Chapter 18
Management of Waiting Lines

True / False Questions 1. Waiting lines occur even in under loaded systems because of variability in service rates and/or arrival rates.
TRUE

Difficulty: Easy
TLO: 1
Taxonomy: Knowledge 2. A system has one service facility that can service 10 customers per hour. The customers arrive at a variable rate, which averages 6 per hour. Since there is excess capacity, no waiting lines will form.
FALSE

Difficulty: Easy
TLO: 1
Taxonomy: Knowledge 3. The goal of queuing analysis is to help eliminate customer waiting lines.
FALSE

Difficulty: Easy
TLO: 2
Taxonomy: Knowledge

4. The cost of customer waiting is easy to estimate, the number waiting multiplied by the wait cost per minute.
FALSE

Difficulty: Easy
TLO: 3
Taxonomy: Knowledge 5. In a theme park like Disney world, reservation systems are a win-lose situation since only those holding reservations are satisfied.
FALSE

Difficulty: Medium
TLO: 2
Taxonomy: Knowledge 6. The point that minimizes total queuing system costs is that point where waiting costs and capacity costs are equal.
FALSE

Difficulty: Medium
TLO: 2
Taxonomy: Knowledge 7. A multiple channel system assumes that each server will have its own waiting line and line changing is not permitted.
FALSE

Difficulty: Medium
TLO: 4
Taxonomy: Knowledge

8. A dental office with two professionals (one dentist, one hygienist) who work together as a team would be an example of a multiple channel system.
FALSE

Difficulty: Medium
TLO: 4
Taxonomy: Knowledge 9. A Single Server, Variable Service Time is known as a M/D/1 system.
FALSE

Difficulty: Hard
TLO: 4
Taxonomy: Knowledge 10. The most commonly used queuing models assume that the arrival rate can be described by a Poisson distribution.
TRUE

Difficulty: Hard
TLO: 4
Taxonomy: Knowledge 11. The goal of queuing analysis is to balance the cost of providing a level of service capacity with the possible loss of business due to customers leaving the line or refusing to wait.
FALSE

Difficulty: Hard
TLO: 2
Taxonomy: Knowledge

12. According to Little's Law, the number of people in line depends on the time of day that they arrive.
FALSE

Difficulty: Medium
TLO: 3
Taxonomy: Knowledge 13. For a system that has a low utilization ratio, decreasing service capacity slightly will have only negligible effect on customer waiting time.
TRUE

Difficulty: Medium
TLO: 3
Taxonomy: Knowledge 14. An approach to reducing the variability in processing times might include greater standardization.
TRUE

Difficulty: Medium
TLO: 2
Taxonomy: Knowledge 15. The queuing models discussed in the text apply only to "steady state" conditions. Steady state exists only when customers arrive at a steady rate; that is, without any variability.
FALSE

Difficulty: Hard
TLO: 4
Taxonomy: Knowledge

16. In an infinite source model, the average number being served is equal to the ratio of the arrival rate to the service rate.
TRUE

Difficulty: Medium
TLO: 4
Taxonomy: Knowledge 17. In an infinite source model, the average time in line is equal to the average number in line divided by the arrival rate.
TRUE

Difficulty: Medium
TLO: 3
Taxonomy: Knowledge 18. In an infinite source model, the system utilization is the ratio of the arrival rate to the service capacity.
TRUE

Difficulty: Hard
TLO: 3
Taxonomy: Knowledge 19. To reduce the average number waiting in line, it is important to increase utilization.
FALSE

Difficulty: Medium
TLO: 3
Taxonomy: Knowledge 20. All infinite source queuing models require the system utilization to be less than 1.0.
TRUE

Difficulty: Medium
TLO: 4
Taxonomy: Knowledge

21. Queue discipline requires a security presence to maintain order.
FALSE

Difficulty: Easy
TLO: 4
Taxonomy: Knowledge 22. Compared to a single channel a system with exponential service time, a single channel system with a constant service time causes a reduction of 50 percent in the average number waiting in line.
TRUE

AACSB: AS
Difficulty: Medium
TLO: 4
Taxonomy: Application 23. Compared to a single channel system with exponential service time, the same system with a constant service time will have an average of one-half the number of customers waiting in the system.
FALSE

Difficulty: Hard
TLO: 4
Taxonomy: Knowledge 24. If variation in arrival and service rates can be reduced, waiting lines will be shortened.
TRUE

