Deck 9: Queuing Models

Question 1

If the service times are not memoryless, then the Erlangian distribution might be a better basis than the exponential distribution for modeling service.

Answer 1

True

Question 2

Where would one most likely face a tandem queue?

A)A local post office.
B)A supermarket.
C)A cafeteria.
D)A bank.

Answer 2

C

Question 3

The average waiting time in the system is less than the sum of the average waiting time in the queue plus the average service time because some customers do not have to wait in the queue.

Answer 3

False

Question 4

Although service times may be relatively constant at sequential work stations in an assembly line, the line still may be treated as a tandem queue.

Question 5

The steady state service measure formulas for the M/G/k/k queue are the same as those for the M/M/k/k queue.

Question 6

One method for reducing the randomness of customer arrivals at a retail business is to allow for customer appointments.

Question 7

Necessary assumptions underlying a Poisson arrival process do not include:

A)orderliness.
B)homogeneity.
C)independence.
D)stationarity.

Question 8

The probability distribution for the arrival process can be estimated if we know either the time between customer arrivals or the number of customers in a given time interval.

Question 9

Airline passenger arrivals at U.S.Customs counters, at a
port-of-entry airport, would not likely be modeled as Poisson due to arriving in groups.

Question 10

"Balking" is:

A)refusing service.
B)leaving the queue.
C)refusing to enter the queue.
D)refusing to leave after service.

Question 11

The size of the population of potential customers may have an impact on the validity of the Poisson arrival pattern assumption.

Question 12

For an M/M/k queue to reach steady state, the service rate for each server μ must be greater than the arrival rate λ.

Question 13

Random arrival processes must be modeled using continuous
probability distributions, not discrete ones.

Question 14

In order to achieve steady state performance in a queuing system, the sum of the effective service rates of all servers must:

A)exceed the effective arrival rate of all customers.
B)equal the effective arrival rate of all customers.
C)be less than the effective arrival rate of all customers.
D)be exponentially distributed.

Question 15

Which of the following would best be characterized as a Poisson arrival process?

A)Football game attendees,
B)Tax returns received by a regional IRS office.
C)Incoming phone calls to a business switchboard.
D)Ladies visiting a hair salon.

Question 16

The optimal situation (in terms of minimizing the average number of customers in the queue) in an M/M/1 queuing system is when the arrival rate λ exactly equals the service rate μ.

Question 17

μ, in the service segment of a queuing process, is the average number of customers actually served per unit of time.

Question 18

The priority rule chosen for determining the next customer to be served affects both waiting time variance and average customer
waiting time.

Question 19

A Poisson distribution with mean λ is equivalent to an
exponential distribution with mean 1/λ.

Question 20

"Jockeying" occurs when a customer in the waiting line gets upset with the time the process is taking and leaves the system.

Question 21

Suppose that an office has one secretary who can put up to two callers on hold while speaking to a third caller.(If two callers are on hold, additional callers will get a busy signal and will not
Call back.) If the arrival rate of calls follows a Poisson
Distribution with a mean rate of 20 per hour and the average length
Of a telephone conversation is 2 minutes, the average number of
Callers who will be on hold is approximately:

A)2.0.
B)1.3333
C)1.0154.
D).4308.

Question 22

What does the Excel formula EXPONDIST(A10,B3,FALSE) return?

Question 23

You go to your local hospital for your complete annual physical exam.From a queuing standpoint, you are facing:

A)single server; single queue.
B)multiple servers; single queue.
C)multiple servers; multiple queues.
D)tandem queues.

Question 24

Using Little's formula, if we know the mean arrival rate, mean service rate, and the average time a customer spends in the queue (W_q), what else can we calculate?

Question 25

In queuing analysis, the exponential distribution is a special case of which other distribution?

A)Erlangian.
B)Poisson.
C)normal.
D)Markovian.

Question 26

In an M/M/1 queuing system, ? = 6, and the system is idle 40%
of the time.What is the average service rate, ??

Question 27

The Pollaczek-Khintchine formula comes from the study of:

A)M/G/1 queues.
B)simulation.
C)M/M/1 queues.
D)Markov chains.

