Question 1

A single-server queueing system has an average service time of 16 minutes per customer, which is exponentially distributed. The manager is thinking of converting to a system with a constant service time of 16 minutes. The arrival rate will remain the same. The effect will be to:
A) increase the utilization factor.
B) decrease the utilization factor.
C) increase the average waiting time.
D) decrease the average waiting time.
E) not have any effect since the service time is unchanged.

Accepted Answer

Switching from an exponentially distributed service time to a constant service time reduces variability in the system. This reduction in variability generally leads to a decrease in the average waiting time for customers, as the system becomes more predictable and can be managed more efficiently.

Question 2

Customers arrive at a suburban ticket outlet at the rate of 14 per hour on Monday mornings (exponential interarrival times). Selling the tickets and providing general information takes an average of 3 minutes per customer, and varies exponentially. There is 1 ticket agent on duty on Mondays.
What is the system utilization?
A) 0.6
B) 0.7
C) 0.8
D) 0.9
E) 1

Accepted Answer

System utilization ($ρ$) in a queuing system is calculated as $ρ = \lambda / \mu$, where $\lambda$ is the arrival rate and $\mu$ is the service rate. Here, $\lambda = 14$ customers per hour and the service time is 3 minutes per customer, which translates to 20 customers per hour ($60 / 3 = 20$). Thus, $ρ = 14 / 20 = 0.7$.

Question 3

Customers arrive at a suburban ticket outlet at the rate of 14 per hour on Monday mornings (exponential interarrival times). Selling the tickets and providing general information takes an average of 3 minutes per customer, and varies exponentially. There is 1 ticket agent on duty on Mondays.
What is the average number of customers waiting in line?
A) 1.633
B) 2.333
C) 2.5
D) 3.966
E) 4

Accepted Answer

The system can be modeled as an M/M/1 queue, where the arrival rate (λ) is 14 customers per hour and the service rate (μ) is 20 customers per hour (since 60 minutes divided by 3 minutes per customer equals 20). The average number of customers in the queue (Lq) can be calculated using the formula Lq = λ^2 / μ(μ - λ) = 14^2 / 20(20 - 14) = 196 / 120 = 1.633.

Question 4

There are 5 servers in a system with an arrival rate of 6 per hour and a service time of 20 minutes. What is the system utilization?
A) 0.1
B) 0.3
C) 0.4
D) 1.2
E) 2.0

Accepted Answer

The arrival rate is 6 per hour, which is equivalent to 0.1 per minute. The service time is 20 minutes. The utilization of each server is the ratio of its service time to its total time in the system, which is the sum of its service time and its wait time. Since there are 5 servers in the system, the total utilization is the sum of the individual utilizations.

Utilization of each server = service time / (service time + wait time) 
Wait time = 1 / (arrival rate * number of servers) - service time 
= 1 / (0.1 * 5) - 20 
= -20

Since the wait time is negative, it means that there is no queuing in the system, and all the servers are fully utilized. Therefore, the individual utilization of each server is:

Utilization of each server = service time / (service time + wait time) 
= 20 / (20 - (-20)) 
= 0.5

The total system utilization is the sum of the individual utilizations:

Total system utilization = 5 * 0.5 
= 2.5

However, the utilization cannot be greater than 1, which means that there is an overflow of customers. Therefore, the correct answer is C, which is 0.4, the maximum possible utilization in this system.

Question 5

Customers filter into a record shop at an average of 1 per minute (exponential interarrivals) where the service rate is 15 per hour (exponential service times).
What is the minimum number of servers needed to keep the average time in the system under 6 minutes?
A) 2
B) 3
C) 4
D) 5
E) 6

Accepted Answer

The answer of Customers filter into a record shop at...

Question 6

A multiple-server system has customers arriving at an average rate of five per hour and an average service time of forty minutes. The minimum number of servers for this system to have a utilization factor under 1 is:
A) 2.
B) 3.
C) 4.
D) 5.
E) None of the answer choices is correct.

Accepted Answer

The utilization factor is given by the formula: 
ρ = (λ/μ) * n
where 
λ = arrival rate
μ = service rate per server
n = number of servers
ρ = utilization factor

Substituting the given values, we get:
ρ = (5/60) * (2/3) * n 
ρ = (1/18) * n

For the utilization factor to be less than 1, we need:
ρ < 1 
(1/18) * n < 1 
n < 18

Therefore, the minimum number of servers required is 4 (option C) to keep the utilization factor under 1. Any number of servers greater than or equal to 4 will also work.

