Question

In a gas station there is one gas pump. Cars arrive at the gas
station according to a Poisson proces. The arrival rate is 20 cars
per hour. An arriving car ﬁnding n cars at the station immediately
leaves with probability qn = n/4, and joins the queue with
probability 1−qn, n = 0,1,2,3,4. Cars are served in order of
arrival. The service time (i.e. the time needed for pumping and
paying) is exponential. The mean service time is 3 minutes.

(i) Determine the stationary distribution of the number of cars at
the gas station.

(ii) Determine the mean number of cars at the gas station.

(iii) Determine the mean sojourn time (waiting time plus service
time) of cars deciding to take gas at the station.

(iv) Determine the mean sojourn time and the mean waiting time of
all cars arriving at the gas station.

Answer #1

A gas station with two pumps does not allow drivers to pump
their gas and has a service attendant for each pump. Potential
customers (i.e. cars) arrive according to KinKo, a process at a
rate of 40 cars per hour. If the two pumps are busy, then arriving
cars wait in a single queue to be served in the order of arrival by
the first available pump. However, cars cannot enter the station to
wait if there are already two...

Eat & Gas convenience store operates a two-pump gas station.
The lane leading to the pumps can house at most 3 cars, excluding
those being served. Arriving cars go elsewhere if the lane is full.
The distribution of arriving cars is Poisson with mean 20 per hour.
The time to fill up and pay for the purchase is exponential with
mean 6 minutes. What is the percentage of cars that will seek
business elsewhere? What is the percentage of time...

Cars arrive to a gas station according to a Poisson
distribution with a mean of 4 cars per
hour. Use Excel or StatCrunch to
solve.
a. What is the expected number of cars arriving
in 2 hours, or λt?
b. What is the probability of 6 or less cars
arriving in 2 hours? ROUND TO FOUR (4) DECIMAL
PLACES.
c. What is the probability of 9 or more cars
arriving in 2 hours? ROUND TO FOUR (4) DECIMAL
PLACES.

A new gas station has only one pump for gasoline. Cars take on
an average 5 minutes to fuel up. The gas station manager calculated
that the pump has an average of 8 cars visiting it every hour. On
an average, how many cars will be waiting in the queue for
refueling? (3 decimals)
The answer is suppose to be 1.333.
I need help figuring out how my professor got this answer.
Please only reply if you know a step-by-step...

The number of cars arriving at a gas station can be modelled by
Poisson distribution with the average rate of 5 cars per 10
minutes. a. The probability that one car will arrive to a gas
station in a 5 -minute interval is _________ b. The probability
that at least one car will arrive to the gas station in a 10 -
minute interval is ______

Exercise 11.2.5 Customers arrive at Bunkey’s car wash service at
a rate of one every 20 minutes and the average time it takes for a
car to proceed through their single wash station is 8 minutes.
Answer the following questions under the assumption of Poisson
arrivals and exponential service.
(a) What is the probability that an arriving customer will have
to wait?
(b) What is the average number of cars waiting to begin their
wash?
(c) What is the probability...

The number of cars arriving at a
petrol station in a period of t minutes may be assumed to
have a Poisson distribution with mean
720t.
Use this information to
answer questions 31-32.
Find the probability that exactly 12 cars will arrive in one
and half hours.
Find the probability that more than 24 cars will arrive in an
hour.

The next four questions are based on the following information.
At a car wash station, on average, there are 4 cars coming in for
the service every 10 minutes. The average wash time is 2 minutes.
The Poisson distribution is appropriate for the arrival rate and
service times are exponentially distributed. Please convert all
rates into cars per hour and answer the following questions.
1.
The average time a car spent in the waiting line is ___________
hours, and the...

In a grocery store, there is one cashier counter. Customers
arrive at the cashier counter according to a Poisson process. The
arrival rate is 30 customers per hour. The service time is
exponentially distributed. The mean service time is 1 minute 30
seconds.
1. What is the expected number of customers waiting in the
system.
2. What is the expected waiting time.( unit in minutes)
3. What is the utilization rate (unit in %) of
the cashier?

The Quick Snap photo machine at the Lemon County bus station
takes snapshots in exactly 80 seconds. Customers arrive at the
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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.

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