Deck 15: Waiting Line Models

Question 1

The manner in which units receive their service,such as FCFS,is the

A)queue discipline.
B)channel.
C)steady state.
D)operating characteristic.

Answer 1

A

Question 2

Queue discipline refers to the assumption that a customer has the patience to remain in a slow moving queue.

Answer 2

False

Question 3

A waiting line situation where every customer waits in the same line before being served by the same server is called a single server waiting line.

Answer 3

False

Question 4

Performance measures dealing with the number of units in line and the time spent waiting are called

A)queuing facts.
B)performance queues.
C)system measures.
D)operating characteristics.

Question 5

For all waiting lines,P₀ + P_w = 1.

Question 6

For many waiting line situations,the arrivals occur randomly and independently of other arrivals and it has been found that a good description of the arrival pattern is provided by

A)a normal probability distribution.
B)an exponential probability distribution.
C)a uniform probability distribution.
D)a Poisson probability distribution.

Question 7

Operating characteristics formulas for the single-channel queue do NOT require

A) $\lambda$ > $\mu$ .
B)Poisson distribution of arrivals.
C)an exponential distribution of service times.
D)an FCFS queue discipline.

Question 8

If arrivals occur according to the Poisson distribution every 20 minutes,then which is NOT true?

A) $\lambda$ = 20 arrivals per hour
B) $\lambda$ = 3 arrivals per hour
C) $\lambda$ = 1/20 arrivals per minute
D) $\lambda$ = 72 arrivals per day

Question 9

In a waiting line situation,arrivals occur,on average,every 10 minutes,and 10 units can be received every hour.What are $\lambda$ and $\mu$ ?

A) $\lambda$ = 10, $\mu$ = 10
B) $\lambda$ = 6, $\mu$ = 6
C) $\lambda$ = 6, $\mu$ = 10
D) $\lambda$ = 10, $\mu$ = 6

Question 10

Which of the following can NOT be found by the queuing formulas presented in the textbook?

A)the probability that no units are in the system.
B)the average number of units in the system.
C)the maximum time a unit spends in the system.
D)the average time a unit spends in the system.

Question 11

The arrival rate in queuing formulas is expressed as

A)the mean time between arrivals.
B)the minimum number of arrivals per time period.
C)the mean number of arrivals per channel.
D)the mean number of arrivals per time period.

Question 12

For an M/M/1 queuing system,if the service rate,µ,is doubled,the average wait in the system,W,is cut in half.

Question 13

What queue discipline is assumed by the waiting line models presented in the textbook?

A)first-come first-served.
B)last-in first-out.
C)shortest processing time first.
D)No discipline is assumed.

Question 14

In a multiple channel system

A)each server has its own queue.
B)each server has the same service rate.
C) $\mu$ > $\lambda$
D)All of the alternatives are correct.

Question 15

The total cost for a waiting line does NOT specifically depend on

A)the cost of waiting.
B)the cost of service.
C)the number of units in the system.
D)the cost of a lost customer.

Question 16

Models with a finite calling population

A)have an arrival rate independent of the number of units in the system.
B)have a service rate dependent on the number of units in the system.
C)use the size of the population as a parameter in the operating characteristics formulas.
D)All of the alternatives are correct.

Question 17

Decision makers in queuing situations attempt to balance

A)operating characteristics against the arrival rate.
B)service levels against service cost.
C)the number of units in the system against the time in the system.
D)the service rate against the arrival rate.

Question 18

The assumption of exponentially distributed service times indicates that

A)37% of the service times are less than the mean service time.
B)50% of the service times are less than the mean service time.
C)63% of the service times are less than the mean service time.
D)service time increase at an exponential rate as the waiting line grows.

Question 19

Use of the Poisson probability distribution assumes that arrivals are not random.

Question 20

Little's flow equations

A)require Poisson and exponential assumptions.
B)are applicable to any waiting line model.
C)require independent calculation of W,L,W_q,and L_q.
D)All of the alternatives are correct.

Question 21

Before waiting lines can be analyzed economically,the arrivals' cost of waiting must be estimated.

Question 22

For an M/M/k system,the average number of customers in the system equals the customer arrival rate times the average time a customer spends waiting in the system.

Question 23

In developing the total cost for a waiting line,waiting cost takes into consideration both the time spent waiting in line and the time spent being served.

Question 24

For a single-server queuing system,the average number of customers in the waiting line is one less than the average number in the system.

Question 25

Waiting line models describe the transient-period operating characteristics of a waiting line.

Question 26

In waiting line systems where the length of the waiting line is limited,the mean number of units entering the system might be less than the arrival rate.

Question 27

The post office uses a multiple channel queue,where customers wait in a single line for the first available window.If the average service time is 1 minute and the arrival rate is 7 customers every five minutes,find,when two service windows are open,
a.the probability both windows are idle.
b.the probability a customer will have to wait.
c.the average time a customer is in line.
d.the average time a customer is in the post office.

Question 28

The time to process a registration at the Sea View Resort follows the exponential distribution and has a mean of 6 minutes.
a.What is the probability of a registration time shorter than 3 minutes?
b.What is the probability of a registration time shorter than 6 minutes?
c.What is the probability of a registration time between 3 and 6 minutes?

Question 29

Adding more channels always improves the operating characteristics of the waiting line and reduces the waiting cost.

Question 30

The Sea View Resort uses a multiple-channel queue registration system.If the average service time is 8 minutes,there are three registration clerks,and guests arrive at the rate of one every 5 minutes,find
a. $\lambda$ and $\mu$ .
b.the probability all three clerks are idle.
c.the probability a guest will have to wait.
d.the average time a customer is in line.
e.the average number of customers in line.