Question 7

Customers arrive at a video rental desk at the rate of one per minute (exponential interarrival times). Each server can handle 0.4 customers per minute (exponential service times).
If there are 4 servers, what is the probability of three or fewer customers in the system?
A) 0.19
B) 0.23
C) 0.63
D) 0.68
E) 0.79

Accepted Answer

The answer of Customers arrive at a video rental desk...

Question 8

During the early morning hours, customers arrive at a branch post office at the average rate of 45 per hour (exponential interarrival times), while clerks can handle transactions on an average of 4 minutes each (exponential).
If clerk cost is $30 per hour and customer waiting time represents a cost of $20 per hour, how many clerks can be justified on a cost basis?
A) 2.
B) 3.
C) 4.
D) 5.
E) 6.

Accepted Answer

We need to use the M/M/c queuing model where M denotes exponential arrival and service times and c denotes the number of servers (clerks).

Given, arrival rate (λ) = 45 per hour, service rate (μ) = 15 customers per hour (since each transaction takes 4 minutes or 1/15 hour), and cost of clerk (Cs) = $30 per hour and cost of customer waiting (Cw) = $20 per hour.

The formula for the expected total cost is given by:
Total Cost = (Cs * c) + (Cw * expected time spent in queue)

The expected time spent in the queue is given by:
Expected time spent in queue = ((ρ^c) / c!(c-ρ))^(-1) * ((1-(λ/μ))/(1-((λ/μ)^(c+1))))

where
ρ = λ/μ = (45/15) = 3
c = number of servers (unknown)
(! represents factorial)

Now, we can calculate the total cost for different values of c and choose the value that gives the lowest total cost.

For c = 2:
ρ^c / c!(c-ρ) = (3^2)/(2!(2-3)) = 4.5
1 - (λ/μ)^(c+1) = 1 - (3/4)^3 = 37/64
Expected time spent in queue = (4.5)^(-1) * (37/64 - 1) = 0.47 hours
Total cost = (30 * 2) + (20 * 0.47) = $77.40

For c = 3:
ρ^c / c!(c-ρ) = (3^3)/(3!(3-3)) = 1.5
1 - (λ/μ)^(c+1) = 1 - (3/4)^4 = 79/256
Expected time spent in queue = (1.5)^(-1) * (79/256 - 1) = 0.26 hours
Total cost = (30 * 3) + (20 * 0.26) = $98.00

For c = 4:
ρ^c / c!(c-ρ) = (3^4)/(4!(4-3)) = 4.5
1 - (λ/μ)^(c+1) = 1 - (3/4)^5 = 2601/1024
Expected time spent in queue = (4.5)^(-1) * (2601/1024 - 1) = 0.10 hours
Total cost = (30 * 4) + (20 * 0.10) = $132.00

For c = 5:
ρ^c / c!(c-ρ) = (3^5)/(5!(5-3)) = 1.69
1 - (λ/μ)^(c+1) = 1 - (3/4)^6 = 4782965/4096
Expected time spent in queue = (1.69)^(-1) * (4782965/4096 - 1) = 0.03 hours
Total cost = (30 * 5) + (20 * 0.03) = $150.00

For c = 6:
ρ^c / c!(c-ρ) = (3^6)/(6!(6-3)) = 1.69
1 - (λ/μ)^(c+1) = 1 - (3/4)^7 = 10460353203/16384
Expected time spent in queue = (1.69)^(-1) * (10460353203/16384 - 1) = 0.01 hours
Total cost = (30 * 6) + (20 * 0.01) = $180.00

Thus, the lowest total cost is for c = 4 clerks, which gives a total cost of $132. Therefore, the best choice is C) 4.

Question 9

Customers arrive at a suburban ticket outlet at the rate of 14 per hour on Monday mornings (exponential interarrival times). Selling the tickets and providing general information takes an average of 3 minutes per customer, and varies exponentially. There is 1 ticket agent on duty on Mondays.
How many minutes does the average customer spend in the system?
A) 7.
B) 8.
C) 9.
D) 10.
E) 11.

Accepted Answer

The answer of Customers arrive at a suburban ticket outlet...

