Deck 13: Queuing Theory

Question 1

What is the mean arrival rate based on the following 8 arrival rate observations?
Number of arrivals per hour: 6, 5, 3, 4, 7, 6, 4, 5

A)3
B)4
C)5
D)6

Answer 1

5

Question 2

If the service rate decreases as the arrival rate remains constant, then, in general

A)customer waiting time increases.
B)customer waiting time decreases.
C)service costs increase.
D)customer dissatisfaction decreases.

Answer 2

A

Question 3

For a Poisson random variable, $\lambda$ represents the ____ number of arrivals per time period

A)maximum
B)minimum
C)average
D)standard deviation of

Answer 3

average

Question 4

A Poisson distribution shape can be described as

A)slightly skewed to the left.
B)symmetric around the parameter $\lambda$ .
C)skewed to the right.
D)discrete so it lacks any definable shape.

Question 5

What is the formula for the probability of x arrivals, p(x), under a Poisson distribution with arrival rate $\lambda$ ?

A)
B)
C)
D)

Question 6

If the arrival process is modeled as a Poisson random variable with arrival rate $\lambda$ , then the average time between arrivals is

A)1/ $\mu$
B)1/ $\lambda$
C)1/ $\lambda$ ²
D) $\sigma$

Question 7

An arrival process is memoryless if

A)the time until the next arrival depends on the time elapsed since the last arrival.
B)the time until the next arrival is based on the time elapsed since the last arrival.
C)the time until the next arrival does not depend on the time elapsed since the last arrival.
D)the time until the next arrival is based on the arrival rate.

Question 8

Which of the following is the typical operating characteristic for average number of units in a queue?

A)W
B)W_q
C)L
D)L_q

Question 9

The memoryless property is also referred to as the ____ property.

A)Markov
B)Erlang
C)Poisson
D)Normal

Question 10

Which type of queuing system are you likely to encounter at a grocery store?

A)Single waiting line, single service station.
B)Multiple waiting lines, single service station.
C)Single waiting line, multiple service stations.
D)Multiple waiting lines, multiple service stations.

Question 11

What is the service policy in the queuing systems presented in this chapter that is considered "fair" by the customers?

A)FIFO
B)LIFO
C)FILO
D)Priority

Question 12

Which of the following best describes queuing theory?

A)The study of arrival rates.
B)The study of service times.
C)The study of waiting lines.
D)The evaluation of service time costs.

Question 13

If cell B2 contains the value for $\mu$ and cell A5 contains the value for T, what formula should go in cell B5 to compute the P(Service time) $\le$ T for this exponential distribution?

A)=1-EXP($B$2*A5)
B)=EXP(-$B$2*A5)
C)=1-EXP(-$B$2)
D)=1-EXP(-$B$2*A5)

Question 14

Which type of queuing system are you likely to encounter at an ATM?

A)Single waiting line, single service station.
B)Multiple waiting lines, single service station.
C)Single waiting line, multiple service stations.
D)Multiple waiting lines, multiple service stations.

Question 15

The number of arrivals to a store follows a Poisson distribution with mean $\lambda$ = 10/hour. What is the mean inter-arrival time?

A)6 seconds
B)6 minutes
C)10 minutes
D)10 hours

Question 16

Which of the following is the typical operating characteristic for average time a unit spends waiting for service?

A)W
B)W_q
C)L
D)L_q

Question 17

If the number of arrivals in a time period follow a Poisson distribution with mean $\lambda$ then the inter-arrival times follow a(n) ____ distribution with mean ____.

A)normal; $\mu$
B)constant; $\lambda$
C)exponential; $\lambda$
D)exponential; 1/ $\lambda$

Question 18

Which of the following is a reason to employ queuing theory?

A)To reduce customer wait time in line.
B)To reduce service times.
C)To generate more arrivals to the system.
D)To reduce worker idle time in line.

Question 19

Which type of queuing system are you likely to encounter at a Wendy's restaurant?

A)Single waiting line, single service station.
B)Multiple waiting lines, single service station.
C)Single waiting line, multiple service stations.
D)Multiple waiting lines, multiple service stations.

Question 20

What is the formula for P(t - T) under the exponential distribution with rate $\mu$ ?

A)1 -e ^{$\mu$ T}
B)e^-T
C)1 -e- ^{$\mu$ T}
D)1 - e^T

Question 21

To find steady-state values for the M/M/S queuing system, which of the following statements must be true about the arrival rate?

A) $\lambda$ < s $\mu$
B) $\lambda$ - s = $\mu$
C) $\lambda$ > s $\mu$
D) $\lambda$ = s $\mu$

Question 22

If a company adds an additional identical server to its M/M/1 system, making an M/M/2 system, what happens to a customer's average service time?