Question 28

Ray's Barber Shop has 3 barber's chairs, and 8 seats for waiting customers.The greatest number of people ever waiting for a haircut, in Ray's experience, is 6.For analytical purposes, then, Ray's queue length is:

A)infinite.
B)6.
C)8.
D)11.

Question 29

Suppose a penny arcade worker is in charge of keeping ten machines in operation.The failure rate of each machine follows an exponential distribution with a mean time of ten hours and the time required to repair each machine follows an exponential distribution with a mean time of thirty minutes.Over the long run, approximately what percentage of machines, on average, will be operating?

A)76.5%
B)92.4%
C)53.8%
D)46.2%

Question 30

If ? is the non-constant average arrival rate, and ? (or k? with multiple servers) is the average service rate, why does a queuing system not approach maximum efficiency when these two values are approximately the same?

Question 31

Which of the following is not a common steady state performance measure?

A)The probability that a customer does not have to wait in the queue.
B)The average number of customers who do not have to wait in the queue.
C)The probability all servers are idle.
D)The average time a customer spends between entering and leaving the system.

Question 32

A Markovian queuing process has a(n) __________ arrival pattern, and a(n)__________ service pattern.

A)Poisson; Poisson.
B)Poisson; exponential.
C)exponential; Poisson.
D)exponential; exponential.

Question 33

In the exponential distribution of service times, the mean customer service time equals the:

A)variance of customer service times.
B)standard deviation of customer service times.
C)median customer service time.
D)most likely customer service time.

Question 34

Homogeneity means:

A)all customers arrive according to the same pattern and receive the same service.
B)past service time does not affect future service time.
C)processes reach steady state.
D)all servers are available as long as the queue functions.

Question 35

If a service facility changes from a first-come, first-served basis to a random basis in selecting the next customer to be served, one should expect average customer waiting time to:

A)decrease slightly.
B)remain constant.
C)increase slightly.
D)increase dramatically.

Question 36

You are studying service times at Derman's Department Store, which is open 7 days a week from 10:00 AM to 9:00 PM.Why might you ignore data from 10:00 to 10:30 AM?

A)Insufficient sample size.
B)Start up bias.
C)The doors do not open at exactly 10:00 AM.
D)The data do not fit the hypothesis.

Question 37

Which of the following is a basic component of a queuing system?

A)Poisson distribution.
B)Waiting in a queue.
C)Priority rules.
D)Exponential distribution.

Question 38

The exponential distribution is:

A)generally discrete.
B)never symmetrical.
C)usually symmetrical.
D)not related to the Poisson distribution.

Question 39

In a situation where the distribution of service times is assumed exponential, suppose the probability that service time is under five minutes is 0.40.If a given customer has already had five minutes of service, the probability that he/she will have service totally completed in less than ten minutes, is:

A)0.40.
B)greater than 0.40.
C)less than 0.40.
D)indeterminate.

Question 40

A local dry cleaning establishment is open from 7
a.m.to 7 p.m., weekdays.The average arrival rate of customers is 15 per hour, both from 7 to 10
a.m., and from 4 to 7 p.m.In between, it is 9 per hour.In all time periods, the Poisson assumption for the arrival process seems to be valid.Can this be treated as a queuing system, with ? = 12 ([15 + 9]/2)?

Question 41

Explain Kendall's notation: G/M/3/10/20.

Question 42

Ajax, Inc.specializes in the maintenance and repair of all types of electronic products.Tools and test equipment needed by its many skilled technicians on various diverse jobs must be drawn from and returned to a centrally-located tool room.Technicians arrive at the tool room at the rate of 20 per hour according to a Poisson process and earn $16 per hour per person.The company can hire tool room clerks at $10 per hour.It is estimated that it takes an average of 4 minutes for a clerk to serve a technician and service time follows an exponential distribution.Determine the optimal number of tool room clerks that Ajax should hire.

Question 43

What is the formula for the effective arrival rate of an M/M/k/F queue?

Question 44

If the standard deviation of service time in an M/G/1 queue is zero, what type of system does it become? What if ? = 1/??