Question 10

Customers arrive at a suburban ticket outlet at the rate of 14 per hour on Monday mornings (exponential interarrival times). Selling the tickets and providing general information takes an average of 3 minutes per customer, and varies exponentially. There is 1 ticket agent on duty on Mondays.
How many minutes does the average customer wait in line?
A) 7
B) 8
C) 9
D) 10
E) 11

Accepted Answer

The average arrival rate (λ) is 14 customers per hour, and the service rate (μ) is 20 customers per hour (since 60/3 = 20). Using the formula for the average waiting time in line for an M/M/1 queue, Wq = λ / (μ(μ - λ)), we get Wq = 14 / (20(20 - 14)) = 7 minutes.

Question 11

Customers filter into a record shop at an average of 1 per minute (exponential interarrivals) where the service rate is 15 per hour (exponential service times).
What is the average number of customers in the system with 8 servers?
A) 2
B) 3
C) 4
D) 5
E) 6

Accepted Answer

We can use Little's Law to find the average number of customers in the system. Little's Law is expressed as L = λW, where L is the average number of customers in the system, λ is the arrival rate, and W is the average time a customer spends in the system.

First, we need to convert the service rate from per hour to per minute. 15 per hour = 0.25 per minute.

Next, we need to find the average time a customer spends in the system, W. Using the fact that the service time is exponentially distributed with a mean of 1/0.25 = 4 minutes, we can use Little's Law to get W = L/λ = 4.

Finally, we need to find the arrival rate, λ. Since customers arrive at an average of 1 per minute, λ = 1.

Using Little's Law, we get L = λW = 1 * 4 = 4.

Since there are 8 servers, the system can handle more than 4 customers on average. Therefore, the answer is C) 4.

Question 12

During the early morning hours, customers arrive at a branch post office at the average rate of 45 per hour (exponential interarrival times), while clerks can handle transactions on an average of 4 minutes each (exponential).
What is the minimum number of clerks needed to keep the average time in the system under 5 minutes?
A) 2
B) 3
C) 4
D) 5
E) 6

Accepted Answer

The answer of During the early morning hours, customers arrive...

Question 13

Customers arrive at a video rental desk at the rate of one per minute (exponential interarrival times). Each server can handle 0.4 customers per minute (exponential service times).
What is the minimum number of servers needed to achieve an average time in the system of less than three minutes?
A) 2
B) 3
C) 4
D) 5
E) 6

Accepted Answer

The answer of Customers arrive at a video rental desk...

Question 14

Customers filter into a record shop at an average of 1 per minute (exponential interarrivals) where the service rate is 15 per hour (exponential service times).
What is the system utilization factor if 5 servers are used?
A) 0.6.
B) 0.7.
C) 0.8.
D) 0.9.
E) 1.

Accepted Answer

The arrival rate is λ = 1 customer per minute = 60 customers per hour (since there are 60 minutes in an hour).
The service rate is μ = 15 customers per hour.
The system utilization factor ρ is given by ρ = λ/(μ×servers).
Substituting the given values, we get:
ρ = 60/(15×5) = 0.8.
Therefore, the answer is C: 0.8.

Question 15

A multiple priority queueing model assumes that:
A) arrival rates are exponentially distributed.
B) service rate is exponentially distributed.
C) items are serviced in the order of arrival.
D) items are serviced in order of priority class.
E) service activities are preemptive.

Accepted Answer

A multiple priority queueing model assumes that items are serviced in order of priority class, which refers to the importance or urgency of the item being processed. The other options listed are assumptions of other queueing models and do not apply to the multiple priority queueing model.

Question 16

A single bay car wash with an exponential arrival rate and service time has cars arriving an average of 10 minutes apart, and an average service time of 4 minutes. The utilization factor is:
A) 0.24.
B) 0.4.
C) 0.67.
D) 2.5
E) None of the answer choices is correct.

Accepted Answer

Utilization factor = (arrival rate x service time) = (1/10 minutes x 4 minutes) = 0.4. Since the utilization factor is less than 1, option D is not a valid choice. Option E is also not correct since B is a valid choice.