A)increases
B)decreases
C)it is unchanged
D)depends on the arrival rate

Question 23

The M in M/G/1 stands for

A)Markovian inter-arrival times.
B)Mendelian inter-arrival times.
C)Mean inter-arrival times.
D)Mathematical inter-arrival times.

Question 24

A store is considering adding a second clerk. The customer arrival rate at this new server will be

A)twice the old rate.
B)half the old rate.
C)the same as the old rate.
D)unpredictable.

Question 25

A store currently operates its service system with 1 operator. Arrivals follow a Poisson distribution and service times are exponentially distributed. The following spreadsheet has been developed for the system.


-Refer to Exhibit 13.1. What is the probability that a customer can go directly into service without waiting in line?

A)0.00
B)0.25
C)0.75
D)1.00

Question 26

Which of the following is the typical operating characteristic for the probability an arriving unit has to wait for service?

A)W_p
B)P₀
C)P_w
D)P_n

Question 27

If a service system has a constant service time, Poisson arrival rates and 2 servers its Kendall notation is

A)P/D/2
B)M/D/2
C)M/D/1
D)G/D/2

Question 28

A balk refers to

A)a customer who refuses to join the queue.
B)a customer who refuses service by a specific server.
C)a customer who joins the queue but leaves before service is complete.
D)a customer who requires extra service time.

Question 29

A store currently operates its service system with 1 operator. Arrivals follow a Poisson distribution and service times are exponentially distributed. The following spreadsheet has been developed for the system.


-Refer to Exhibit 13.1. What is the probability that a customer must wait in queue before being served?

A)0.00
B)0.25
C)0.75
D)1.00

Question 30

A doctor's office only has 8 chairs. The doctor's service times and customer inter-arrival times are exponentially distributed. What type of system is it?

A)M/M/1
B)M/M/8
C)M/M/1 with Finite Queue
D)M/M/1 with Finite Population

Question 31

Joe's Copy Center has 10 copiers. They break down at a rate of 0.02 copiers per hour and are sent to the service facility. What is the average arrival rate of broken copiers to the service facility?

A)0.02
B)0.2
C)10
D)It cannot be determined from the information provided.

Question 32

The amount of time a customer spends with the server is referred to as

A)system time.
B)queue time.
C)service time.
D)served time.

Question 33

A store currently operates its service system with 1 operator. Arrivals follow a Poisson distribution and service times are exponentially distributed. The following spreadsheet has been developed for the system.


-Refer to Exhibit 13.1. How many customers will be in the store on average at any one time?

A)0.375
B)0.50
C)2.25
D)3.00

Question 34

What is the probability that it will take less than or equal to 0.25 hours to service any call based on the following exponential probability distribution with rate $\mu$ = 5?

A)0.00
B)0.71
C)0.92
D)1.00

Question 35

A barber shop has one barber, a Poisson arrival rate and exponentially distributed service times. What is the Kendall notation for this system?

A)M/M/E
B)M/M/1
C)M/E/1
D)P/M/1

Question 36

Joe's Copy Center has 10 copiers. They break down and require service quite often. Time between breakdowns follows an exponential distribution for each copier. The repair person services machines as quickly as possible, but the service time follows an exponential distribution. What type of system is it?

A)M/M/1 with Finite Population
B)M/M/1 with Finite Queue
C)M/M/1
D)M/M/10

Question 37

The service times for a grocery store with one checkout line have a mean of 3 minutes and a standard deviation of 20 seconds. Customer arrivals at the checkout stand follow a Poisson distribution. What type of system is it?

A)M/G/1
B)M/D/1
C)G/M/1
D)M/M/1

Question 38

A store currently operates its service system with 1 operator. Arrivals follow a Poisson distribution and service times are exponentially distributed. The following spreadsheet has been developed for the system.


-Refer to Exhibit 13.1. What is average amount of time spent waiting in line?

A)0.375
B)0.50
C)2.25
D)3.00

Question 39

The standardized queuing system notation such as M/M/1 or M/G/2 is referred to as

A)Kendall notation.
B)Erlang notation.
C)Poisson notation.
D)Queuing notation.

Question 40

A company has recorded the following list of service rates (customers/hour) for one of its servers. What is the mean service time for this server? Customers / hour: 4, 4, 5, 6, 5, 4, 3, 4, 3, 5, 5, 6

A)0.22 min
B)1.11 min
C)4.5 min
D)13.3 min

Question 41

A grocery clerk can serve 20 customers per hour on average and the service time follows an exponential distribution. What is the probability that a customer's service time is more than 4 minutes?

