ 0
McGraw-Hill/Irwin
14.26
© The McGraw-Hill Companies, Inc., 2003
M/M/1 Queueing Model for the Dupit’s Current Policy
B
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
l
m
s=
Pr(W > t) =
when t =
Prob(W q > t) =
when t =
McGraw-Hill/Irwin
C
Data
3
4
1
D
(mean arrival rate)
(mean service rate)
(# servers)
0.368
1
0.276
1
14.27
E
G
L=
Lq =
H
Results
3
2.25
W=
Wq =
1
0.75
r
0.75
n
0
1
2
3
4
5
6
7
8
9
10
Pn
0.25
0.1875
0.1406
0.1055
0.0791
0.0593
0.0445
0.0334
0.0250
0.0188
0.0141
© The McGraw-Hill Companies, Inc., 2003
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 r = l / m, until Wq ≤ 1/4 day,
where
l = (Number of machines assigned to tech rep) / 50.
McGraw-Hill/Irwin
14.28
© The McGraw-Hill Companies, Inc., 2003
M/M/1 Model for John Phixitt’s Suggested Approach
(Reduce Machines/Rep)
B
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
l
m
s=
C
Data
2
4
1
Pr(W > t) =
when t =
0.135
1
Prob(W q > t) =
0.068
1
when t =
McGraw-Hill/Irwin
D
E
(mean arrival rate)
(mean service rate)
(# servers)
14.29
G
L=
Lq =
H
Results
1
0.5
W=
Wq =
0.5
0.25
r
0.5
n
0
1
2
3
4
5
6
7
8
9
10
Pn
0.5
0.25
0.1250
0.0625
0.0313
0.0156
0.0078
0.0039
0.0020
0.0010
0.0005
© The McGraw-Hill Companies, Inc., 2003
The M/G/1 Model
•
Assumptions
1. Interarrival times have an exponential distribution with a mean of 1/l.
2. Service times can have any probability distribution. You only need the mean (1/m)
and standard deviation (s).
3. The queueing system has one server.
•
The probability of zero customers in the system is
P0 = 1 – r
•
The expected number of customers in the queue is
Lq = [l2s2 + r2] / [2(1 – r)]
•
The expected number of customers in the system is
L = Lq + r
•
The expected waiting time in the queue is
Wq = Lq / l
•
The expected waiting time in the system is
W = Wq + 1/m
McGraw-Hill/Irwin
14.30
© The McGraw-Hill Companies, Inc., 2003
The Values of s and Lq for the M/G/1 Model
with Various Service-Time Distributions
Distribution
Mean
s
Model
Lq
Exponential
1/m
1/m
M/M/1
r2 / (1 – r)
Degenerate (constant)
1/m
0
M/D/1
(1/2) [r2 / (1 – r)]
Erlang, with shape parameter k
1/m
(1/k) (1/m)
M/Ek/1
(k+1)/(2k) [r2 / (1 – r)]
McGraw-Hill/Irwin
14.31
© The McGraw-Hill Companies, Inc., 2003
VP for Engineering Approach (New Equipment)
•
The proposed new service standard is that the average waiting time before
service begins be two hours (i.e., Wq ≤ 1/4 day).
•
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.
McGraw-Hill/Irwin
14.32
© The McGraw-Hill Companies, Inc., 2003
M/G/1 Model for the VP of Engineering Approach
(New Equipment)
B
3
4
5
6
7
8
9
10
11
12
l
1/m 
s
s=
McGraw-Hill/Irwin
C
Data
3
0.2
0.1
1
D
(mean arrival rate)
(expected service time)
(standard deviation)
(# servers)
14.33
E
F
L=
Lq =
G
Results
1.163
0.563
W=
Wq =
0.388
0.188
r
0.6
P0 =
0.4
© The McGraw-Hill Companies, Inc., 2003
The M/M/s Model
•
Assumptions
1. Interarrival times have an exponential distribution with a mean of 1/l.
2. Service times have an exponential distribution with a mean of 1/m.
3. Any number of servers (denoted by s).
•
With multiple servers, the formula for the utilization factor becomes
r = l / sm
but still represents that average fraction of time that individual servers are
busy.
McGraw-Hill/Irwin
14.34
© The McGraw-Hill Companies, Inc., 2003
Values of L for the M/M/s Model for Various Values of s
Steady-state expected number of customers in the queueing system
100
10
s = 25
s = 20
s = 15
s = 10
s =7
s =5
s =4
s =3
0.5
s =2
s =1
0.2
0.1
McGraw-Hill/Irwin
0
0.1
0.3
14.35
0.5
0.7
Utilization factor
0.9
1.0
rl
sm
© The McGraw-Hill Companies, Inc., 2003
CFO Suggested Approach (Combine Into Teams)
•
The proposed new service standard is that the average waiting time before
service begins be two hours (i.e., Wq ≤ 1/4 day).
