1. 1
Technical Note 7
Waiting Line
Management
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2. 2
OBJECTIVES
 Waiting Line Characteristics
 Suggestions for Managing Queues
 Examples (Models 1, 2, 3, and 4)
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3. Components of the Queuing 3
System
Servicing System
Servers
Queue or
Customer Waiting Line
Arrivals Exit
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4. Customer Service Population 4
Sources
Population Source
Finite Infinite
Example: Number of
Example: Number of Example: The
Example: The
machines needing
machines needing number of people
number of people
repair when a
repair when a who could wait in
who could wait in
company only has
company only has a line for
a line for
three machines.
three machines. gasoline.
gasoline.
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5. 5
Service Pattern
Service
Pattern
Constant Variable
Example: Items
Example: Items Example: People
Example: People
coming down an
coming down an spending time
spending time
automated
automated shopping.
shopping.
assembly line.
assembly line.
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6. 6
The Queuing System
Length
Queuing Number of Lines &
Queue Discipline
System Line Structures
Service Time
Distribution
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7. 7
Examples of Line
Structures
Single
Multiphase
Phase
One-person
Single Channel Car wash
barber shop
Bank tellers’ Hospital
Multichannel
windows admissions
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8. 8
Degree of Patience
No Way! No Way!
BALK RENEG
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9. Suggestions for Managing 9
Queues
1. Determine an acceptable waiting
time for your customers
2. Try to divert your customer’s
attention when waiting
3. Inform your customers of what to
expect
4. Keep employees not serving the
customers out of sight
5. Segment customers
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10. 10
Suggestions for Managing Queues
(Continued)
6. Train your servers to be friendly
7. Encourage customers to come
during the slack periods
8. Take a long-term perspective
toward getting rid of the queues
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11. 11
Waiting Line Models
Source
Model Layout Population Service Pattern
1 Single channel Infinite Exponential
2 Single channel Infinite Constant
3 Multichannel Infinite Exponential
4 Single or Multi Finite Exponential
These four models share the following characteristics:
• Single phase
• Poisson arrival
• FCFS
• Unlimited queue length
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12. 12
Notation: Infinite Queuing:
λ = Arrival rate
Models 1-3
µ = Service rate
1
= Average service time
µ
1
= Average time between arrivals
λ
λ
ρ = = Ratio of total arrival rate to sevice rate
µ
for a single server
Lq = Average number waiting in line
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13. Infinite Queuing Models 1-3
13
Ls = Average(Continued)
number in system
(including those being served)
Wq = Average time waiting in line
Ws = Average total time in system
(including time to be served)
n = Number of units in the system
S = Number of identical service channels
Pn = Probability of exactly n units in system
Pw = Probability of waiting in line
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14. 14
Example: Model 1
Assume a drive-up window at a fast food restaurant.
Customers arrive at the rate of 25 per hour.
The employee can serve one customer every two
minutes.
Assume Poisson arrival and exponential service
rates.
Determine:
Determine:
A) What is the average utilization of the employee?
A) What is the average utilization of the employee?
B) What is the average number of customers in line?
B) What is the average number of customers in line?
C) What is the average number of customers in the
C) What is the average number of customers in the
system?
system?
D) What is the average waiting time in line?
D) What is the average waiting time in line?
E) What is the average waiting time in the system?
E) What is the average waiting time in the system?
F) What is the probability that exactly two cars will be
F) What is the probability that exactly two cars will be
in the system?
in the system?
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15. 15
Example: Model 1
A) What is the average utilization of the
employee?
λ = 25 cust / hr
1 customer
µ = = 30 cust / hr
2 mins (1hr / 60 mins)
λ 25 cust / hr
ρ = = = .8333
µ 30 cust / hr
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16. 16
Example: Model 1
B) What is the average number of customers in
line?
λ 2
(25) 2
Lq = = = 4.167
µ ( µ - λ ) 30(30 - 25)
C) What is the average number of customers in the
system?
λ 25
Ls = = =5
µ - λ (30 - 25)
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17. 17
Example: Model 1
D) What is the average waiting time in line?
Lq
Wq = = .1667 hrs = 10 mins
λ
E) What is the average waiting time in the system?
Ls
Ws = = .2 hrs = 12 mins
λ
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18. 18
Example: Model 1
F) What is the probability that exactly two cars
will be in the system (one being served and the
other waiting in line)?
λ λ n
pn = (1- )( )
µ µ
25 25 2
p 2 = (1- )( ) = .1157
30 30
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19. 19
Example: Model 2
An automated pizza vending machine
heats and
dispenses a slice of pizza in 4 minutes.
Customers arrive at a rate of one every 6
minutes with the arrival rate exhibiting a
Poisson distribution.
Determine:
Determine:
A)
A) The average number of customers in line.
The average number of customers in line.
B)
B) The average total waiting time in the system.
The average total waiting time in the system.
