Supplement
B Waiting Lines
PROBLEMS
Structure of Waiting-Line Problems
1. Wingard Credit Union
or 13.5%
or 27.1%
or 27.1%
or 18.0%
or 9.0%
The probability that between 1 and 4 customers arrive equals
(0.271+0.271+0.180+0.090=0.812)or 81.2%.
2. Wingard Credit Union part 2
The probability a customer will take less than half a minute is calculated as follows
or 28.3%
The probability that a customer will take more than 3 minutes is calculated as follows
or 13.5%
Using Waiting-Line Models to Analyze Operations
3. Solomon, Smith and Samson
a. Single-server model, average utilization rate.
λ 8
ρ = = = 0.8 or 80% utilization
µ 10
b. The probability of four or fewer documents in the system is 0.6723 as shown
following. Therefore, the probability of more than four documents in the system
is 1 – 0.6723 = 0.3277.
B-1
Copyright © 2016 Pearson Education, Inc.
, B-2 l PART 1 l Managing Processes
n
Pn = (1 − ρ )( ρ )
4
P4 = (1 − 0.8 )( 0.8 ) = 0.0819
3
P3 = (1 − 0.8 )( 0.8 ) = 0.1024
2
P2 = (1 − 0.8 )( 0.8 ) = 0.1280
1
P1 = (1 − 0.8 )( 0.8 ) = 0.1600
0
P0 = (1 − 0.8 )( 0.8 ) = 0.2000
= 0.6723
c. The average number of pages of documents waiting to be typed,
⎛ λ ⎞⎛ λ ⎞ ⎛ 8 ⎞⎛ 8 ⎞
Lq = ρ L = ⎜ ⎟⎜ ⎟ = ⎜ ⎟⎜ ⎟ = 3.2 pages
⎝ µ ⎠⎝ µ − λ ⎠ ⎝ 10 ⎠⎝ 10 − 8 ⎠
4. Benny’s Arcade
Because there are only six machines, we must use the finite source model.
Solver - Waiting Lines
Enter data in yellow shaded areas.
Single-server model Multiple-server model Finite-source model
Customers 6
Arrival Rate (λ) 0.02
Service Rate (µ) 0.0667
Probability of zero customers in the system (P 0) 0.0719
Probability of at most 4 customers in the system (P n) #N/A
ρ
Average utilization of the server ( ) 0.9281
Average number of customers in the system (L) 2.9048
Average number of customers in line (Lq) 1.9766
Average waiting/service time in the system (W) 46.9227
Average waiting time in line (W q) 31.9302
a. Jimmy’s utilization is 0.9281
b. Average number of machines out of service: L = 2.9048
c. Average time a machine is out of service: W = 46.9227 hours
5. Moore, Akin, and Payne (dental clinic). Multiple-server model. This problem is
solved with the help of the Waiting Line Analysis module in POM for Windows
a. Operating characteristics when 3 chairs are staffed
Parameter Value
Copyright © 2016 Pearson Education, Inc.
B Waiting Lines
PROBLEMS
Structure of Waiting-Line Problems
1. Wingard Credit Union
or 13.5%
or 27.1%
or 27.1%
or 18.0%
or 9.0%
The probability that between 1 and 4 customers arrive equals
(0.271+0.271+0.180+0.090=0.812)or 81.2%.
2. Wingard Credit Union part 2
The probability a customer will take less than half a minute is calculated as follows
or 28.3%
The probability that a customer will take more than 3 minutes is calculated as follows
or 13.5%
Using Waiting-Line Models to Analyze Operations
3. Solomon, Smith and Samson
a. Single-server model, average utilization rate.
λ 8
ρ = = = 0.8 or 80% utilization
µ 10
b. The probability of four or fewer documents in the system is 0.6723 as shown
following. Therefore, the probability of more than four documents in the system
is 1 – 0.6723 = 0.3277.
B-1
Copyright © 2016 Pearson Education, Inc.
, B-2 l PART 1 l Managing Processes
n
Pn = (1 − ρ )( ρ )
4
P4 = (1 − 0.8 )( 0.8 ) = 0.0819
3
P3 = (1 − 0.8 )( 0.8 ) = 0.1024
2
P2 = (1 − 0.8 )( 0.8 ) = 0.1280
1
P1 = (1 − 0.8 )( 0.8 ) = 0.1600
0
P0 = (1 − 0.8 )( 0.8 ) = 0.2000
= 0.6723
c. The average number of pages of documents waiting to be typed,
⎛ λ ⎞⎛ λ ⎞ ⎛ 8 ⎞⎛ 8 ⎞
Lq = ρ L = ⎜ ⎟⎜ ⎟ = ⎜ ⎟⎜ ⎟ = 3.2 pages
⎝ µ ⎠⎝ µ − λ ⎠ ⎝ 10 ⎠⎝ 10 − 8 ⎠
4. Benny’s Arcade
Because there are only six machines, we must use the finite source model.
Solver - Waiting Lines
Enter data in yellow shaded areas.
Single-server model Multiple-server model Finite-source model
Customers 6
Arrival Rate (λ) 0.02
Service Rate (µ) 0.0667
Probability of zero customers in the system (P 0) 0.0719
Probability of at most 4 customers in the system (P n) #N/A
ρ
Average utilization of the server ( ) 0.9281
Average number of customers in the system (L) 2.9048
Average number of customers in line (Lq) 1.9766
Average waiting/service time in the system (W) 46.9227
Average waiting time in line (W q) 31.9302
a. Jimmy’s utilization is 0.9281
b. Average number of machines out of service: L = 2.9048
c. Average time a machine is out of service: W = 46.9227 hours
5. Moore, Akin, and Payne (dental clinic). Multiple-server model. This problem is
solved with the help of the Waiting Line Analysis module in POM for Windows
a. Operating characteristics when 3 chairs are staffed
Parameter Value
Copyright © 2016 Pearson Education, Inc.