•
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
Mean arrival rate = l = 6
Mean service rate = m = 4
Number of servers = s = 2
Utilization factor = r = l/sm = 0.75
McGraw-Hill/Irwin
(versus 150 before)
(versus l = 3 before)
(as before)
(versus s = 1 before)
(as before)
14.36
© The McGraw-Hill Companies, Inc., 2003
M/M/s Model for the CFO’s Suggested Approach
(Combine Into Teams of Two)
B
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
l
m
s=
C
Data
6
4
2
Pr(W > t) =
when t =
0.169
1
Prob(W q > t) =
McGraw-Hill/Irwin
when t =
D
(mean arrival rate)
(mean service rate)
(# servers)
0.087
1
14.37
E
G
L=
Lq =
H
Results
3.4286
1.9286
W=
Wq =
0.5714
0.3214
r
0.75
n
0
1
2
3
4
5
6
7
8
9
10
Pn
0.1429
0.2143
0.1607
0.1205
0.0904
0.0678
0.0509
0.0381
0.0286
0.0215
0.0161
© The McGraw-Hill Companies, Inc., 2003
CFO Suggested Approach (Teams of Three)
•
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 three tech reps:
–
–
–
–
–
Number of machines = 450
Mean arrival rate = l = 9
Mean service rate = m = 4
Number of servers = s = 3
Utilization factor = r = l/sm = 0.75
McGraw-Hill/Irwin
(versus 150 before)
(versus l = 3 before)
(as before)
(versus s = 1 before)
(as before)
14.38
© The McGraw-Hill Companies, Inc., 2003
M/M/s Model for the CFO’s Suggested Approach
(Combine Into Teams of Three)
B
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
l
m
s=
C
Data
9
4
3
Pr(W > t) =
when t =
0.090
1
Prob(W q > t) =
McGraw-Hill/Irwin
when t =
D
(mean arrival rate)
(mean service rate)
(# servers)
0.028
1
14.39
E
G
L=
Lq =
H
Results
3.9533
1.7033
W=
Wq =
0.4393
0.1893
r
0.75
n
0
1
2
3
4
5
6
7
8
9
10
Pn
0.0748
0.1682
0.1893
0.1419
0.1065
0.0798
0.0599
0.0449
0.0337
0.0253
0.0189
© The McGraw-Hill Companies, Inc., 2003
Comparison of Wq with Territories of Different Sizes
Number of
Tech Reps
Number of
Machines
l
m
s
r
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)
McGraw-Hill/Irwin
14.40
© The McGraw-Hill Companies, Inc., 2003
Values of L for the M/D/s Model for Various Values of s
Steady-state expected number of customers in the queueing system
100
s = 25
10
s = 20
s = 15
s = 10
s =7
s =5
1.0
s =4
s =3
s =2
s =1
0.1
McGraw-Hill/Irwin
0
0.1
0.3
0.5
0.7
Utilization factor
14.41
0.9
1.0
rl
sm
© The McGraw-Hill Companies, Inc., 2003
Values of L for the M/Ek/2 Model for Various Values of k
Steady-state expected number of customers in the queueing system
100
k=1
10
k=2
k=8
1.0
0.1
McGraw-Hill/Irwin
0
0.2
0.4
14.42
0.6
0.8
Utilization factor
1.0
rl
sm
© The McGraw-Hill Companies, Inc., 2003
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.
McGraw-Hill/Irwin
14.43
© The McGraw-Hill Companies, Inc., 2003
Preemptive Priorities Queueing Model
•
Additional Assumptions
1. 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/li.
3. All service times have an exponential distribution with a mean of 1/m, regardless of
the priority class involved.
4. The queueing system has a single server.
•
The utilization factor for the server is
r = (l1 + l2 + … + ln) / m
McGraw-Hill/Irwin
14.44
© The McGraw-Hill Companies, Inc., 2003
Nonpreemptive Priorities Queueing Model
•
Additional Assumptions
1. 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/li.
3. All service times have an exponential distribution with a mean of 1/m, regardless of
the priority class involved.
4. The queueing system can have any number of servers.
•
The utilization factor for the servers is
r = (l1 + l2 + … + ln) / sm
McGraw-Hill/Irwin
14.45
© The McGraw-Hill Companies, Inc., 2003
VP of Marketing Approach (Priority for New Copiers)
•
The proposed new service standard is that the average waiting time before
service begins be two hours (i.e., Wq ≤ 1/4 day).