Question 42

A barber shop has one barber who can give 12 haircuts per hour. Customers arrive at a rate of 8 customers per hour. Customer inter-arrival times and service times are exponentially distributed. The following queuing analysis spreadsheet was developed from this information.


-Refer to Exhibit 13.2. What is the Kendall notation for this system?

Question 43

A grocery clerk can serve 20 customers per hour on average and the service time follows an exponential distribution. What is the probability that a customer's service time is less than 2 minutes?

Question 44

Customers arrive at a store randomly, following a Poisson distribution at an average rate of 90 per hour. How many customers would you expect to arrive in a 20 minute period?

Question 45

The M/D/1 model results can be derived from which of the following systems?

A)M/M/1 with $\lambda$ = 0
B)M/G/1 with $\mu$ = 0
C)M/G/1 with $\sigma$ = 0
D)M/M/2 with finite queue length.

Question 46

A company has recorded the following list of service rates (customers/hour) for one of its servers. What is the mean service time for this server?

Question 47

A barber shop has one barber who can give 12 haircuts per hour. Customers arrive at a rate of 8 customers per hour. Customer inter-arrival times and service times are exponentially distributed. The following queuing analysis spreadsheet was developed from this information.


-Refer to Exhibit 13.2. Based on this report what is the average total time spent waiting for a haircut and getting a haircut?

Question 48

The customer service desk at Joe's Discount Electronics store receives 5 customers per hour on average. On average, each customer requires 10 minutes for service. The customer service desk is staffed by a single person. What is the average time a customer spends in the customer service area if modeled as an M/M/1 queuing system?

Question 49

A renege refers to

A)a customer who refuses to join the queue.
B)a customer who refuses service by a specific server.
C)a customer who joins the queue but leaves before service is complete.
D)a customer who requires extra service time.

Question 50

A barber shop has one barber who can give 12 haircuts per hour. Customers arrive at a rate of 8 customers per hour. Customer inter-arrival times and service times are exponentially distributed. The following queuing analysis spreadsheet was developed from this information.


-Refer to Exhibit 13.2. Based on this report what percent of the time is the barber busy cutting hair?

Question 51

A jockey refers to

A)a customer who refuses to join the queue.
B)a customer who refuses service by a specific server.
C)a customer who joins the queue but leaves before service is complete.
D)a customer who switches between queues in the system.

Question 52

The customer service desk at Joe's Discount Electronics store receives 5 customers per hour on average. On average, each customer requires 10 minutes for service. The customer service desk is staffed by a single person. What is the average number of customers in the customer service area, if modeled as an M/M/1 queuing system?

Question 53

Customers arrive at a store randomly, following a Poisson distribution at an average rate of 20 per hour. What is the probability of exactly 0, 1 2, and 3 arrivals in a 15 minute period?

Question 54

A company has recorded the following customer inter-arrival times and service times for 10 customers at one of its single teller service lines. Assume the data are exponentially distributed and the 10 data points represent a reasonable sample.

-Refer to Exhibit 13.3. What is the mean arrival rate per hour?

Question 55

What is the mean arrival rate based on the following 10 arrival rate observations?

Question 56

A barber shop has one barber who can give 12 haircuts per hour. Customers arrive at a rate of 8 customers per hour. Customer inter-arrival times and service times are exponentially distributed. The following queuing analysis spreadsheet was developed from this information.


-Refer to Exhibit 13.2. Based on this report what is the average number of customers waiting for a haircut?

Question 57

A grocery clerk can serve 20 customers per hour on average and the service time follows an exponential distribution. What is the expected service time per customer?

Question 58

A company has recorded the following customer inter-arrival times and service times for 10 customers at one of its single teller service lines. Assume the data are exponentially distributed and the 10 data points represent a reasonable sample.

-Refer to Exhibit 13.3. What is the mean service rate per hour?

Question 59

Customers arrive at a store randomly, following a Poisson distribution at an average rate of 90 per hour. How many customers arrive per minute, on average?

Question 60

A barber shop has one barber who can give 12 haircuts per hour. Customers arrive at a rate of 8 customers per hour. Customer inter-arrival times and service times are exponentially distributed. The following queuing analysis spreadsheet was developed from this information.


-Refer to Exhibit 13.2. Based on this report what is the average waiting time before the barber begins a customer's haircut?

Question 61

The university computer lab has 10 computers which are constantly being used by students. Users need help from the one lab assistant fairly often. Students ask for help at a Poisson rate of with an average of 4 requests per hour for any one computer. The assistant answers questions as quickly as possible and the service time follows an exponential distribution with mean of 1 minute per help session. The following queuing analysis spreadsheet was developed from this information.


