WAITING-LINE MODELS
WAITING LINES
• Waiting line system includes the customer population source as well
as the process or service system.
• Queuing system is another name to define a waiting line.
• Waiting lines are common situations
• Useful in both manufacturing and service areas
COMMON QUEUING SITUATIONS
SITUATION
ARRIVALS IN QUEUE
SERVICE PROCESS
Supermarket
Grocery shoppers
Checkout clerks at cash register
Highway toll booth
Automobiles
Collection of tolls at booth
Doctor’s office
Patients
Treatment by doctors and nurses
Computer system
Programs to be run
Computer processes jobs
Telephone company
Callers
Switching equipment to forward calls
Bank
Customer
Transactions handled by teller
Machine maintenance
Broken machines
Repair people fix machines
Harbor
Ships and barges
Dock workers load and unload
WAITING COST AND SERVICE LEVEL TRADEOFF
ELEMENTS OF WAITING LINES
• customer population source,
• the service system,
• the arrival and service patterns, and
• the priorities used for controlling the line.
THE CUSTOMER POPULATION
• Finite customer population
• The number of potential new customers is affected by the number of
customers already in the system.
• Infinite customer population
• The number of potential new customers is not affected by the number of
customers already in the system.
CUSTOMERS POSSIBLE ACTIONS
• Balking
• The customer decides not to enter the waiting line.
• Reneging
• The customer enters the line but decides to exit before being served.
• Jockeying
• The customer enters one line and then switches to a different line in an effort
to reduce the waiting time.
THE SERVICE SYSTEM
• The number of waiting lines
• The number of servers
• The arrangement of the servers
• Arrival and service patterns
• Service priority rules
CHARACTERISTICS OF WAITING-LINE
SYSTEMS
• Arrivals or inputs to the system
• Population size, behavior, statistical distribution
• Queue discipline, or the waiting line itself
• Limited or unlimited in length, discipline of people or items in it
• The service facility
• Design, statistical distribution of service times
PARTS OF A WAITING LINE
Population of
dirty cars
Arrivals
from the
general
population …
Queue
(waiting line)
Service
facility
Dave’s
Car Wash
Enter
Arrivals to the system
Arrival Characteristics
 Size of the population
 Behavior of arrivals
 Statistical distribution of
arrivals
Exit the system
In the system
Waiting Line
Characteristics
 Limited vs.
unlimited
 Queue discipline
Exit
Exit the system
Service Characteristics
 Service design
 Statistical distribution of
service
QUEUING SYSTEM DESIGNS
A family dentist’s office
Queue
Arrivals
Service
facility
Departures
after service
Phase 2
service
facility
Departures
after service
Single-channel, single-phase system
A McDonald’s dual window drive-through
Queue
Arrivals
Phase 1
service
facility
Single-channel, multiphase system
QUEUING SYSTEM DESIGNS
Most bank and post office service windows
Service
facility
Channel 1
Queue
Service
facility
Channel 2
Arrivals
Service
facility
Channel 3
Multi-channel, single-phase system
Departures
after service
QUEUING SYSTEM DESIGNS
Some college registrations
Queue
Phase 1
service
facility
Channel 1
Phase 2
service
facility
Channel 1
Phase 1
service
facility
Channel 2
Phase 2
service
facility
Channel 2
Arrivals
Multi-channel, multiphase system
Departures
after service
WAITING LINE PERFORMANCE MEASURES
• The average number of customers waiting in line and in the system
• The average time customers spend waiting, and the average time a
customer spends in the system
• The system utilization rate
SINGLE-SERVER WAITING LINE MODEL
• The customers are patient and come from a population that can be
considered infinite
• Customer arrivals are described by a Poisson distribution with a mean
arrival rate of 𝜆. This means that the time between successive
customer arrivals follows an exponential distribution with an average
1
of .
𝜆
• The customer service rate is described by a Poisson distribution with
a mean service rate of µ. This means that the service time for one
1
customer follows an exponential distribution with an average of .
𝜇
• The waiting line priority rule used is FIFO.
MODEL A – SINGLE-CHANNEL
 = Mean number of arrivals per time period
µ = Mean number of units served per time period
Ls = Average number of units (customers) in the system (waiting and being
served)
=
𝜆
𝜇−𝜆
Ws = Average time a unit spends in the system (waiting time plus service time)
1
=
𝜇−𝜆
MODEL A – SINGLE-CHANNEL
LQ = Average number of units waiting in the queue
=
𝜆2
𝜆 𝜇−𝜆
or we an also use 𝑝𝐿𝑠
WQ = Average time a unit spends waiting in the queue
=
𝜆
𝜇 𝜇−𝜆
or we can also use 𝑝𝑊𝑠
p = Utilization factor for the system
=
𝜆
𝜇
MODEL A – SINGLE-CHANNEL
P0 = Probability of 0 units in the system (that is, the service unit is idle)
= 1−
𝜆
𝜇
Pn = 1 − 𝑝 𝑝𝑛 the probability that n customers are in the service system at a given
time
Pn > k= Probability of more than k units in the system, where n is the number of
units in the system
=
𝜆 𝑘+1
𝜇
SINGLE-CHANNEL EXAMPLE 1
 = 2 cars arriving/hour
Ls=
𝜆
𝜇−𝜆
1
=
2
3−2
µ = 3 cars serviced/hour
= 2 cars in the system on average
1
Ws = 𝜇−𝜆 = 3−2 = 1 hour average waiting time in the system
Lq =
𝜆2
22
=
𝜇(𝜇−𝜆)
3(3−2)
= 1.33 cars waiting in line
SINGLE-CHANNEL EXAMPLE
Wq =
p
𝜆
𝜇(𝜇−𝜆)
𝜆
2
𝜇
3
=
2
3(3−2)
2
= hour or 40 minutes
3
= = = 66.67% of time, the mechanic is busy
𝜆
2
P0 = 1 − = 1 − =0.3333 or 33.33% probability there are 0 cars in the
𝜇
3
system
SINGLE-CHANNEL EXAMPLE
Probability of more than k Cars in the System
k
0
1
2
3
4
5
6
7
Pn > k = (2/3)k + 1
.667  Note that this is equal to 1 - P0 = 1 - .33
.444
.296
.198  Implies that there is a 19.8% chance that more than 3
cars are in the system
.132
.088
.058
.039
SINGLE-CHANNEL ECONOMICS
Customer dissatisfaction and lost goodwill = $10 per hour
2
3
Wq = hour or 40 minutes
Total arrivals = 16 per day
Mechanic’s salary = $56 per day
Total hours customers spend waiting per day
= Wq x Total arrivals =
2
3
Customer waiting-time cost =
32
x 16 = 3 or 10.67 hours
32
$10 x 3 = $106.67
Total expected cost = customer waiting time cost + mechanic’s salary
= $106.67 + $56 = $162.67
SINGLE-CHANNEL EXAMPLE 2
The computer lab at State University has a help desk to assist students
working on computer spreadsheet assignments. The students patiently
form a single line in front of the desk to wait for help. Students are
served based on a first-come, first-served priority rule. On average, 15
students per hour arrive at the help desk. Student arrivals are best
described using a Poisson distribution. The help desk server can help an
average of 20 students per hour, with the service rate being de-scribed
by an exponential distribution. Calculate the following operating
characteristics of the service system.
• The average utilization of the help desk server
• The average number of students in the system
• The average number of students waiting in line
• The average time a student spends in the system
• The average time a student spends waiting in line
SOLUTION:
𝜆 = 15 customers arrive each hour
𝜇 = 20 customers per hour
• The average utilization of the help desk server
𝑝=
𝜆
𝜇
=
15
20
= 0.75 or 75%
• The average number of students in the system
Ls=
𝜆
𝜇−𝜆
=
15
20−15
=3
• The average number of students waiting in line
Lq= pLs = 0.75 x 3 = 2.25
𝜆 = 5 customers arrive each hour
𝜇 = 20 customers per hour
• The average time a student spends in the system
Ws
1
𝜇−𝜆
=
1
20−15
= 0.2 hours or 12 minutes
• The average time a student spends waiting in line
WQ= pWs = 0.75 x 0.2 = 0.15 hours or 9 minutes
MULTISERVER WAITING LINE MODEL
s = the number of servers in the system
p=
𝜆
𝑠𝜇
= the average utilization of the system
𝑛
𝑠−1 𝜆/𝜇
𝑛=0 𝑛!
Po =
in the system
LQ =
𝑃0 𝜆/𝜇 𝑠 𝑝
𝑠! 1−𝑝
2
+
𝜆 𝑠
𝜇
𝑠!
−1
(
1
1−𝑝
)
= the probability that no customers are
= the average number of customers waiting in line
WQ =
𝐿𝑄
𝜆
= the average time spent waiting in line
1
W = WQ + 𝜇 = the average time spent in the system, including service
L = 𝜆𝑊 = the average number of customers in the service system
𝜆 𝑛
𝜇
𝑃𝑛 = {
𝑛!
𝑃0 𝑓𝑜𝑟 𝑛 ≤ 𝑠
𝜆 𝑛
𝜇
𝑠!𝑠𝑛−1
𝑃0 𝑓𝑜𝑟 𝑛 > 𝑠
= the probability that n customers are in the system at a given time
MULTISERVER WAITING LINE MODEL EXAMPLE
State University has decided to increase the number of computer
assignments in its curriculum and is concerned about the impact on the
help desk. Instead of a single person working at the help desk, the
university is considering a plan to have three identical service
providers. It expects that students will arrive at a rate of 45 per hour,
according to a Poisson distribution. The service rate for each of the
three servers is 18 students per hour, with exponential service times.