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20. 20
Example: Model 2
A) The average number of customers in line.
λ2 (10) 2
Lq = = = .6667
2 µ ( µ - λ ) (2)(15)(15 - 10)
B) The average total waiting time in the system.
Lq .6667
Wq = = = .06667 hrs = 4 mins
λ 10
1 1
Ws = Wq + = .06667 hrs + = .1333 hrs = 8 mins
µ 15/hr
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21. 21
Example: Model 3
Recall the Model 1 example:
Drive-up window at a fast food restaurant.
Customers arrive at the rate of 25 per
hour.
The employee can serve one customer
every two minutes.
Assume Poisson arrival and exponential
service rates.
If an identical window (and an identically trained
If an identical window (and an identically trained
server) were added, what would the effects be on
server) were added, what would the effects be on
the average number of cars in the system and the
the average number of cars in the system and the
total time customers wait before being served?
total time customers wait before being served?
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22. 22
Example: Model 3
Average number of cars in the system
Lq = 0.176
(Exhibit TN7.11 - -using linear interpolation)
λ 25
Ls = Lq + = .176 + = 1.009
µ 30
Total time customers wait before being served
Lq .176 customers
Wq = = = .007 mins ( No Wait! )
λ 25 customers/min
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23. 23
Notation: Finite Queuing:
Model 4
D = Probability that an arrival must wait in line
F = Efficiency factor, a measure of the effect of
having to wait in line
H = Average number of units being served
J = Population source less those in queuing
system ( N - n)
L = Average number of units in line
S = Number of service channels
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24. 24
Finite Queuing: Model 4
(Continued)
n = Average number of units in queuing system
(including the one being served)
N = Number of units in population source
Pn = Probability of exactly n units in queuing system
T = Average time to perform the service
U = Average time between customer service requirements
W = Average waiting time in line
X = Service factor, or proportion of service time required
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25. 25
Example: Model 4
The copy center of an electronics firm has four copy
The copy center of an electronics firm has four copy
machines that are all serviced by a single technician.
machines that are all serviced by a single technician.
Every two hours, on average, the machines require
Every two hours, on average, the machines require
adjustment. The technician spends an average of 10
adjustment. The technician spends an average of 10
minutes per machine when adjustment is required.
minutes per machine when adjustment is required.
Assuming Poisson arrivals and exponential service,
Assuming Poisson arrivals and exponential service,
how many machines are “down” (on average)?
how many machines are “down” (on average)?
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26. 26
Example: Model 4
N, the number of machines in the population = 4
M, the number of repair people = 1
T, the time required to service a machine = 10 minutes
U, the average time between service = 2 hours
T 10 min
X= = = .077
T+ U 10 min + 120 min
From Table TN7.11, F = .980 (Interpolation)
From Table TN7.11, F = .980 (Interpolation)
L, the number of machines waiting to be
L, the number of machines waiting to be
serviced = N(1-F) = 4(1-.980) = .08 machines
serviced = N(1-F) = 4(1-.980) = .08 machines
H, the number of machines being
H, the number of machines being
serviced = FNX = .980(4)(.077) = .302 machines
serviced = FNX = .980(4)(.077) = .302 machines
Number of machines down = L + H = .382 machines
Number of machines down = L + H = .382 machines
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27. 27
Queuing
Approximation
 This approximation is quick way to analyze a queuing
situation. Now, both interarrival time and service time
distributions are allowed to be general.
 In general, average performance measures (waiting time in
queue, number in queue, etc) can be very well
approximated by mean and variance of the distribution
(distribution shape not very important).
 This is very good news for managers: all you need is
mean and standard deviation, to compute average waiting
time
Define:
Standard deviation of X
Cx = coefficient of variation for r.v. X =
Mean of X
Variance
Cx = squared coefficient of variation (scv) = ( Cx ) =
2 2
mean2
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28. Queue 28
Approximation
Inputs: S, λ, µ, Ca ,Cs2
2
(Alternatively: S, λ, µ, variances of interarrival and service time distributions)
λ
Compute ρ =
Sµ
ρ 2( S +1) Ca + Cs2
2
Lq
Lq = ⋅ Ls
1− ρ 2 as before, Wq = , and Ws =
λ λ
Ls = Lq + S ρ
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29. 29
Approximation
 Example
Consider a manufacturing process (for example making
plastic parts) consisting of a single stage with five
machines. Processing times have a mean of 5.4 days
and standard deviation of 4 days. The firm operates
make-to-order. Management has collected date on
customer orders, and verified that the time between
orders has a mean of 1.2 days and variance of 0.72
days. What is the average time that an order waits
before being worked on?
Using our “Waiting Line Approximation” spreadsheet we
get:
Lq = 3.154 Expected number of orders waiting to be
completed.
Wq = 3.78 Expected number of days order waits.
Ρ = 0.9 Expected machine utilization.
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30. 30
End of Technical
Note 7
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