•
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
–
–
•
l1 = 1 customer (printer-copier) per workday
l2 = 2 customers (other machines) per workday
(now)
(now)
The proportion of printer-copiers is expected to increase, so in a couple years
–
–
l1 = 1.5 customers (printer-copiers) per workday
l2 = 1.5 customers (other machines) per workday
McGraw-Hill/Irwin
14.46
(later)
(later)
© The McGraw-Hill Companies, Inc., 2003
Nonpreemptive Priorities Model for
VP of Marketing’s Approach (Current Arrival Rates)
B
3
4
5
6
7
8
n=
m
s=
D
E
F
G
(# of priority classes)
(mean service rate)
(# servers)
Results
9
10
11
12
13
14
15
16
17
C
Data
2
4
1
Priority
Priority
Priority
Priority
Priority
McGraw-Hill/Irwin
Class
Class
Class
Class
Class
1
2
3
4
5
l
r
li
L
Lq
W
Wq
1
2
1
1
1
0.5
2.5
#DIV/0!
#DIV/0!
1.75
0.25
2
#DIV/0!
#DIV/0!
1.5
0.5
1.25
#DIV/0!
#DIV/0!
1.75
0.25
1
#DIV/0!
#DIV/0!
1.5
3
0.75
14.47
© The McGraw-Hill Companies, Inc., 2003
Nonpreemptive Priorities Model for
VP of Marketing’s Approach (Future Arrival Rates)
B
3
4
5
6
7
8
n=
m
s=
D
E
F
G
(# of priority classes)
(mean service rate)
(# servers)
Results
li
L
Lq
W
Wq
1
2
3
4
5
1.5
1.5
1
1
1
0.825
2.175
#DIV/0!
#DIV/0!
1.75
0.45
1.8
#DIV/0!
#DIV/0!
1.5
0.55
1.45
#DIV/0!
#DIV/0!
1.75
0.3
1.2
#DIV/0!
#DIV/0!
1.5
l
r
3
0.75
9
10
11
12
13
14
15
16
17
C
Data
2
4
1
Priority
Priority
Priority
Priority
Priority
McGraw-Hill/Irwin
Class
Class
Class
Class
Class
14.48
© The McGraw-Hill Companies, Inc., 2003
Expected Waiting Times with Nonpreemptive Priorities
s
When
l1
l2
m
r
1
Now
1
2
4
0.75
0.25 workday (2 hrs.)
1 workday (8 hrs.)
1
Later
1.5
1.5
4
0.75
0.3 workday (2.4 hrs.)
1.2 workday (9.6 hrs.)
2
Now
2
4
4
0.75
0.107 workday (0.86 hr.)
0.439 workday (3.43 hrs.)
2
Later
3
3
4
0.75
0.129 workday (1.03 hrs.)
0.514 workday (4.11 hrs.)
3
Now
3
6
4
0.75
0.063 workday (0.50 hr.)
0.252 workday (2.02 hrs.)
3
Later
4.5
4.5
4
0.75
0.076 workday (0.61 hr.)
0.303 workday (2.42 hrs.)
McGraw-Hill/Irwin
Wq for Printer Copiers
14.49
Wq for Other Machines
© The McGraw-Hill Companies, Inc., 2003
The Four Approaches Under Considerations
Proposer
Proposal
Additional Cost
John Phixitt
Maintain one-person territories, but $300 million per year
reduce number of machines assigned
to each from 150 to 100
VP for Engineering
Keep current one-person territories,
but provide new state-of-the-art
equipment to the tech-reps
One-time cost of $500
million
Chief Financial Officer
Change to three-person territories
None, except
disadvantages of larger
territories
VP for Marketing
Change to two-person territories
with priority given to the printercopiers for repairs
None, except
disadvantages of larger
territories
Decision: Adopt fourth proposal (except for sparsely populated areas where
second proposal should be adopted).
McGraw-Hill/Irwin
14.50
© The McGraw-Hill Companies, Inc., 2003
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.