Calculate the following operating characteristics of the service system:
• The average utilization of the help desk
• The probability that there are no students in the system
• The average number of students waiting in line
• The average time a student spends waiting in line
• The average time a student spends in the system
• The average number of students in the system
SOLUTION:
The average utilization of the help desk
Average utilization: p =
𝜆
𝑠𝜇
=
45
3𝑥18
= 0.833 𝑜𝑟 83.3%
The probability that there are no students in the system
Po =
=
𝑛
𝑠−1 𝜆/𝜇
𝑛=0 𝑛!
45/18
0!
0
+
1
( )
𝑠! 1−𝑝
+
45/18
1!
−1
𝜆 𝑠
𝜇
1
+
45/18
2!
2
+
45/18
3!
3
−1
1
1−0.833
= 0.045 or 4.5%
The average number of students waiting in line
LQ =
𝑃0 𝜆/𝜇 𝑠 𝑝
𝑠! 1−𝑝 2
=
0.045 45/18 3 𝑥 0.833
3! 1−0.833 2
= 3.5 students
The average time a student spends waiting in line
WQ =
𝐿𝑄
𝜆
=
3.5
45
=0.078 hour or 4.68 minutes
The average time a student spends in the system
1
1
𝜇
18
W = WQ + = W = 0.078 +
= 0.134 hour or 4.68 minutes
The average number of students in the system
L = 𝜆𝑊 = 45 0.134 = 6.03 students
UNDEFINED SERVICE TIMES MODEL
• 𝑃0 = 1 −
• 𝐿𝑄 =
𝜆
𝜇
𝜆2 𝜎 2 + 𝜆/𝜇 2
2 1−𝜆/𝜇
𝜆
• 𝐿 = 𝐿𝑄 +
• 𝑊𝑄 =
𝜇
𝐿𝑄
𝜆
• 𝑊 = 𝑊𝑄 +
•𝑈=
𝜆
𝜇
1
𝜇
UNDEFINED SERVICE TIMES EXAMPLE
• Consider a business firm with a single fax machine. Employees arrive
randomly to use the fax machine, at an average rate of 20 per hour,
according to Poisson distribution. The time an employee spends using
the machine is not defined by any probability distribution but has a
mean of 2 minutes and a standard deviation of 4 minutes. The
operating characteristics for this system are computed as follows:
𝜆
𝜇
• 𝑃0 = 1 − = 1
•
•
•
•
•
20
− 30
= 0.33 probability that no one is using the machine
𝜆2 𝜎 2 + 𝜆/𝜇 2
(20)2 (1/15)2 + 20/30 2
𝐿𝑄 = 2 1−(𝜆/𝜇 ) =
= 3.33 employees waiting in line
2 1−(20/30 )
𝜆
20
𝐿 = 𝐿𝑄 + 𝜇 = 3.33 + 30 = 4 employees in line and using the machine
𝐿𝑄 3.33
𝑊𝑄 = 𝜆 = 20 = 0.1665 hour or 10 minutes waiting in line
1
1
𝑊 = 𝑊𝑄 + 𝜇 = 0.1665 + 30= 0.1998 hour or 12 minutes in the system
𝜆 20
𝑈 = 𝜇= 30=0.67 or 67% machine utilization
CONSTANT SERVICE TIMES MODEL
• 𝐿𝑄 =
𝜆2
2𝜇(𝜇−𝜆)
𝐿𝑄
• 𝑊𝑄 =
𝜆
CONSTANT SERVICE TIMES MODEL EXAMPLE
• The Petroco Service Station has an automatic carwash, and motorists
purchasing gas at the station receive a discounted car wash,
depending on the number of gallons of gas they buy. The carwash can
accommodate one car at a time, and it requires a constant time of 4.5
minutes for a wash. Cars arrive at the carwash at an average rate of
10 per hour (Poisson distributed). The service station manager wants
to determine the average length of the waiting line and the average
waiting time at the carwash.
• 𝜆 = 10
•𝜇=
60
4.5
• 𝐿𝑄 =
= 13.3 cars per hour
𝜆2
=
102
=1.14 cars waiting
2𝜇(𝜇−𝜆) 2(13.3)(13.3−10)
𝐿𝑄 1.14
• 𝑊𝑄 =
𝜆
=
10
= 0.114 hour or 6.84 minutes waiting in line
CHANGING OPERATIONAL CHARACTERISTICS
• Customer arrival rates
• Number and type of service facilities
• Changing the number of phases
• Server efficiency
• Changing the priority rule
• Changing the number of lines
WAITING LINE MODELS WITHIN OM
• Accounting
• Marketing
• Purchasing
• Operations
FORMULA REVIEW