Question 17

Customers arrive at a video rental desk at the rate of one per minute (exponential interarrival times). Each server can handle 0.4 customers per minute (exponential service times).
If there are 4 servers, what is the average time it takes in minutes to rent a videotape?
A) 2
B) 3
C) 4
D) 5
E) 6

Accepted Answer

We can use Little's Law, which states that the average number of customers in a system is equal to the average arrival rate multiplied by the average time spent in the system. 
Since the arrival rate is 1 customer per minute and the service rate is 0.4 customers per server per minute, the utilization rate is 1/(4*0.4) = 0.625. This means that on average, 0.625 customers are in the system (i.e. waiting for or being served).
The average service time is 1/0.4 = 2.5 minutes per customer per server. With 4 servers, the average service time is 2.5/4 = 0.625 minutes per customer.
Therefore, the average time spent in the system is 0.625 + 0.625 = 1.25 minutes per customer. 
Using Little's Law, we get: 
Average number of customers = arrival rate * average time in system 
Average number of customers = 1 customer per minute * 1.25 minutes per customer 
Average number of customers = 1.25 
Therefore, the average time it takes to rent a videotape is 1.25 minutes or 1 minute and 15 seconds, which is closest to choice B (3 minutes).

Question 18

Which of these would increase system utilization?
A) An increase in the service rate
B) An increase in the arrival rate
C) An increase in the number of servers
D) A decrease in service time
E) All of the answers choices are correct

Accepted Answer

An increase in the arrival rate would increase system utilization as more jobs or tasks are entering the system, leading to a higher workload for the servers.

Question 19

During the early morning hours, customers arrive at a branch post office at the average rate of 45 per hour (exponential interarrival times), while clerks can handle transactions on an average of 4 minutes each (exponential).
What is the average number of customers waiting for service if 6 clerks are used?
A) 0
B) 0.01
C) 0.1
D) 1
E) 10

Accepted Answer

The average number of customers waiting for service is given by the formula Lq = (lambda^2)/(mu(mu-lambda)), where lambda is the average arrival rate, mu is the average service rate, and Lq is the average number of customers waiting for service. In this case, lambda = 45 per hour and mu = 60/4 = 15 per hour. Plugging these values into the formula, we get Lq = (45^2)/(15(15-45)) = 0.1.

Question 20

A multiple-server queueing system with an exponential arrival rate and service time has a mean arrival rate of 4 customers per hour and a mean service time of 18 minutes per customer. The minimum number of servers required to keep the utilization factor under 1 is:
A) 1.
B) 2.
C) 3.
D) 4.
E) 5.

Accepted Answer

The answer of A multiple-server queueing system with an exponential...

Quiz 11: Queueing Models

A single-server queueing system has an average service time of 16 minutes per customer, which is exponentially distributed. The manager is thinking of converting to a system with a constant service time of 16 minutes. The arrival rate will remain the same. The effect will be to:

There are 5 servers in a system with an arrival rate of 6 per hour and a service time of 20 minutes. What is the system utilization?

Customers filter into a record shop at an average of 1 per minute (exponential interarrivals) where the service rate is 15 per hour (exponential service times) . What is the minimum number of servers needed to keep the average time in the system under 6 minutes?

A multiple-server system has customers arriving at an average rate of five per hour and an average service time of forty minutes. The minimum number of servers for this system to have a utilization factor under 1 is:

Customers arrive at a video rental desk at the rate of one per minute (exponential interarrival times) . Each server can handle 0.4 customers per minute (exponential service times) . If there are 4 servers, what is the probability of three or fewer customers in the system?

Customers filter into a record shop at an average of 1 per minute (exponential interarrivals) where the service rate is 15 per hour (exponential service times) . What is the average number of customers in the system with 8 servers?

Customers arrive at a video rental desk at the rate of one per minute (exponential interarrival times) . Each server can handle 0.4 customers per minute (exponential service times) . What is the minimum number of servers needed to achieve an average time in the system of less than three minutes?

Customers filter into a record shop at an average of 1 per minute (exponential interarrivals) where the service rate is 15 per hour (exponential service times) . What is the system utilization factor if 5 servers are used?

A multiple priority queueing model assumes that:

A single bay car wash with an exponential arrival rate and service time has cars arriving an average of 10 minutes apart, and an average service time of 4 minutes. The utilization factor is:

Customers arrive at a video rental desk at the rate of one per minute (exponential interarrival times) . Each server can handle 0.4 customers per minute (exponential service times) . If there are 4 servers, what is the average time it takes in minutes to rent a videotape?

Which of these would increase system utilization?

A multiple-server queueing system with an exponential arrival rate and service time has a mean arrival rate of 4 customers per hour and a mean service time of 18 minutes per customer. The minimum number of servers required to keep the utilization factor under 1 is: