COMMERCE 2OC3 Final Exam Practice Questions Winter 2018
Notes:
• The format of the practice questions is NOT the same as the final exam. The
final exam will include between 20 and 25 MC questions and 3
computational/short answer.
• During the semester, we managed to solve many more problems that have
not been included here in this document. They are all also excellent
questions for practice. So please in addition to the problems included in this
document, make sure you study those other questions and examples for
practice as well (i.e., problems on the handouts, problems on the slides,
suggested textbook problems, etc.).
Questions:
1- Patients arrive at a clinic with inter-arrival times that have an average and standard
deviation equal to 8 and 4 minutes, respectively. The clinic has 4 doctors and the service
time of each doctor has mean and standard deviation equal to 24 and 14 minutes,
respectively. Patients form one line to go into any of the 4 doctor’s offices. Find the
following values. (Note: All numbers are rounded to 2 decimal places).
A) Average inter-arrival times E(A)
B) Average service time E(S)
C) Standard deviation of inter-arrival times 𝜎(𝐴)
D) Standard deviation of service times 𝜎(𝑆)
E) Coefficient of variation for inter-arrival times 𝐶𝐴
F) Coefficient of variation for service times 𝐶𝑆
G) Average arrival rate into the clinic
H) Average aggregate service rate of the clinic
I) Utilization 𝜌
J) Average number of patients waiting in the queue
K) Average number of patients receiving service (in the service area)
L) Average number of patients in the clinic
M) Average waiting time in the queue
N) Average service time
O) Average time in the system
Solution:
A) 𝐸(𝐴) = 8
B) 𝐸(𝑆) = 24
C) 𝜎(𝐴) = 4
D) 𝜎(𝑆) = 14
𝜎(𝐴)
E) 𝐶𝐴 = 𝐸(𝐴) = 0.5
𝜎(𝑆)
F) 𝐶𝑆 = 𝐸(𝑆) = 0.58
,COMMERCE 2OC3 Final Exam Practice Questions Winter 2018
1
G) 𝜆 = 𝐸(𝐴) = 0.125 ≅ 0.13 patients per minute
1
H) 𝜇 = 𝐸(𝑆) = 0.041667 ≅ 0.04 and for the who clinic the average aggregate service rate is
𝑐𝜇 = 4 ∗ 0.04 = 0.16 patients per minute
𝜆 0.13
I) 𝜌 = 𝑐𝜇 = 0.16 ≅ 0.81
J) We should use the PK formula, this formula in the general form is,
𝜌√2(𝑐+1) 𝐶𝐴2 + 𝐶𝑆2 0.51 0.59
𝐿𝑞 = × ≅ × ≅ 0.8
1−𝜌 2 0.19 2
Substituting the numbers into the formula we get,
𝐿𝑞 = 0.8
On average, there are 0.8 patients waiting in the queue.
K) On average, 𝑐𝜌 patients are in the service area, we have,
𝑐𝜌 = 4 ∗ 0.81 = 3.24
On average, 3.24 patients are receiving service.
L) This is 𝐿𝑠 . This is the total number of patients in the system on average. This equals the
average number of patients in the queue plus the average number of patients receiving
service. We have,
𝐿𝑠 = 𝐿𝑞 + 𝑐𝜌 = 4.04
On average, there are 4.04 patients in the clinic.
M) Average waiting time in the queue is,
𝐿𝑞
𝑊𝑞 = ≅ 6.15
𝜆
On average, patients wait 6.15 minutes waiting in the queue.
N) Average service time is,
1 1
= ≅ 24
𝜇 0.0416
On average, each service takes 24 minutes.
O) Average total time in the system is the average waiting time in the queue plus the average
service time. We have,
1
𝑊𝑠 = 𝑊𝑞 + = 30.15
𝜇
On average, patients spend 30.15 minutes at the clinic.
2- Consider a system as shown in the picture below. Assume the arrival rate into the system
is exponential with rate 6 customers per hour. There are 7 servers each with a service rate
of 1 customers per hour. Both inter-arrival time and service time are assumed to be
exponential.
