Deck 13: Waiting Line Models

Question 1

Waiting lines will never form in a service facility if the overall system capacity is much more than overall processing requirements.

Answer 1

False

Question 2

Variability in demand and/or time to service is the main reason for the formation of waiting lines in front of service facilities.

Answer 2

True

Question 3

If the overall system capacity is less than the overall processing requirements, and if the facility is open continuously, then very long waiting lines will be formed as time passes.

Answer 3

True

Question 4

The combined cost of service capacity and customer waiting cost is generally ' $U$ ' shaped.

Question 5

Calling population' refers to customers who will definitely arrive during the day, though precise time of arrival may be unknown.

Question 6

Customers arrive at a bus station at the rate of 5 per minute following Poisson distribution. If one looks at the distribution of inter-arrival times of customers, it would also follow Poisson distribution.

Question 7

If arrivals of customers to a grocery store follow Poisson distribution, time between successive arrivals to the store follows exponential distribution.

Question 8

A random variable that follows Poisson distribution refers to the number of events per unit time.

Question 9

A histogram of probability against the number of events in a given time unit is unimodal.

Question 10

A statistical test known as t-test can be used to check whether the given data follows exponential distribution.

Question 11

In a single server queuing problem with Poisson arrival and exponential service, first come, first served service discipline will produce a shorter queue length than last come, first served service discipline.

Question 12

Multiple phase and multiple channel are two different ways of looking at the same waiting line problems.

Question 13

The average waiting time for a single channel, two-phase system will be the same as a two-channel, single phase system as long as the arrival and service distributions are identical.

Question 14

Service time at a hair salon follows exponential with rate 2 per hour. Probability of service time taking anywhere from 0 to 0.5 hour will be greater than the probability that the service time is between 0.5 to 1.0 hour.

Question 15

In a single server waiting line system with Poisson arrival and exponential service, the average number in the system will always be greater than the average number in the queue.

Question 16

In a single server waiting line system with Poisson arrival and exponential service, the average waiting time in the system will always be less than the average waiting time in the queue.

Question 17

In a single server waiting line system with Poisson arrival and exponential service, as the average utilization increases, the average number in the system will increase.

Question 18

In theory, for a single server Poisson arrival exponential service waiting line system with server utilization less than 1 and unlimited source, the number of units in the system can be infinite.

Question 19

Typically, the number of waiting spaces needed in a waiting line system may be found by finding the number of spaces needed to make the probability that the waiting line will exceed the space available to less than $5 \%$ or $1 \%$ , as desired by the manager.

Question 20

In a single server Poisson arrival general service system, with server utilization less than 1 , the average number in the system goes up as the variance of service time goes down.

Question 21

A single server Poisson arrival constant service time system, with server utilization less than 1 , will always have a smaller queue length as compared to a single server Poisson arrival general service system with same server utilization.

Question 22

A single server Poisson arrival exponential service system with $\lambda / \mu<1.0$ will always have a smaller queue length, as compared to a single server Poisson arrival exponential service system with same $\lambda$ and $\mu$ but limited system capacity.

Question 23

In a single server Poisson arrival exponential service system with $\lambda / \mu<1$ but limited system capacity, the average queue length will increase as the system capacity increases.

Question 24

In the multi-channel priority service model, service on low priority unit can be interrupted when a high priority unit arrives.

Question 25

In the multi-channel, priority service model, if the arrival rate of the highest priority class decreases, the average waiting time for units of that class will also decrease.

Question 26

In the queuing systems with Poisson arrival and exponential service (single or multi-channel), the average time in the system is equal to sum of the average time in waiting line and average service time.

Question 27

Unlike the basic single channel Poisson arrival and exponential service model, the customer arrival rate in the finite number in the system model, depends on the number in the system.

Question 28

In queuing systems with variable arrival and service rate model with $\lambda / \mu<1.0$ , the average waiting time in the system increases rapidly as $\lambda / \mu$ approaches 1.0.

Question 29

Assumptions of Poisson distribution include all of the following except

A) expected number of occurrences of an event is proportional to the length of the interval
B) probability of occurrence of events in two intervals of the same length will be equal
C) probability of occurrence of events in the interval $t$ to $t+1$ is not affected by occurrence history in the interval $\mathrm{t}-1$ to $\mathrm{t}$
D)probability of occurrence of events in the interval $t$ to $t+3$ is not affected by the occurrence of events in the interval $t-1$ to $t+1$

Question 30

Customers arrive at a bus station at the rate of 5 per minute following Poisson distribution. What is the probability of 3 arrivals in a one-minute interval?

A) $\frac{e^{-5} 5^{3}}{3 !}$
B) $\frac{e^{-3} 5^{5}}{3 !}$
C) $\frac{e^{-5} 5^{3}}{5 !}$
D) $\frac{e^{5} 5^{-3}}{5 !}$

Question 31

Customers arrive at a bus station at the rate of 5 per minute following Poisson distribution. What is the probability of 6 arrivals in a two-minute interval?

A) $\frac{e^{-5} 5^{3}}{3 !}$
B) $\frac{e^{-10} 10^{6}}{6 !}$
C) $\frac{e^{-5} 5^{3}}{5 !}$
D) $\frac{e^{-6} 10^{-10}}{10 !}$

Question 32

Time between consecutive arrivals of customers to a grocery store is exponential with a rate of 3 per minute. What is the probability that a time of 2 minutes will elapse between the arrivals of two consecutive customers?