Difficulty: Medium
TLO: 1
Taxonomy: Knowledge

25. The finite-source queuing model is appropriate when the potential calling population is relatively large.
FALSE

Difficulty: Easy
TLO: 4
Taxonomy: Knowledge 26. Balking is when customers grow impatient and leave a line.
FALSE

Difficulty: Medium
TLO: 4
Taxonomy: Knowledge 27. In a single-channel system, the utilization is equal to the arrival rate divided by the service rate.
TRUE

Difficulty: Medium
TLO: 3
Taxonomy: Knowledge 28. The goal of waiting line management is to minimize waiting time.
FALSE

Difficulty: Medium
TLO: 2
Taxonomy: Knowledge

Multiple Choice Questions

29. Why is there waiting in an infinite source queuing system?
A. poor scheduling of servers
B. slow service
C. low utilization
D. variability in arrival and service rates
E. multiple phase processing

Difficulty: Easy
TLO: 1
Taxonomy: Knowledge 30. The goal of queuing analysis is to minimize:
A. the sum of customer waiting costs and capacity costs
B. the sum of customer waiting time and service time
C. capacity costs
D. customer waiting time
E. idle servers

Difficulty: Easy
TLO: 2
Taxonomy: Knowledge 31. A basic difference between infinite source and finite source queuing models is:
A. the number of servers
B. the average waiting time
C. the arrival distribution
D. size of potential calling population
E. processing rate

Difficulty: Easy
TLO: 4
Taxonomy: Knowledge

32. If a manager increases system utilization (assuming no change in the customer arrival rate) what happens to the customer waiting time?
A. It increases exponentially.
B. It increases proportionally.
C. It decreases proportionally.
D. It decreases exponentially.
E. no change

Difficulty: Medium
TLO: 3
Taxonomy: Knowledge 33. A single phase queuing system is one which has a single:
A. channel
B. server
C. customer being served
D. operation
E. waiting line

Difficulty: Hard
TLO: 4
Taxonomy: Knowledge 34. A single channel queuing system has an average service time of eight minutes and an average time between arrivals of 10 minutes. The arrival rate is:
A. 6 per hour
B. 7.5 per hour
C. 8 per hour
D. 10 per hour
E. 12.5 per hour

AACSB: AS
Difficulty: Easy
TLO: 4
Taxonomy: Application

35. In a _______ system, customers enter the waiting line, receive service and leave.
A. Fast-track
B. Simulated
C. Queuing
D. Random
E. Non-random

Difficulty: Easy
TLO: 1
Taxonomy: Knowledge 36. Which of the following is not generally considered as a measure of system performance in a queuing analysis?
A. the average number waiting in line
B. the average number in the system
C. system utilization
D. the cost of servers plus customer waiting cost
E. average serving time

Difficulty: Medium
TLO: 3
Taxonomy: Knowledge 37. The total cost curve:
A. Starts at zero and increases as service capacity increases.
B. Begins high and decreases as service capacity increases.
C. Starts high, declines, then increases again.
D. Remains relatively flat regardless of service capacity.
E. Starts at zero, increases rapidly, then declines slowly back to zero.

Difficulty: Medium
TLO: 3
Taxonomy: Knowledge

38. Little's Law states that the number of people in a waiting line is the average customer arrival rate multiplied by the
A. average time in the system
B. average waiting time
C. service time minus the waiting time
D. average number in line
E. waiting time

Difficulty: Hard
TLO: 3
Taxonomy: Knowledge 39. As the ratio of arrival rate to service rate is increased, which of the following is likely?
A. Customers move through the system in less time because utilization is increased.
B. Customers move through the system more slowly because utilization is increased.
C. Utilization is decreased because of the added strain on the system.
D. The average number in the system decreases.
E. There really is no change since arrival rates are offset by service rates.