, COMMERCE 2OC3 Final Exam Practice Questions Winter 2018
A) Which queueing model is this? (e.g., M/M/1, M/D/1 or M/M/c)
B) What is the utilization of the system 𝜌?
C) What is the average inter-arrival time?
D) What is the average service time?
E) On average, how many customers are in the queue?
F) On average, how many customers are in the service area?
G) On average, how many customers are in the system?
H) What is the average waiting time (time spent in the queue)?
I) What is the average system time (time spent in the system)?
J) What is the average service time?
K) If we increase the service rate of each server to 2 customers per hour, will that make
the average queue size bigger or smaller? Show this using the formula.
Solution:
A) This is M/M/c because, inter-arrival times are exponential (hence the first M), service
times are exponential (hence the second M) and we have more than one servers (hence
the 𝑐).
B) From the general formula presented we know the utilization of this queueing system is
𝜆
equal to 𝜌 = 𝑐𝜇. We know, 𝑐 = 7, 𝜆 = 6, and 𝜇 = 1. Therefore, utilization is,
𝜆 6
𝜌= = = 0.857143 ≅ 0.86
𝑐𝜇 7 ∗ 1
C) That is equal to,
1
𝐸(𝐴) = = 0.166667 ≅ 0.17 ℎ𝑜𝑢𝑟𝑠
𝜆
D) Average service time is equal to,
1
𝐸(𝑆) = = 1 ℎ𝑜𝑢𝑟𝑠
𝜇
E) This is 𝐿𝑞 . We know from the slides that we should be using the PK formula to find the
𝐿𝑞 . This formula is,
𝜌√2(𝑐+1) 𝐶𝐴2 + 𝐶𝑆2
𝐿𝑞 = ×
1−𝜌 2
We already know that 𝑐 = 7. Also, since both inter-arrival times and service times are
exponential, then we know that their coefficients of variation should be equal to 1.
Therefore, we also have, 𝐶𝐴2 = 1 and 𝐶𝑆2 = 1. As a result, for this specific M/M/c system,
the formula for 𝐿𝑞 becomes,
Notes:
• The format of the practice questions is NOT the same as the final exam. The
final exam will include between 20 and 25 MC questions and 3
computational/short answer.
• During the semester, we managed to solve many more problems that have
not been included here in this document. They are all also excellent
questions for practice. So please in addition to the problems included in this
document, make sure you study those other questions and examples for
practice as well (i.e., problems on the handouts, problems on the slides,
suggested textbook problems, etc.).
Questions:
1- Patients arrive at a clinic with inter-arrival times that have an average and standard
deviation equal to 8 and 4 minutes, respectively. The clinic has 4 doctors and the service
time of each doctor has mean and standard deviation equal to 24 and 14 minutes,
respectively. Patients form one line to go into any of the 4 doctor’s offices. Find the
following values. (Note: All numbers are rounded to 2 decimal places).
A) Average inter-arrival times E(A)
B) Average service time E(S)
C) Standard deviation of inter-arrival times 𝜎(𝐴)
D) Standard deviation of service times 𝜎(𝑆)
E) Coefficient of variation for inter-arrival times 𝐶𝐴
F) Coefficient of variation for service times 𝐶𝑆
G) Average arrival rate into the clinic
H) Average aggregate service rate of the clinic
I) Utilization 𝜌
J) Average number of patients waiting in the queue
K) Average number of patients receiving service (in the service area)
L) Average number of patients in the clinic
M) Average waiting time in the queue
N) Average service time
O) Average time in the system
Solution:
A) 𝐸(𝐴) = 8
B) 𝐸(𝑆) = 24
C) 𝜎(𝐴) = 4
D) 𝜎(𝑆) = 14
𝜎(𝐴)
E) 𝐶𝐴 = 𝐸(𝐴) = 0.5
𝜎(𝑆)
F) 𝐶𝑆 = 𝐸(𝑆) = 0.58
,COMMERCE 2OC3 Final Exam Practice Questions Winter 2018
1
G) 𝜆 = 𝐸(𝐴) = 0.125 ≅ 0.13 patients per minute
1
H) 𝜇 = 𝐸(𝑆) = 0.041667 ≅ 0.04 and for the who clinic the average aggregate service rate is
𝑐𝜇 = 4 ∗ 0.04 = 0.16 patients per minute
𝜆 0.13
I) 𝜌 = 𝑐𝜇 = 0.16 ≅ 0.81
J) We should use the PK formula, this formula in the general form is,
𝜌√2(𝑐+1) 𝐶𝐴2 + 𝐶𝑆2 0.51 0.59
𝐿𝑞 = × ≅ × ≅ 0.8
1−𝜌 2 0.19 2
Substituting the numbers into the formula we get,
𝐿𝑞 = 0.8
On average, there are 0.8 patients waiting in the queue.