-Refer to Exhibit 13.6. What is the Kendall notation for this system?

Question 62

A tax accountant has found that the time to serve a customer has a mean of 30 minutes (or 0.5 hours) and a standard deviation of 6 minutes (or 0.1 hours). Customer arrivals follow a Poisson distribution with an average of 60 minutes between arrivals. The following queuing analysis spreadsheet was developed from this information.


-Refer to Exhibit 13.7. Based on this report how long does a customer spend at the tax accountant's office?

Question 63

Project 13.1 - Internet Sales, Inc.Internet Sales, Inc. is establishing a new ordering system to handle its on-line sales. Customers arrive randomly at an average of one every minute. On average it takes 45 seconds to process a customer's order. The current system employs one network server to process orders. This server processes orders requests one at a time in the order received. The Internet Sales, Inc., Analysis Team has determined that customer orders arrive according to a Poisson process and the average time spent processing an order is exponentially distributed.Internet Sales' management knows that offering prompt, reliable on-line service is critical to their continued success. Upgrading the current on-line ordering capacity will establish their image as a "cutting edge" on-line sales company. On the other hand, too much unused capacity is inefficient and will ultimately cause company profits to decline.Recently, the management group established a goal of serving an on-line customer immediately, 90% of the time. This means that 90% of the time a customer does not have to wait to place an order on-line after using the Internet Sales, Inc. web page. A heated discussion ensued over the current system's capability of meeting this goal.
Under the current system, a customer waits if the system is busy processing another order. Network server upgrades are available in increments of $5000. Each upgrade adds the capability to process an additional order in parallel. For example if two upgrades were purchased at a cost of $10,000, orders will be processed by three independent network servers each serving at an exponential rate of one order per 45 seconds. The Analysis Team was asked to consider the alternatives and recommend a course of action to the management group. What upgrade (if any) will meet the service goal set by the management group? What are the cost and performance trade-offs?

Question 64

A tax accountant has found that the time to serve a customer has a mean of 30 minutes (or 0.5 hours) and a standard deviation of 6 minutes (or 0.1 hours). Customer arrivals follow a Poisson distribution with an average of 60 minutes between arrivals. The following queuing analysis spreadsheet was developed from this information.


-Refer to Exhibit 13.7. Based on this report what is the average number of customers waiting to be helped?

Question 65

A company has recorded the following customer inter-arrival times and service times for 10 customers at one of its single teller service lines. Assume the data are exponentially distributed and the 10 data points represent a reasonable sample.

-Refer to Exhibit 13.3. What is the average number of customers in the service line?

Question 66

A grocery store can serve an average of 360 customers per hour. The service times are exponentially distributed. The store has 4 checkout lines each of which serves 90 customers per hour. Customers arrive at the store at a Poisson rate of 240 customers per hour. The following queuing analysis spreadsheet was developed from this information.


-Refer to Exhibit 13.4. Based on this report what percent of the time is a grocery clerk busy serving a customer?

Question 67

The university computer lab has 10 computers which are constantly being used by students. Users need help from the one lab assistant fairly often. Students ask for help at a Poisson rate of with an average of 4 requests per hour for any one computer. The assistant answers questions as quickly as possible and the service time follows an exponential distribution with mean of 1 minute per help session. The following queuing analysis spreadsheet was developed from this information.


-Refer to Exhibit 13.6. Based on this report what is the average number of students waiting to be helped?

Question 68

A computer printer in a large administrative office has a printer buffer (memory to store printing jobs) capacity of 3 jobs. If the buffer is full when a user wants to print a file the user is told that the job cannot be printed and to try again later. There are so many users in this office that we can assume that there is an infinite calling population. Jobs arrive at the printer at a Poisson rate of 55 jobs per hour and take an average of 1 minute to print. Printing times are exponentially distributed. The following queuing analysis spreadsheet was developed from this information.


-Refer to Exhibit 13.5. What is the Kendall notation for this system?

Question 69

The university computer lab has 10 computers which are constantly being used by students. Users need help from the one lab assistant fairly often. Students ask for help at a Poisson rate of with an average of 4 requests per hour for any one computer. The assistant answers questions as quickly as possible and the service time follows an exponential distribution with mean of 1 minute per help session. The following queuing analysis spreadsheet was developed from this information.


-Refer to Exhibit 13.6. Based on this report what is the probability that a student will not get instantaneous help?