Difficulty: Medium
TLO: 3
Taxonomy: Knowledge 40. Servers, channels are described with many acronyms except
A. M/P/S
B. M/M/1
C. M/D/1
D. M/M/S

Difficulty: Hard
TLO: 4
Taxonomy: Knowledge

41. A single bay car wash with a Poisson arrival rate and an exponential service time has cars arriving an average of 10 minutes apart, and an average service time of four minutes. The system utilization is:
A. .24
B. .40
C. .67
D. 2.50
E. none of these

AACSB: AS
Difficulty: Medium
TLO: 5
Taxonomy: Application 42. A multiple channel queuing system with a Poisson arrival rate and exponential service time has an average arrival rate of 4 customers per hour and an average service time of 18 minutes per customer. The minimum number of servers required to avoid an overloaded system is:
A. 1
B. 2
C. 3
D. 4
E. 5

AACSB: AS
Difficulty: Hard
TLO: 5
Taxonomy: Application 43. Which one of the following measures of system performance is a key measure?
A. average number of customers waiting in line
B. average time customers wait in line
C. average time customers are in the system
D. average number of customers in the system
E. probability that an arrival will have to wait for service

Difficulty: Hard
TLO: 3
Taxonomy: Knowledge

44. A single channel queuing 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 utilization
B. decrease utilization
C. increase the average waiting time
D. decrease the average waiting time
E. not have any effect since the service time is unchanged

AACSB: AS
Difficulty: Hard
TLO: 4
Taxonomy: Application 45. When the cost incurred by all customers varies, an appropriate queuing model is:
A. single channel, single phase
B. single channel, multiple phase
C. multiple channel, single priority
D. multiple channel, multiple phase
E. multiple channel, multiple priority

Difficulty: Medium
TLO: 4
Taxonomy: Knowledge 46. A ________ is one way of reducing perceived waiting time.
A. Bonus
B. Gatekeeper
C. Reservation
D. Diversion
E. Number calling system

Difficulty: Medium
TLO: 1
Taxonomy: Knowledge

47. An alternative strategy to increase the capacity of a service system is:
A. Reducing the number of arrivals
B. Increasing variability
C. Increasing the processing rate
D. Reducing the number of servers.
E. All of the above are viable options.

Difficulty: Hard
TLO: 1
Taxonomy: Knowledge 48. A queuing system has four crews with three members each. The number of "servers" is:
A. 3
B. 4
C. 7
D. 12
E. 1

Difficulty: Medium
TLO: 4
Taxonomy: Knowledge 49. A multiple channel 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 be under loaded is:
A. 2
B. 3
C. 4
D. 5
E. none of these

AACSB: AS
Difficulty: Hard
TLO: 5
Taxonomy: Application

Essay Questions

50. Customers arrive at a suburban ticket outlet at the rate of 14 per hour on Monday mornings. This can be described by a Poisson distribution. 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. Determine each of the following:
(A) system utilization
(B) average number in line
(C) average time in line
(D) average time in the system

A)

B)

C)

D)

AACSB: AS
Difficulty: Medium
TLO: 5
Taxonomy: Application

51. During the early morning hours, customers arrive at a branch post office at an average rate of 45 per hour (Poisson), while clerks can handle transactions in an average time (exponential) of 4 minutes each. Find:
(A) the average number of customers waiting for service if 6 clerks are used
(B) the minimum number of clerks needed to keep the average time in the system to under 5 minutes
(C) 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) With 6 clerks, Lq is equal to 0.099.
B) Average time in the system for the following staffing levels is as follows:

At least five clerks are needed if the average time in the system is to be less than 5 minutes.
C) Total cost equals clerk cost + waiting cost. Total cost for 4, 5 and 6 clerks is as follows:

The optimum number of clerks is 4.

AACSB: AS
Difficulty: Medium
TLO: 5
Taxonomy: Application

52. Customers filter into a record shop at an average of 1 per minute (Poisson) where the service rate is 15 per hour (Poisson).
Determine the following:
(A) the average number of customers in the system with 8 servers
(B) the minimum number of servers needed to keep the average time in the system to under 6 minutes
A)

B) Average time in the system for the following staffing levels is as follows:

A minimum of six servers is required if average time in the system is to be kept under 6 minutes.

AACSB: AS
Difficulty: Hard
TLO: 5
Taxonomy: Application

53. A department has 5 semiautomatic pieces of equipment which operate for an average of 79 minutes before they must be reloaded. The reloading operation takes an average of 21 minutes per machine. Assume exponential distributions.
(A) What is the minimum number of servers needed to keep the average downtime per cycle to less than 25 minutes?
(B) If 1 server is used, what percentage of time will the machine be down?

A)

At least two servers will be needed to keep average downtime below 25 minutes.