K) On average, 𝑐𝜌 patients are in the service area, we have,
𝑐𝜌 = 4 ∗ 0.81 = 3.24
On average, 3.24 patients are receiving service.
L) This is 𝐿𝑠 . This is the total number of patients in the system on average. This equals the
average number of patients in the queue plus the average number of patients receiving
service. We have,
𝐿𝑠 = 𝐿𝑞 + 𝑐𝜌 = 4.04
On average, there are 4.04 patients in the clinic.
M) Average waiting time in the queue is,
𝐿𝑞
𝑊𝑞 = ≅ 6.15
𝜆
On average, patients wait 6.15 minutes waiting in the queue.
N) Average service time is,
1 1
= ≅ 24
𝜇 0.0416
On average, each service takes 24 minutes.
O) Average total time in the system is the average waiting time in the queue plus the average
service time. We have,
1
𝑊𝑠 = 𝑊𝑞 + = 30.15
𝜇
On average, patients spend 30.15 minutes at the clinic.
2- Consider a system as shown in the picture below. Assume the arrival rate into the system
is exponential with rate 6 customers per hour. There are 7 servers each with a service rate
of 1 customers per hour. Both inter-arrival time and service time are assumed to be
exponential.
, COMMERCE 2OC3 Final Exam Practice Questions Winter 2018
A) Which queueing model is this? (e.g., M/M/1, M/D/1 or M/M/c)
B) What is the utilization of the system 𝜌?
C) What is the average inter-arrival time?
D) What is the average service time?
E) On average, how many customers are in the queue?
F) On average, how many customers are in the service area?
G) On average, how many customers are in the system?
H) What is the average waiting time (time spent in the queue)?
I) What is the average system time (time spent in the system)?
J) What is the average service time?
K) If we increase the service rate of each server to 2 customers per hour, will that make
the average queue size bigger or smaller? Show this using the formula.
Solution:
A) This is M/M/c because, inter-arrival times are exponential (hence the first M), service
times are exponential (hence the second M) and we have more than one servers (hence
the 𝑐).
B) From the general formula presented we know the utilization of this queueing system is
𝜆
equal to 𝜌 = 𝑐𝜇. We know, 𝑐 = 7, 𝜆 = 6, and 𝜇 = 1. Therefore, utilization is,
𝜆 6
𝜌= = = 0.857143 ≅ 0.86
𝑐𝜇 7 ∗ 1
C) That is equal to,
1
𝐸(𝐴) = = 0.166667 ≅ 0.17 ℎ𝑜𝑢𝑟𝑠
𝜆
D) Average service time is equal to,
1
𝐸(𝑆) = = 1 ℎ𝑜𝑢𝑟𝑠
𝜇
E) This is 𝐿𝑞 . We know from the slides that we should be using the PK formula to find the
𝐿𝑞 . This formula is,
𝜌√2(𝑐+1) 𝐶𝐴2 + 𝐶𝑆2
𝐿𝑞 = ×
1−𝜌 2
We already know that 𝑐 = 7. Also, since both inter-arrival times and service times are
exponential, then we know that their coefficients of variation should be equal to 1.
Therefore, we also have, 𝐶𝐴2 = 1 and 𝐶𝑆2 = 1. As a result, for this specific M/M/c system,
the formula for 𝐿𝑞 becomes,