McGraw-Hill/Irwin
14.51
© The McGraw-Hill Companies, Inc., 2003
Effect of High-Utilization Factors (Insight 1)
B
3
4
5
l
m
A
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
C
D
E
G
(mean arrival rate)
(mean service rate)
D
H
Results
L=
Lq =
1
0.5
E
Data Table Demonstrating the Effect of
Increasing r on Lq and L for M/M/1
l r
1
0 0.01
0 0.25
0
0.5
0
0.6
0
0.7
0 0.75
0
0.8
0 0.85
0
0.9
0 0.95
0 0.99
0 0.999
Lq
0.5
0.0001
0.0833
0.5
0.9
1.6333
2.25
3.2
4.8167
8.1
18.05
98.01
998.001
McGraw-Hill/Irwin
Average Line Length (L)
9
10
B
C
Data
0.5
1
L
1
0.0101
0.3333
1
1.5
2.3333
3
4
5.6667
9
19
99
999
100
80
60
40
20
0
0
0.2
0.4
0.6
0.8
System Utilization (r)
14.52
© The McGraw-Hill Companies, Inc., 2003
1
Effect of Decreasing s (Insight 2)
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
B
C
D
E
F
G
H
Template for the M/G/1 Queueing Model
l
1/m 
s
s=
Data
0.5
1
0.5
1
(mean arrival rate)
(expected service time)
(standard deviation)
(# servers)
Results
L=
0.8125
Lq =
0.3125
W=
Wq =
1.625
0.625
r
0.5
P0 =
0.5
Data Table Demonstrating the Effect of Decreasing s on Lq for M/G/1
15
16
17
18
19
20
21
22
23
McGraw-Hill/Irwin
Body of Table Shows L q Values
r (l)
0.3125
0.5
0.75
0.9
0.99
1
0.500
2.250
8.100
98.010
s
0.5
0.313
1.406
5.063
61.256
14.53
0
0.250
1.125
4.050
49.005
© The McGraw-Hill Companies, Inc., 2003
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 time
SC = Expected service cost per unit time
WC = Expected waiting cost per unit time
The objective is then to choose the number of servers so as to
Minimize 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
McGraw-Hill/Irwin
14.54
© The McGraw-Hill Companies, Inc., 2003
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 l and the mean service rate (per server)
m are
l = 120 customers per hour
m = 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.
McGraw-Hill/Irwin
14.55
© The McGraw-Hill Companies, Inc., 2003
Excel Template for Choosing the Number of Servers
B
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
l
m
s=
C
Data
120
80
3
D
E
(mean arrival rate)
(mean service rate)
(# servers)
Pr(W > t) = 0.02581732
when t =
0.05
Prob(W q > t) = 0.00058707
when t =
0.05
Economic Analysis:
Cs =
Cw =
Cost of Service
Cost of Waiting
Total Cost
McGraw-Hill/Irwin
$20.00
$48.00
(cost / server / unit time)
(waiting cost / unit time)
$60.00
$83.37
$143.37
14.56
F
L=
Lq =
G
Results
1.736842105
0.236842105
W=
Wq =
0.014473684
0.001973684
r
0.5
n
0
1
2
3
4
5
6
7
Pn
0.210526316
0.315789474
0.236842105
0.118421053
0.059210526
0.029605263
0.014802632
0.007401316
© The McGraw-Hill Companies, Inc., 2003
Comparing Expected Cost vs. Number of Clerks
H
J
K
L
M
N
Data Table for Expected Total Cost of Alternatives
s
1
2
3
4
5
r
0.50
1.50
0.75
0.50
0.38
0.30
L
1.74
#N/A
3.43
1.74
1.54
1.51
Cost of
Service
$60.00
$20.00
$40.00
$60.00
$80.00
$100.00
Cost of
Waiting
$83.37
#N/A
$164.57
$83.37
$74.15
$72.41
Total
Cost
$143.37
#N/A
$204.57
$143.37
$154.15
$172.41
$250
Cost ($/hour)
1
2
3
4
5
6
7
8
9
10
I
Cost of
Service
$200
$150
Cost of
Waiting
$100
Total Cost
$50
$0
0
1
2
3
4
5
Number of Servers (s)
McGraw-Hill/Irwin
14.57
© The McGraw-Hill Companies, Inc., 2003
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
McGraw-Hill/Irwin
14.58
© The McGraw-Hill Companies, Inc., 2003
Why is There Waiting?
•
Example #1: McDonalds
– 50 customers arrive per hour
– Service rate is 60 customers per hour
•
Example #2: Doctor’s Office
– Arrivals are scheduled to arrive every 20 minutes.
– The doctor spends an average of 18 minutes with each patient.
McGraw-Hill/Irwin
14.59
© The McGraw-Hill Companies, Inc., 2003
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?
McGraw-Hill/Irwin
14.60
© The McGraw-Hill Companies, Inc., 2003
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?
McGraw-Hill/Irwin
14.61
© The McGraw-Hill Companies, Inc., 2003
Number of Servers
•
Single Server
...