Question 70

A computer printer in a large administrative office has a printer buffer (memory to store printing jobs) capacity of 3 jobs. If the buffer is full when a user wants to print a file the user is told that the job cannot be printed and to try again later. There are so many users in this office that we can assume that there is an infinite calling population. Jobs arrive at the printer at a Poisson rate of 55 jobs per hour and take an average of 1 minute to print. Printing times are exponentially distributed. The following queuing analysis spreadsheet was developed from this information.


-Refer to Exhibit 13.5. Based on this report how long does a computer user have to wait for his/her job to be completed?

Question 71

A grocery store can serve an average of 360 customers per hour. The service times are exponentially distributed. The store has 4 checkout lines each of which serves 90 customers per hour. Customers arrive at the store at a Poisson rate of 240 customers per hour. The following queuing analysis spreadsheet was developed from this information.


-Refer to Exhibit 13.4. What is the Kendall notation for this system?

Question 72

A computer printer in a large administrative office has a printer buffer (memory to store printing jobs) capacity of 3 jobs. If the buffer is full when a user wants to print a file the user is told that the job cannot be printed and to try again later. There are so many users in this office that we can assume that there is an infinite calling population. Jobs arrive at the printer at a Poisson rate of 55 jobs per hour and take an average of 1 minute to print. Printing times are exponentially distributed. The following queuing analysis spreadsheet was developed from this information.


-Refer to Exhibit 13.5. Based on this report what is the average number of jobs waiting to be printed?

Question 73

A tax accountant has found that the time to serve a customer has a mean of 30 minutes (or 0.5 hours) and a standard deviation of 6 minutes (or 0.1 hours). Customer arrivals follow a Poisson distribution with an average of 60 minutes between arrivals. The following queuing analysis spreadsheet was developed from this information.


-Refer to Exhibit 13.7. What is the Kendall notation for this system?

Question 74

The university computer lab has 10 computers which are constantly being used by students. Users need help from the one lab assistant fairly often. Students ask for help at a Poisson rate of with an average of 4 requests per hour for any one computer. The assistant answers questions as quickly as possible and the service time follows an exponential distribution with mean of 1 minute per help session. The following queuing analysis spreadsheet was developed from this information.


-Refer to Exhibit 13.6. Based on this report how much time do students spend getting help before they can resume work on their computers?

Question 75

A tax accountant has found that the time to serve a customer has a mean of 30 minutes (or 0.5 hours) and a standard deviation of 6 minutes (or 0.1 hours). Customer arrivals follow a Poisson distribution with an average of 60 minutes between arrivals. The following queuing analysis spreadsheet was developed from this information.


-Refer to Exhibit 13.7. Based on this report what is the probability that a customer does not have to wait for assistance with his or her taxes?

Question 76

A grocery store can serve an average of 360 customers per hour. The service times are exponentially distributed. The store has 4 checkout lines each of which serves 90 customers per hour. Customers arrive at the store at a Poisson rate of 240 customers per hour. The following queuing analysis spreadsheet was developed from this information.


-Refer to Exhibit 13.4. Based on this report how long does a customer wait before the checker begins serving them?

Question 77

A computer printer in a large administrative office has a printer buffer (memory to store printing jobs) capacity of 3 jobs. If the buffer is full when a user wants to print a file the user is told that the job cannot be printed and to try again later. There are so many users in this office that we can assume that there is an infinite calling population. Jobs arrive at the printer at a Poisson rate of 55 jobs per hour and take an average of 1 minute to print. Printing times are exponentially distributed. The following queuing analysis spreadsheet was developed from this information.


-Refer to Exhibit 13.5. Based on this report what is the probability that a computer user will be told to resubmit a print job at a later time?

Question 78

A grocery store can serve an average of 360 customers per hour. The service times are exponentially distributed. The store has 4 checkout lines each of which serves 90 customers per hour. Customers arrive at the store at a Poisson rate of 240 customers per hour. The following queuing analysis spreadsheet was developed from this information.


-Refer to Exhibit 13.4. Based on this report what is the average number of customers waiting for a checker?

Question 79

A company has recorded the following customer inter-arrival times and service times for 10 customers at one of its single teller service lines. Assume the data are exponentially distributed and the 10 data points represent a reasonable sample.

-Refer to Exhibit 13.3. What is the average time a customer spends in the service line?

Question 80

A grocery store can serve an average of 360 customers per hour. The service times are exponentially distributed. The store has 4 checkout lines each of which serves 90 customers per hour. Customers arrive at the store at a Poisson rate of 240 customers per hour. The following queuing analysis spreadsheet was developed from this information.


-Refer to Exhibit 13.4. Based on this report what is the average total time spent in line and being checked out?