AACSB: AS
Difficulty: Medium
TLO: 5
Taxonomy: Application

54. Two troubleshooters handle service calls for 10 machines. The average time between service requirements is 18 days, and service time averages 2 days. Assume exponential distributions. While running, each machine can produce 1,500 pieces per day. Determine:
(A) the percentage of time troubleshooters are idle
(B) each machine's net productivity
(C) If troubleshooters represent a cost of $150 per day, and machine downtime cost is $600 per day, would another troubleshooter be justified? Explain.

A) Troubleshooters are idle (1 – D) x 100% of time = (1 – 0.258) x 100% = 74.2%
B) J = NF x (1 – X) = 10 x 0.981 x 0.90 = 8.829
Per-machine productivity is J/10 x 1,500 units per day = 1324.35 units per day
C) Adding an additional server would cost $150. The average number of machines down (N – J) would fall from 1.17 to 1.02, representing a savings of $90. The additional server would not be justified.

AACSB: AS
Difficulty: Medium
TLO: 5
Taxonomy: Application

55. A department has 5 machines that each run for an average of 8.4 hours (exponential) before service is required. Service time average is 1.6 hours (exponential).
(A) While running, each machine can produce 120 pieces per hour. With one server, what is the average hourly output actually achieved?
(B) With 2 servers, what is the probability that a machine would be served immediately when it requires service?
(C) If machine downtime cost is $100 per hour per machine, and server time costs $30 per hour, how many servers would be optimal?

A) Given M = 1, J = NF x (1 – X) = 5 x 0.869 x 0.84 = 3.65. Hourly output equals 3.65 x 120 = 438 units.
B) With M = 2, the probability that a machine will have to wait is 0.130. Therefore, the probability that a machine will be served immediately is 1 – 0.130 or 0.87.
C)

The optimal number of servers is 2.

AACSB: AS
Difficulty: Medium
TLO: 5
Taxonomy: Application

56. Customers arrive at a video rental desk at the rate of one per minute (Poisson). Each server can handle 0.40 customers per minute (Poisson).
(A) If there are four servers, determine:
(1) The average time it takes to rent a video tape
(2) The probability of three or fewer customers in the system
(B) What is the minimum number of servers needed to achieve an average time in the system of less than three minutes?
A)
1)

2) If there are four or more customers in the system, customers will have to wait. Therefore, the probability that there will be three or fewer customers in the system equals 1 – Pw.

Therefore, the probability that there will be three or fewer customers in the systems equals 1 – 0.32 or 0.68.
B) Average time in the system for the following staffing levels is as follows:

A minimum of five servers is required if average time in the system is to be kept under three minutes.

AACSB: AS
Difficulty: Medium
TLO: 5
Taxonomy: Application

The following questions refer to this data for a multiple server, priority service queuing model:

Service Rate: 2 per hour (Poisson)
Number of Servers: 5 57. What is the overall average arrival rate?
8 per hour

AACSB: AS
Difficulty: Medium
TLO: 5
Taxonomy: Application 58. What is system utilization?
80 percent

AACSB: AS
Difficulty: Medium
TLO: 5
Taxonomy: Application 59. What is average time in line for a high priority item?
.079 hour

AACSB: AS
Difficulty: Medium
TLO: 5
Taxonomy: Application

60. What is average time in line for a low priority item?
.396 hour

AACSB: AS
Difficulty: Medium
TLO: 5
Taxonomy: Application 61. What is average time in the system for a high priority item?
.579 hour

AACSB: AS
Difficulty: Medium
TLO: 5
Taxonomy: Application 62. What is average time in the system for a low priority item?
.896 hour

AACSB: AS
Difficulty: Medium
TLO: 5
Taxonomy: Application 63. What is the average number of high priority items waiting in line for service?
.237

AACSB: AS
Difficulty: Medium
TLO: 5
Taxonomy: Application

64. What is the average number of low priority items waiting in line for service?
1.98

AACSB: AS
Difficulty: Medium
TLO: 5
Taxonomy: Application 65. What is the average number of all items waiting in line for service?
2.217

AACSB: AS
Difficulty: Medium
TLO: 5
Taxonomy: Application

66. A manager assembled the following information about an infinite source waiting line system: 5 servers, an arrival rate of 6 per hour, and a service time of 20 minutes. The manager has determined that the average number of customers waiting for service is 0.04. Determine each of the following:
(A) the system utilization
(B) the average waiting time in line in minutes
(C) the average time in the system
(D) the average number in the system