Customers
•
Service
Center
Multiple Servers
...
...
...
Customers
...
Customers
Service
Centers
McGraw-Hill/Irwin
14.62
Service
Centers
© The McGraw-Hill Companies, Inc., 2003
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.
Relative
Frequency
.18
.16
.14
.12
.10
.08
.06
.04
.02
0 1 2 3 4 5 6 7 8 9 10 11 12 13
Customers per time unit
McGraw-Hill/Irwin
14.63
© The McGraw-Hill Companies, Inc., 2003
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.
Relative
Frequency (%)
Service Time
•
Or any distribution
– Only single-server model is easily solved.
McGraw-Hill/Irwin
14.64
© The McGraw-Hill Companies, Inc., 2003
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.
McGraw-Hill/Irwin
14.65
© The McGraw-Hill Companies, Inc., 2003
Queue Discipline
•
Most queueing systems assume customers are served first-come first-served.
..
.
•
Customers
Service
Center
If certain customers are given priority, a priority queueing model can be
used.
– Nonpreemptive: Finish customer in service before taking a new one.
– Preemptive: If priority customer arrives, any regular customer in service is
preempted (put back in the queue).
McGraw-Hill/Irwin
14.66
© The McGraw-Hill Companies, Inc., 2003
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.
McGraw-Hill/Irwin
14.67
© The McGraw-Hill Companies, Inc., 2003
Models
1.
2.
3.
4.
5.
6.
Single server, exponential service time (M/M/1)
Single server, general service time (M/G/1)
Multiple servers, exponential service time (M/M/s)
Finite queue (M/M/s/K)
Priority queue (nonpreemptive and preemptive)
Finite calling population
A Taxonomy
— / — / — (and an optional fourth element / —)
Arrival
Distribution
Service
Distribution
Number of
Servers
Maximum
in Queue
where
M = Exponential (Markovian)
D = deterministic (constant)
G = general distribution
McGraw-Hill/Irwin
14.68
© The McGraw-Hill Companies, Inc., 2003
Notation
•
Parameters:
l = customer arrival rate
m = service rate (1/m = average service time)
s = number of servers
•
Performance Measures
Lq = average number of customers in the queue
L = average number of customers in the system
Wq = average waiting time in the queue
W = average waiting time (including service)
Pn = probability of having n customers in the system
r = system utilization
McGraw-Hill/Irwin
14.69
© The McGraw-Hill Companies, Inc., 2003
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).
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
B
C
D
E
G
H
I
Template for the M/M/s Queueing Model
l
m
s=
Data
10
12
1
Results
(mean arrival rate)
(mean service rate)
(# servers)
Pr(W > t) = 0.13533528
when t =
1
Prob(W q > t) =
McGraw-Hill/Irwin
when t =
0.1127794
1
L=
Lq =
5
4.166666667
W=
Wq =
0.5
0.416666667
r
0.833333333
n
0
1
2
3
4
5
6
7
8
9
10
14.70
minutes
30
25
Pn
0.166666667
0.138888889
0.115740741
0.096450617
0.080375514
0.066979595
0.055816329
0.046513608
0.03876134
0.032301117
0.026917597
© The McGraw-Hill Companies, Inc., 2003
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.
A
1
2
3
4
5
6
7
8
9
10
11
12
B
C
D
E
F
G
H
Template for the M/G/1 Queueing Model
McGraw-Hill/Irwin
l
1/m 
s
s=
Data
20
0.03333333
0
1
(mean arrival rate)
(expected service time)
(standard deviation)
(# servers)
14.71
L=
Lq =
Results
1.333
0.667
W=
Wq =
0.067
0.033
r
0.666666667
P0 =
0.333333333
minutes
4
2
© The McGraw-Hill Companies, Inc., 2003
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).
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
B
C
D
E
G
H
Template for the M/M/s Queueing Model
McGraw-Hill/Irwin
l
m
s=
Data
1
0.5
3
(mean arrival rate)
(mean service rate)
(# servers)
Pr(W > t) = 0.74131525
when t =
1
Prob(W q > t) = 0.26956918
when t =
1
L=
Lq =
Results
2.888888889
0.888888889
W=
Wq =
2.888888889
0.888888889
r
0.666666667
n
0
1
2
3
4
5
6
7
8
9
10
14.72
Pn
0.111111111
0.222222222
0.222222222
0.148148148
0.098765432
0.065843621
0.043895748
0.029263832
0.019509221
0.013006147
0.008670765
© The McGraw-Hill Companies, Inc., 2003
Model 4 (M/M/s/K)
A call center that handles the tech support for a software manufacturer currently
has 10 telephone lines, with three people fielding the calls. Customers call at an
average rate of 40 per hour (Poisson). A customer can be served in an average of
four minutes (exponential).
