Question
In: Statistics and Probability
Calls to an agency come according to a rate λ Poisson process. The agency has s...
Calls to an agency come according to a rate λ Poisson process. The agency has s telephone lines and the duration of each call is an exponential random variable with parameter µ. If a call comes when all lines are busy, it is not taken or put on hold. Find the stationary distribution of the number of busy lines.
Solutions
orchestra answered 2 years agoRelated Solutions
Calls to an agency come according to a rate λ Poisson process. The agency has s...
Calls to an agency come according to a rate λ Poisson process.
The agency has s telephone lines and the duration of each call is
an exponential random variable with parameter µ. If a call comes
when all lines are busy, it is not taken or put on hold. Find the
stationary distribution of the number of busy lines.
Emails arrive in an inbox according to a Poisson process with rate λ (so the number...
Emails arrive in an inbox according to a Poisson process with
rate λ (so the number of emails in a time interval of length t is
distributed as Pois(λt), and the numbers of emails arriving in
disjoint time intervals are independent). Let X, Y, Z be the
numbers of emails that arrive from 9 am to noon, noon to 6 pm, and
6 pm to midnight (respectively) on a certain day. (a) Find the
joint PMF of X, Y, Z....
Suppose people get infected by Coronavirus according to a Poisson process with rate λ > 0...
Suppose people get infected by Coronavirus according to a
Poisson process with rate λ > 0 and
λ denoting the average number of infected people per
day.
(a) (10 points) Find the expected time the 100th infected
person will be identified.
(b) (10 points) Find the probability that the elapsed time
between infected persons 10th and 11th exceeds two days.
Cars get parked in a lot (with infinite capacity) according to a λ-rate Poisson process, and...
Cars get parked in a lot (with infinite capacity) according to a
λ-rate Poisson process, and indepen- dently stay parked for a
random duration. The parking time duration of a car follows a
common distribution X, with cdf F(x) = P(X ≤ x). Let N(t) be the
number of cars parked at time t.
1a. What is the distribution of N(t)?
1b. Assuming a car arrival rate of 1 per minute, and X (in min)
∼ Gamma(3, 1) (as defined...
Starting at time 0, a red bulb flashes according to a Poisson process with rate λ=1....
Starting at time 0, a red bulb flashes according to a Poisson
process with rate λ=1. Similarly, starting at time 0, a blue bulb
flashes according to a Poisson process with rate λ=2, but only
until a nonnegative random time X, at which point the blue bulb
“dies." We assume that the two Poisson processes and the random
variable X are (mutually) independent.
a) Suppose that X is equal to either 1 or 2, with equal
probability. Write down an...
Customers arrive at a two-server system according to a Poisson process having rate λ = 5....
Customers arrive at a two-server system according to a Poisson
process having rate λ = 5. An arrival finding server 1 free will
begin service with that server. An arrival finding server 1 busy
and server 2 free will enter service with server 2. An arrival
finding both servers busy goes away. Once a customer is served by
either server, he departs the system. The service times at server i
are exponential with rates μi, where μ1 = 4, μ2...
People arrive at a party according to a Poisson process of rate 30 per hour and...
People arrive at a party according to a Poisson process of rate
30 per hour and remain for an independent exponential time of mean
2 hours. Let X(t) be the number of people at the party at time t
(in hours) after it started. Compute E[X(t)] and determine how long
it takes to have on average more than 40 people at the party.
Students enter the bathroom according to a Poisson process at a rate of 7.5 arrivals per...
Students enter the bathroom according to a Poisson
process at a rate of 7.5 arrivals per minute.
What is the probability that exactly 46 students enter
between 3:00 and 3:05?
Given that 6 students enter the bathroom between 4:00
and 4:01, what is the probability that exactly 36 students enter
between 4:00 and 4:07?
Each student entering the bathroom has a .15 probability
of wearing a hoodie, independent of other students. What is the
probability that exactly 10 students wearing...
A radioactive mass emits particles according to a Poisson process at a mean rate of 2.5...
A radioactive mass emits particles according to a Poisson
process at a mean rate of 2.5 per second. Let T be the
waiting time, in seconds, between emissions.
1-What is the median waiting time?
2-Find P(0.3 < T < 1.5).
3-If 3 seconds have elapsed with no emission, what is the
probability that there will be an emission within the next
second
Small aircraft arrive at a private airport according to a given Poisson process at a rate...
Small aircraft arrive at a private airport according to a given
Poisson process at a rate of 6 per hour.
What is the probability that exactly 5 small aircraft arrive in
the first hour (after opening) and exactly 7 arrive in the hour
after that?
What is the probability that fewer than 5 small aircraft arrive
in the first hour and at least 10 arrive in the hour after
that?
What is the probability that exactly 5 small aircraft arrive...
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