A) $e^{-6}$
B) $e^{-3}$
C) $1-e^{-6}$
D) $1-e^{-3}$

Question 33

Two key parameters in any single channel, single phase queuing system are

A) waiting time and service time
B) arrival rate and number of servers
C) system utilization and average queue length
D) mean arrival rate and mean service rate

Question 34

Service time in a coffee shop follows exponential distribution with an average service time of 6 minutes. Probability that a service takes between 4 and 8 minutes is equal to

A) 0.248
B) 0.513
C) 0.736
D) 0.263

Question 35

Service time in a coffee shop follows exponential distribution with an average service time of 6 minutes. Probability that a service takes more than 6 minutes is equal to

A) 0.500
B) 0.368
C) 0.632
D) 0.263

Question 36

In a single channel manual soft cloth wash, cars arrive following Poisson distribution at the rate of 8 per hour. It takes an average of 6 minutes to wash a car, and the service time follows exponential distribution. The system utilization would be

A) $75 \%$
B) $80 \%$
C) $133.33 \%$
D) $125 \%$

Question 37

In a single channel manual soft cloth wash, cars arrive following Poisson distribution at the rate of 8 per hour. It takes an average of 6 minutes to wash a car, and the service time follows exponential distribution. The average number of cars in the system will be

A) 0.8
B) 3.2
C) 5
D) 4

Question 38

In a single channel manual soft cloth wash, cars arrive following Poisson distribution at the rate of 8 per hour. It takes an average of 6 minutes to wash a car and the service time follows exponential distribution. The probability of exactly 2 customers in the system will be

A) 0.8
B) 0.2
C) 0.16
D) 0.128

Question 39

A grocery store has two identical checkouts; each can serve an average of 20 customers per hour following an exponential distribution. Customers ready to check out arrive at a common checkout line following Poisson distribution with a rate of 30 per hour. The utilization of the system will be:

A) 1.5
B) 0.75
C) 0.667
D) 0.25

Question 40

A grocery store has two identical checkouts each can serve an average of 20 customers per hour following an exponential distribution. Customers ready to checkout arrive at a common checkout line following Poisson distribution with a rate of 30 per hour. For this system, it is known that $\mathrm{L}_{\mathrm{q}}=1.929$ and $\mathrm{P}_{0}=0.143$ . The average number in the system will be

A) 1.929
B) 2
C) 1.643
D) 3.429

Question 41

In Mr. and Mrs. Phillips' grocery store, customers join the single checkout line following Poisson distribution at a rate of 15 per hour. The checkout clerk can handle 20 customers per hour, and the service time follows exponential distribution. The average number in the queue $\left(\mathrm{L}_{\mathrm{q}}\right)=2.25$ and utilization is 0.75 . The maximum number of customers waiting in line with a probability of 0.95 will be

A) 8
B) 9
C) 18
D) 19

Question 42

Speedy Delivery, a new mail service firm, rents docking and unloading bays from its friendly competitor at a cost of $\$ 200$ per hour, which includes the cost of unloading the crew. Each bay can unload at a rate of 2 trucks per hour. Trucks arrive at its facility at the rate of 4 per hour, and the trucking company charges $\$ 300$ per hour of time spent by the truck in the system. The arrivals are Poisson, and service is exponential. At present the firm rents 3 bays. From appropriate table, $\mathrm{L}_{\mathrm{q}}$ was found to be 0.45 . The total hourly cost of bay and truck unloading and waiting cost will be

A) $\$ 735$
B) $\$ 1,035$
C) $\$ 600$
D) $\$ 435$

Question 43

In a single server, Poisson arrival, constant service time system with $\lambda=9$ per hour and ${ }^{\mu}=10$ per hour, $\mathrm{L}_{\mathrm{q}}$ will be equal to

A) 8.1
B) 4.05
C) 5.0
D) 9.0

Question 44

In a single server, Poisson arrival, constant service time system with $\lambda=9$ per hour and ${ }^{\mu}=10$ per hour, $\mathrm{L}$ will be equal to

A) 8.1
B) 4.05
C) 5.0
D) 9.0

Question 45

In a single server, Poisson arrival, constant service time system with $\lambda=9$ per hour and ${ }^{\mu}=10$ per hour, $\mathrm{W}$ (in minutes) will be equal to

A) $5 / 9$
B) $4.05 / 9$
C) 33.3
D) 60

Question 46

In a single server, Poisson arrival, exponential service system with $\lambda=8$ and ${ }^{\mu}=10$ and a system capacity of $3, \mathrm{P}_{0}$ will be equal to

A) 0.2
B) 0.338
C) 0.8
D) 0.173

Question 47

In a single server, Poisson arrival, exponential service system with $\lambda=8, \mu=10$ , and a system capacity of $3, \mathrm{P}_{3}$ will be equal to

A) 0.2
B) 0.338
C) 0.8
D) 0.173

Question 48

In a single channel automatic car wash, cars arrive following Poisson distribution at the rate of 20 per hour. It takes an average of 2 minutes to wash a car, and the service time follows exponential distribution. Find the following operating characteristics of the system
(A) average number of cars in the system.
(B) average waiting time in the system
(C) the probability of having 3 cars in the system

Question 49

A new 24-hour pharmacy is opened in a busy intersection by the Blue-Green Pharmacy chain. Customers come into the system at a rate of 15 per hour; at all times, there are 2 pharmacists who can each serve at the rate of 10 per hour. Arrivals are Poisson, and service is exponential. Find the following operating characteristics of the system
(A) average number of customers in the system
(B) average waiting time in the system
(C) the probability of having 3 customers in the system

Question 50

A new 24-hour gas station located in a crowded intersection in Piscataway, New Jersey, has cars coming in at the rate of 30 per hour. Each of the two attendants can service cars at the rate of 30 per hour. The owner would like to know the probability distribution of the number of cars in the system, so that plans can be launched for adding staff (if necessary) if the system has a $5 \%$ chance of having 5 or more cars. Find Pi for $\mathrm{I}=0,1,2,3 \ldots$ And find the probability of the system having 5 or more cars.