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
B
C
D
E
F
G
Template for M/M/s Finite Queue Model
McGraw-Hill/Irwin
l
m
s=
K=
Data
40
15
3
10
(mean arrival rate)
(mean service rate)
(# servers)
(max customers)
Results
L = 4.5577194
Lq = 2.0413889
W=
0.1208
Wq = 0.0540838
r  0.8888889
n
0
1
2
3
4
5
6
7
8
9
10
14.73
Pn
0.0406825
0.1084866
0.1446488
0.1285767
0.1142904
0.1015915
0.0903035
0.0802698
0.071351
0.0634231
0.0563761
© The McGraw-Hill Companies, Inc., 2003
Model 5a (Nonpreemptive Priority Queue)
Consider a small-town hospital emergency room (ER) that has just one doctor on
duty. When patients arrive, they are classified as either critical or non-critical.
When the doctor is finished treating a patient, she takes the next critical patient. If
there are no critical patients, then she takes the next non-critical patient. The ER
doctor spends an average of 10 minutes (exponential) treating each patient before
they are either released or admitted to the hospital. An average of 1 critical patient
and 3 non-critical patients arrive each hour (Poisson).
A
1
2
3
4
5
6
7
8
B
C
D
E
F
G
H
Template for M/M/s Nonpreemptive Priorities Queueing Model
1
0
0
0
0
9
0
10
11
12
13
14
15
16
17
0
0
0
0
0
0
0
0
n=
m
s=
Data
2
6
1
(# of priority classes)
(mean service rate)
(# servers)
Results
Priority
Priority
Priority
Priority
Priority
McGraw-Hill/Irwin
Class
Class
Class
Class
Class
1
2
3
4
5
li
L
Lq
W
Wq
Wq (minutes)
1
3
1
1
1
0.3
1.7
2.166666667
#DIV/0!
#DIV/0!
0.133333333
1.2
2
#DIV/0!
#DIV/0!
0.3
0.566666667
2.166666667
#DIV/0!
#DIV/0!
0.133333333
0.4
2
#DIV/0!
#DIV/0!
8
24
l
4
r  0.666666667
14.74
© The McGraw-Hill Companies, Inc., 2003
Model 5b (Preemptive Priority Queue)
Reconsider the same small-town hospital emergency room (ER). Now suppose
they change their policy so that if a critical patient arrives while a non-critical
patient is being treated, the doctor stops treating the non-critical patient, and
immediately starts treating the critical patient. Only when there are no critical
patients to be treated does the doctor start treating non-critical patients.
A
1
2
3
4
5
6
7
8
B
D
E
F
G
H
Template for M/M/1 Preemptive Priorities Queueing Model
n=
m
s=
Data
2
6
1
(# of priority classes)
(mean service rate)
(# servers)
Results
li
9
10
11
12
13
14
15
16
C
Priority
Priority
Priority
Priority
Priority
McGraw-Hill/Irwin
Class 1
1
Class 2
3
Class 3
1
Class 4
1
Class 5
1
l
4
r  0.666666667
L
Lq
W
Wq
Wq (minutes)
0.2
1.8
3
#DIV/0!
#DIV/0!
0.033333333
1.3
2.833333333
#DIV/0!
#DIV/0!
0.2
0.6
3
#DIV/0!
#DIV/0!
0.033333333
0.433333333
2.833333333
#DIV/0!
#DIV/0!
2
26
14.75
© The McGraw-Hill Companies, Inc., 2003
Model 6 (Finite Calling Population)
Consider a PC-Board assembly facility. There are six automated component
insertion machines. Unfortunately, they are very prone to break down. Each
operating machine breaks down every eight hours or so (exponential distribution).
Because these machines are so prone to break down, a full-time repairperson is
kept on staff just to repair these machines. Each repair takes an average of one
hour (exponential distribution). On average, how many machines are operating at
a time?
G
F
E
D
C
B
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
McGraw-Hill/Irwin
Template for M/M/s Finite Calling Population Model
l
m
s=
N=
Data
0.75
1
1
6
(max arrival rate)
(mean service rate)
(# servers)
(size of population)
0
1
2
3
4
5
6
7
8
9
10
14.76
L=
Lq =
Results
1.118015082
0.507766967
W=
Wq =
1.832066425
0.832066425
r
l-bar =
0.75
0.610248115
n
0
1
2
3
4
5
6
7
8
9
0.389751885
0.292313914
0.182696196
0.091348098
0.034255537
0.008563884
0.001070486
0
0
0
Pn
© The McGraw-Hill Companies, Inc., 2003
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.