Question 45

The computer help desk at Averill University receives an average of 40 calls per hour and calls come in according to a Poisson distribution. The average time a technician takes to diagnose a problem is three minutes and twenty seconds, and the service time follows an exponential distribution. For 60% of the callers the diagnosis is satisfactory, but for 40% of the callers, the technician must transfer the call to a specialist. The time a
caller speaks to a specialist follows an exponential distribution
with a mean of five minutes. For ninety five percent of callers
transferred to a specialist, the problem is solved, but five percent
of callers will need an on-site visit from a service technician.
The average time a technician takes to fix a computer on-site is
forty minutes with a standard deviation of ten minutes.

The help desk operation has three technicians answering the incoming
calls, two specialists handling calls, and one on-site technician.
If a caller needs to have on-site service, determine the average
time it will take to have the service completed form the time the
initial call is made.

Question 46

For the post-Christmas gift returns, Paver's store added a second clerk on December 26.Paver's expects 10 customers an hour, and the mean service time is 10 minutes.Assuming an M/M/2 queue and a single waiting line, compute the average time a customer spends in the queue.What if there were two lines with jockeying allowed?

Question 47

Harry and Larry have opened up an automated carwash near the edge of town.There is a single service lane, and cars line up in asingle line to enter the carwash.No special services are offered,and the time required for each vehicle to go through the facility is exactly three minutes.Throughout the day, customers arrive
independently and largely at random at an average rate of twelve per hour.Because of the location, even when the queue is full,potential customers may line up in the adjacent side street.
A.What percentage of time is the carwash idle?
B.What is the average number of waiting vehicles?
C.What will be the average elapsed time between when a vehicle enters and leaves the system?
D.What percentage of time is the carwash idle?
E.What is the average number of waiting vehicles?
F.What will be the average elapsed time between when a vehicle enters and leaves the system?

Question 48

What is the Kendall notation for a queue withmaximum queue length equal to the number of servers; multiple servers working at the same, non-exponential rate;and customer interarrival time exponential.

Question 49

For an M/G/1 queue, ? = 12 per hour, ? = 24 per hour, and the standard deviation of the service time ? = .05.Calculate the average number of customers in the system, average time a customer spends in the system, and the probability that there are exactly two customers in the system.

Question 50

For many queue types, Pw, the probability a customer must wait for service, = ?, the server utilization rate.For what types of queues is this not true?

Question 51

The new community of Lemon Heights is planning to set up a paramedic station.It is estimated that calls will come into this station according to a Poisson distribution, and the station receives an average of twenty calls a day.The time an ambulance is out responding to a call follows an exponential distribution with a mean time of one hour and thirty minutes.If no ambulance is available, an ambulance from a nearby town will be dispatched, but this will significantly increase the response time.Due to the potentially tragic consequences associated with not having an ambulance readily available when a call comes in, the city council has mandated that the probability of this happening should be no more than .005.Determine how many ambulances the paramedic station should purchase.

Question 52

The Red Rock School District has six buses in its fleet.Maintenance of the fleet is handled by Joe Clem, a mechanic who works for the district.The time between bus failures follows an exponential distribution with a mean of twenty days.During the time a bus is out of commission, the district must lease another bus at a cost of $80 per day.Joe has put in for retirement and the district is considering hiring
either Tom Meyers or Andy Johnson.Based on a skills assessment test, it is estimated that Tom can repair a bus in an average of 10
hours, while Andy will take an average of 8 hours.Tom wants a
salary equivalent to $160 per working day, while Andy wants a salary
equivalent to $170 per working day.A working day lasts 8 hours, and
there are 200 working days per year.Which employee should the
district hire? Give your reasons.

Question 53

Customers arrive at the First Fidelity Bank branch on Fridayafternoon according to a Poisson process at a mean rate of 45 per
hour.The average time a teller takes to serve a customer is two
and a half minutes and service time follows an exponential
distribution.At present, the bank has two tellers on staff to
serve customers during this time.
A.Determine the average time a customer will wait in line before
being served.
B.Determine the probability an arriving customer will have to wait in line.
C.Determine the average number of customers in the system.
D.Suppose that the bank would like the average time a customer
spends waiting in line to be three minutes or less.What is the
fewest number of tellers they should employ to meet this goal?

Question 54

Given these parameters: ? = 25 per hour, ? = 30 per hour, and W_q = .3 hours, calculate the average number of customers in the system, average number of customers in the queue, and the average time a customer spends in the system.