Question

Consider a customer arrival process that is a Poisson process. To find the probabilities described below, which of the following random variable selections (as Poisson, Exponential or k-Erlang) is correct?

to find the probability that the time between the 2nd and 3rd customer arrivals is 5 minutes, use a k-Erlang random variable with k>1 |
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to find the probability that 10 customers arrive during a 30-minute period, use a k-Erlang random variable |
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to find the probability that the total time elapsed until the 5th customer arrival is 30 minutes, use an Exponential random variable |
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to find the probability that more than 7 minutes pass before the 3rd customer arrival, use a k-Erlang random variable |

Answer #1

1. Here we will use an exponential distribution. When the arrival process is poisson, then the time between two arrivals follows an exponential distribution.

2. Here we will use a poisson distribution. In the customer arrival poisson process we will be told the average no. Of customers arriving in a particular time (say 1 hour). We will convert it to 30 minute time period and then find P(X=10) by putting it in the poisson PDF.

3. This will be found using k Erlang random variable. It is the sum of the exponential random variables.

4. This will be found using k Erlang random variable.

Select an arrival (Poisson) process on any time interval (eg.:
second, minute, hour, day, week, month, etc….) as you like.
Possible arrival processes could be arrival of signal, click,
broadcast, defective product, customer, passenger, patient, rain,
storm, earthquake etc.[Hint: Poisson and exponential distributions
exits at the same time.]
Collect approximately n=30 observations per unit time interval.
.[Hint: Plot your observations. If there is sharp increase or
decrease then you could assume that you are observing arrivals
according to proper Poisson...

Q1.
Select an arrival (Poisson) process on any time interval (eg.:
second, minute, hour, day, week, month, etc….) as you like.
Possible arrival processes could be
arrival of signal, click, broadcast, defective product, customer,
passenger, patient, rain, storm, earthquake etc.[Hint: Poisson and
exponential distributions exits at the same time.]
Collect approximately n=30
observations per unit time interval. .[Hint: Plot your
observations. If there is sharp increase or decrease then you could
assume that you are observing arrivals according to proper...

Every day, patients arrive at the dentist’s office. If the
Poisson distribution were applied to this process:
a.) What would be an appropriate random variable? What would be
the exponential-distribution counterpart to the random
variable?
b.)If the random discrete variable is Poisson distributed with λ
= 10 patients per hour, and the corresponding exponential
distribution has x = minutes until the next arrival, identify the
mean of x and determine the following:
1. P(x less than or equal to 6)...

Starting at noon, diners arrive at a restaurant according to a
Poisson process at
the rate of five customers per minute. The time each customer
spends eating at
the restaurant has an exponential distribution with mean 40
minutes, independent
of other customers and independent of arrival times. Find the
distribution,
as well as the mean and variance, of the number of diners in the
restaurant at
2 p.m.

Vehicles arrive at a small bridge according to a Poisson process
with arrival rate l = 900 veh/hr. a. What are the mean and the
variance of the number of arrivals during a 30 minutes interval? b.
What is the probability of 0 cars arriving during a 4 second
interval? c. A pedestrian arrives at a crossing point just after a
car passed by. If the pedestrian needs 4 sec to cross the street,
what is the probability that a...

Consider a waiting line system with the following
parameters:
number of servers = 4
customer arrival rate = 7 per minute
customer service rate (per server) = 2 per minute
coefficient of variation of interarrival times = 0.8
coefficient of variation of service times = 0.6
Find the expected length of the waiting line. (Do not assume
Poisson arrivals and exponential service times).
(Provide two significant digits to the right of the decimal
point)

Consider the assumptions of a Poisson process given in class.
Suppose the arrival of cars to a restaurant follows a Poisson
process. Which of the following examples do not violate an
assumption of the Poisson process?
two cars arrive at a restaurant at exactly the same time
the probability of one car arriving between 9:10 and 9:20 is the
same as the probability of one car arriving between 9:25 and
9:35
the probability of one car arriving between 9:04 and...

Suppose that phone calls arrive at a switchboard according to a
Poisson Process at a rate of 2 calls per minute.
(d)What is the probability that the next call comes in 30
seconds and the second call comes at least 45 seconds after
that?
(e) Let T4 be the time between 1st and 5th calls. What is the
distribution of T4?
(f) What is the probability that the time between 1st and 5th
call is longer than 5 minutes?
Please...

A university professor observes that the arrival process of
students to her office follows a Poisson distribution. Students’
questions are answered individually in the professor’s office. This
professor has calculated that the number of students who arrive
during the time she is answering the questions of a single student
has the following distribution:
Prob{0 students arrive} = 0.45,
Prob{1 student arrives} = 0.40,
Prob{2 students arrive} = 0.10,
Prob{3 students arrive} = 0.05
Using the fact that the traffic intensity...

customers arrive at a nail salon according to a poisson process
with rate 10 per hour. 30% of customers come in for only a
manicure. 30% only for a pedicure, and 40% for both.
what is the expe ted waiting time in MINUTES between manicure
oy customer arrivals?

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