McGraw-Hill/Irwin
14.77
© The McGraw-Hill Companies, Inc., 2003
Total Cost
•
The goal is to minimize total cost = cost of servers + cost of waiting
Cost
Total Cost
Cost of Service
Capacity
Cost of customers
waiting
Optimum
Service Capacity
McGraw-Hill/Irwin
14.78
© The McGraw-Hill Companies, Inc., 2003
Example #1: How Many Servers?
• The MIS department of a high tech company handles employee requests for
assistance when computer questions arise. Employees requiring assistance
phone the MIS department with their questions (but may have to wait on hold
if all of the tech support staff are busy).
• The MIS department receives an average of 40 requests for assistance per hour
(Poisson).
• The average question can be answered in 3 minutes (exponential).
• The MIS staff is paid an average of $15 per hour.
• The average employee earns $25 per hour.
Question: What is the optimal size of the MIS tech support staff?
McGraw-Hill/Irwin
14.79
© The McGraw-Hill Companies, Inc., 2003
Example #1: How Many Servers
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
B
C
D
E
F
G
Template for Economic Analysis of M/M/s Queueing Model
l
m
s=
Pr(W > t) =
when t =
Data
40
20
4
(mean arrival rate)
(mean service rate)
(# servers)
0.4083219
0.05
Prob(W q > t) = 0.02353657
when t =
0.05
1
2 Economic Analysis:
2
Cs =
1
Cw =
0
0
Cost of Service
0
Cost of Waiting
0
Total Cost
0
0
0
0
0
0
0
0
0
0
0
0
McGraw-Hill/Irwin
$15.00
$25.00
(cost / server / unit time)
(waiting cost / unit time)
$60.00
$54.35
$114.35
14.80
L=
Lq =
Results
2.173913043
0.173913043
W=
Wq =
0.054347826
0.004347826
r
0.5
n
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
0.130434783
0.260869565
0.260869565
0.173913043
0.086956522
0.043478261
0.02173913
0.010869565
0.005434783
0.002717391
0.001358696
0.000679348
0.000339674
0.000169837
8.49185E-05
4.24592E-05
2.12296E-05
1.06148E-05
5.3074E-06
2.6537E-06
Pn
© The McGraw-Hill Companies, Inc., 2003
A Data Table for Example #1: How Many Servers?
I
1
2
3
4
5
6
7
8
J
K
L
M
N
Cost of Service
$60.00
$45.00
$60.00
$75.00
$90.00
Cost of Waiting
$54.35
$72.22
$54.35
$51.00
$50.23
Total Cost
$114.35
$117.22
$114.35
$126.00
$140.23
Data Table for Example #1
s
3
4
5
6
Lq
0.174
0.889
0.174
0.040
0.009
McGraw-Hill/Irwin
Wq (min)
0.261
1.333
0.261
0.060
0.014
14.81
© The McGraw-Hill Companies, Inc., 2003
Example #2: How Many Servers?
•
A McDonalds franchise is trying to decide how many registers to have open
during their busiest time, the lunch hour.
•
Customers arrive during the lunch hour at a rate of 98 customers per hour
(Poisson distribution).
•
Each service takes an average of 3 minutes (exponential distribution).
Question #1: If management would not like the average customer to wait
longer than five minutes in line, how many registers should they open?
Question #2: If management would like no more than 5% of customers to
wait more than 5 minutes, how many registers should they open?
McGraw-Hill/Irwin
14.82
© The McGraw-Hill Companies, Inc., 2003
Example #2: How Many Servers?
A
1
2
3
4
5
6
7
8
9
10
11
12
B
C
D
E
G
H
I
Template for the M/M/s Queueing Model
l
m
s=
Data
98
20
6
(mean arrival rate)
(mean service rate)
(# servers)
Pr(W > t) = 0.34895764
when t = 0.08333333
minutes
5
Prob(W q > t) = 0.08826736
when t = 0.08333333
minutes
5
K
1
2
3
4
5
6
7
8
9
McGraw-Hill/Irwin
L=
Lq =
W=
Wq =
0.075094814
0.025094814
r
0.816666667
n
L
Results
7.359291808
2.459291808
minutes
4.51
1.51
Pn
M
Data Table for Example #2
s
5
6
7
8
9
Wq (min)
1.5057
28.510
1.506
0.430
0.148
0.053
14.83
Pr(Wq > 5 min)
0.08827
0.80443
0.08827
0.00908
0.00087
0.00008
© The McGraw-Hill Companies, Inc., 2003
Example #3: How Much Waiting Space?