Question 31

During summer weekdays,boats arrive at the inlet drawbridge according to the Poisson distribution at a rate of 3 per hour.In a 2-hour period,
a.what is the probability that no boats arrive?
b.what is the probability that 2 boats arrive?
c.what is the probability that 8 boats arrive?

Question 32

A multiple-channel system has more than one waiting line.

Question 33

Queue discipline refers to the manner in which waiting units are arranged for service.

Question 34

If some maximum number of customers is allowed in a queuing system at one time,the system has a finite calling population.

Question 35

When a waiting system is in steady-state operation,the number of units in the system is not changing.

Question 36

Little's flow equations indicate that the relationship of L to L_q is the same as that of W to W_q.

Question 37

In a multiple channel system it is more efficient to have a separate waiting line for each channel.

Question 38

For a single-channel waiting line,the utilization factor is the probability that an arriving unit must wait for service.

Question 39

When blocked customers are cleared,an important decision is how many channels to provide.

Question 40

If service time follows an exponential probability distribution,approximately 63% of the service times are less than the mean service time.

Question 41

Two new checkout scanning systems are under consideration by a retail store.Arrivals to the checkout stand follow the Poisson distribution with $\lambda$ = 2 per minute.The cost for waiting is $18 per hour.The first system has an exponential service rate of 5 per minute and costs $10 per hour to operate.The second system has an exponential service rate of 8 per minute and costs $20 per hour to operate.Which system should be chosen?

Question 42

Andy Archer,Ph.D. ,is a training consultant for six mid-sized manufacturing firms.On the average,each of his six clients calls him for consulting assistance once every 25 days.Andy typically spends an average of five days at the client's firm during each consultation.
Assuming that the time between client calls follows an exponential distribution,determine the following:
a.the average number of clients Andy has on backlog
b.the average time a client must wait before Andy arrives to it
c.the proportion of the time Andy is busy

Question 43

Circle Electric Supply is considering opening a second service counter to better serve the electrical contractor customers.The arrival rate is 10 per hour.The service rate is 14 per hour.If the cost of waiting is $30 and the cost of each service counter is $22 per hour,then should the second counter be opened?

Question 44

The postmaster at the Oak Hill Post Office expects the mean arrival rate of people to her customer counter will soon increase by fifty percent due to a large apartment complex being built.Currently,the mean arrival rate is 15 people per hour.The postmaster can serve an average of 25 people per hour.By what percentage must the postmaster's mean service rate increase when the apartment complex is completed in order that the average time spent at the post office remains at its current value?

Question 45

For an M/G/1 system with $\lambda$ = 20 and $\mu$ = 35,with $\sigma$ = .005,find
a.the probability the system is idle.
b.the average length of the queue.
c.the average number in the system.

Question 46

For an M/G/1 system with $\lambda$ = 6 and $\mu$ = 9,with $\sigma$ = .03,find
a.the probability the system is idle.
b.the average length of the queue.
c.the average number in the system.

Question 47

The insurance department at Shear's has two agents,each working at a mean speed of 8 customers per hour.Customers arrive at the insurance desk at a mean rate of one every six minutes and form a single queue.
Management feels that some customers are going to find the wait at the desk too long and take their business to Word's,Shear's competitor.
In order to reduce the time required by an agent to serve a customer Shear's is contemplating installing one of two minicomputer systems: System A which leases for $18 per day and will increase an agent's efficiency by 25%;or,System B which leases for $23 per day and will increase an agent's efficiency by 50%.Agents work 8-hour days.
If Shear's estimates its cost of having a customer in the system at $3 per hour,determine if Shear's should install a new minicomputer system,and if so,which one.

Question 48

Arrivals at a box office in the hour before the show follow the Poisson distribution with $\lambda$ = 7 per minute.Service times are constant at 7.5 seconds.Find the average length of the waiting line.

Question 49

A company has tool cribs where workmen draw parts.Two men have applied for the position of distributing parts to the workmen.George Fuller is fresh out of trade school and expects a $6 per hour salary.His average service time is 4 minutes.John Cox is a veteran who expects $12 per hour.His average service time is 2 minutes.A workman's time is figured at $10 per hour.Workmen arrive to draw parts at an average rate of 12 per hour.
a.What is the average waiting time a workman would spend in the system under each applicant?
b.Which applicant should be hired?

Question 50

The Quick Snap photo machine at the Lemon County bus station takes four snapshots in exactly 75 seconds.Customers arrive at the machine according to a Poisson distribution at the mean rate of 20 per hour.On the basis of this information,determine the following:
a.the average number of customers waiting to use the photo machine
b.the average time a customer spends in the system
c.the probability an arriving customer must wait for service.

Question 51

The 8 students in a seminar class must come to the professor's office to turn in a paper and give a 5-minute oral summary.Assume there is a service rate of 10 per hour and adequate time is available for all.The arrival rate for each unit is 5 per hour.What is the probability there is no one in the office or waiting when you come?

Question 52

Quick Clean Rooter cleans out clogged drains.Due to the competitive nature of the drain cleaning business,if a customer calls Quick Clean and finds the line busy,they immediately try another company and Quick Clean loses the business.
Quick Clean management estimates that on the average,a customer tries to call Quick Clean every three minutes and the average time to take a service order is 200 seconds.The company wishes to hire enough operators so that at most 4% of its potential customers get the busy signal.
a.How many operators should be hired to meet this objective?
b.Given your answer to a),what is the probability that all the operators are idle?