A)

B)

C)

D)

AACSB: AS
Difficulty: Medium
TLO: 5
Taxonomy: Application

67. Tourists stop at an information desk at a rate of one every 2 minutes, and answering their questions takes an average of 5 minutes each. There are three employees on duty.
(A) What percentage of time will all three employees be idle at the same time?
(B) If a tourist isn't served immediately, how long on average would the tourist have to wait for service?
(C) What is the probability that there will be at least three tourists "in the system?"
(D) How many employees would be needed to maintain an average waiting time in line of no more than 4 minutes?
\

A)

B)

C) If there are at least three tourists in the system, then other arrivals will have to wait. Therefore, the probability of at least three tourists in the system is equal to

D) Average time in the system for the following staffing levels is as follows:

A minimum of four servers is required if average time in line is to be kept under four minutes.

AACSB: AS
Difficulty: Hard
TLO: 5
Taxonomy: Application

Multiple Choice Questions A bank of ten machines requires regular periodic service. Machine running time and service time are both exponential. Machines run for an average of 44 minutes between service requirements, and service time averages 6 minutes per machine. 68. What is the probability that a machine will have to wait for service with two operators?
A. .654
B. .090
C. .346
D. .910
E. .016

AACSB: AS
Difficulty: Medium
TLO: 5
Taxonomy: Application 69. What is the average machine downtime with two operators?
A. 1.71 minutes
B. 3.46 minutes
C. 6.25 minutes
D. 7.71 minutes
E. 9.46 minutes

AACSB: AS
Difficulty: Medium
TLO: 5
Taxonomy: Application

70. What is the average number of machines down with one operator?
A. 1.49
B. 3.35
C. 4.40
D. 6.65
E. 8.51

AACSB: AS
Difficulty: Medium
TLO: 5
Taxonomy: Application 71. If operators cost $15 per hour in wages and fringe benefits and machine downtime costs $75 per hour in lost production, what is the optimal number of operators for this bank of machines?
A. 1
B. 2
C. 3
D. 4
E. 5

AACSB: AS
Difficulty: Medium
TLO: 5
Taxonomy: Application A bank is designing a new branch office and needs to determine how much driveway space to allow for cars waiting for drive up teller service. The drive up service will have three tellers and a single waiting line. At another branch of the bank in a similar setting, the average service time for drive up tellers is 4 minutes per customer and average arrival rate is 36 customers per hour. It is expected that the new bank will have similar characteristics.

72. How many spaces should be provided to have a 96% probability of accommodating all of the waiting cars?
A. 10
B. 11
C. 12
D. 13
E. 14

AACSB: AS
Difficulty: Medium
TLO: 5
Taxonomy: Application A casino restaurant at popular Colorado casino provides priority service to player's card holders. The restaurant has 10 tables or booths where customers may be seated. The service time (time a booth or table is occupied) averages 40 minutes once a party is seated. The customer arrival rate is 12 parties per hour, with the parties being equally divided between card holders and people without player's cards. 73. What is the average time that player's card holders wait to be seated?
A. approx. 4 minutes
B. approx. 8 minutes
C. approx. 12 minutes
D. approx. 15 minutes
E. approx. 2 minutes

AACSB: AS
Difficulty: Medium
TLO: 5
Taxonomy: Application

74. What is the average time that parties without player's cards wait to be seated?
A. approx. 4 minutes
B. approx. 8 minutes
C. approx. 12 minutes
D. approx. 15 minutes
E. approx. 2 minutes

AACSB: AS
Difficulty: Medium
TLO: 5
Taxonomy: Application 75. On average, how many parties with player's cards are waiting to be seated?
A. 0.52
B. 0.41
C. 0.88
D. 1.23
E. 1.75

AACSB: AS
Difficulty: Medium
TLO: 5
Taxonomy: Application 76. On average, how many parties without player's cards are waiting to be seated?
A. 0.52
B. 0.41
C. 0.88
D. 1.23
E. 1.75

AACSB: AS
Difficulty: Medium
TLO: 5
Taxonomy: Application

77. On average, how much longer do parties without player's cards spend in the system, compared to parties with the player's cards?
A. 2 minutes
B. 4 minutes
C. 8 minutes
D. 10 minutes
E. It is impossible to say without more information.

AACSB: AS
Difficulty: Medium
TLO: 5
Taxonomy: Application