•
A photo development shop operates a drive-through lane where customers can
drop off film to be developed and pick up developed photos.
•
Customers arrive at an average rate of 40 per hour (Poisson).
•
Each service takes an average of 1 minute (exponential).
•
They are remodeling the parking area and drive-through lane. They would like
the drive-through lane to hold all of the customers at least 95% of the time.
Question: How many cars must the drive-through lane be able to hold?
McGraw-Hill/Irwin
14.84
© The McGraw-Hill Companies, Inc., 2003
Example #3: How Much Waiting Space?
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
B
C
D
E
G
H
I
Template for the M/M/s Queueing Model
l
m
s=
Data
40
60
1
Results
(mean arrival rate)
(mean service rate)
(# servers)
Pr(W > t) = 2.0612E-09
when t =
1
Prob(W q > t) = 1.3741E-09
when t =
1
McGraw-Hill/Irwin
14.85
L=
Lq =
2
1.333333333
W=
Wq =
0.05
0.033333333
r
0.666666667
n
0
1
2
3
4
5
6
7
8
9
10
0.333333333
0.222222222
0.148148148
0.098765432
0.065843621
0.043895748
0.029263832
0.019509221
0.013006147
0.008670765
0.00578051
Pn
cumulative
0.3333
0.5556
0.7037
0.8025
0.8683
0.9122
0.9415
0.9610
0.9740
0.9827
0.9884
© The McGraw-Hill Companies, Inc., 2003
Example #4: 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?
McGraw-Hill/Irwin
14.86
© The McGraw-Hill Companies, Inc., 2003
3 Slow Servers (McDonalds)
B
3
4
5
6
7
8
9
10
l
m
s=
C
Data
50
20
3
D
E
(mean arrival rate)
(mean service rate)
(# servers)
G
L=
Lq =
Pr(W > t) = 0.46194225
when t =
0.1
H
Results
6.011235955
3.511235955
W=
Wq =
0.120224719
0.070224719
r
0.833333333
I
minutes
7.21
4.21
1 Fast Server (Wendys)
B
3
4
5
6
7
8
9
10
l
m
s=
Pr(W > t) =
when t =
McGraw-Hill/Irwin
C
Data
50
60
1
D
E
(mean arrival rate)
(mean service rate)
(# servers)
4.54E-05
1
14.87
G
H
Results
L=
Lq =
5
4.166666667
W=
Wq =
0.1
0.083333333
r
0.833333333
I
minutes
6
5
© The McGraw-Hill Companies, Inc., 2003
Example #5: General or Specific Servers
•
A small bank in a mall has two tellers.
•
The bank handles two kinds of customers: merchant customers and regular
customers. Each arrive at an average rate of 20 customers per hour (for a total
arrival rate of 40 customers per hour).
•
Current System (Specific Servers): Currently one teller handles only
merchant customers and one teller handles only regular customers. The service
time for both tellers averages 2 minutes (exponential).
•
Proposed System (General Servers): The bank manager is considering
changing the setup to allow each teller to handle both merchant customers and
regular customers. Since the tellers would have to handle both types of jobs,
their efficiency would decrease to a mean service time of 2.2 minutes.
Question: Should they switch to the proposed system?
McGraw-Hill/Irwin
14.88
© The McGraw-Hill Companies, Inc., 2003
Current (Specific Servers)
B
3
4
5
6
7
8
9
10
l
m
s=
Pr(W > t) =
when t =
C
Data
20
30
1
D
E
(mean arrival rate)
(mean service rate)
(# servers)
4.54E-05
1
G
H
Results
L=
Lq =
2
1.333333333
W=
Wq =
0.1
0.066666667
r
0.666666667
I
total in bank
4
2.667
minutes
6
4
Proposed (General Servers)
B
3
4
5
6
7
8
9
10
l
m
s=
C
Data
40
27.27
2
D
E
(mean arrival rate)
(mean service rate)
(# servers)
Pr(W > t) = 6.4082E-07
when t =
1
McGraw-Hill/Irwin
14.89
G
L=
Lq =
H
Results
3.173076923
1.706410256
W=
Wq =
0.079326923
0.042660256
r
0.733333333
I
total in bank
3.173
1.706
minutes
4.760
2.560
© The McGraw-Hill Companies, Inc., 2003
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
McGraw-Hill/Irwin
14.90
© The McGraw-Hill Companies, Inc., 2003