Question
In: Statistics and Probability
A queueing system serves two types of customers. Type 1 customers arrive according to a Poisson...
A queueing system serves two types of customers. Type 1 customers arrive according to a Poisson process with a mean rate of 5 per hour. Type 2 customers arrive according to a Poisson process at a mean rate of 3 per hour. The system has two servers, both of which serve both types of customer. All service times have exponential distribution with a mean of 10 minutes. Service is provided on a first-come-first-served basis.
a. What is the probability distribution of the time between consecutive arrivals of customers of any type, what is its mean?
b. Assume that when a Type 2 customer arrives, he finds two Type 1 customers being served and no other customers in the system. What is the probability distribution of this Type 2 customer’s waiting time in the queue and its mean?
Solutions
orchestra answered 2 years agoRelated Solutions
A queueing system serves two types of customers. Type 1 customers arrive according to a Poisson...
A queueing system serves two types of customers. Type 1
customers arrive according to a Poisson process with a mean rate of
5 per hour. Type 2 customers arrive according to a
Poisson process at a mean rate of 3 per hour. The
system has two servers, both of which serve both types of customer.
All service times have exponential distribution with a mean of 10
minutes. Service is provided on a first-come-first-served
basis.
a. What is the probability distribution...
A queueing system serves two types of customers. Type 1 customers arrive according to a Poisson...
A queueing system serves two types of customers. Type 1
customers arrive according to a Poisson process with a mean rate of
5 per hour. Type 2 customers arrive according to a
Poisson process at a mean rate of 3 per hour. The
system has two servers, both of which serve both types of customer.
All service times have exponential distribution with a mean of 10
minutes. Service is provided on a first-come-first-served
basis.
a. What is the probability distribution...
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...
Customers arrive at a two-server system at a Poisson rate λ=5. An arrival finding the system...
Customers arrive at a two-server system at a Poisson rate λ=5.
An arrival finding the system empty is equally likely to enter
service with either server. An arrival finding one customer in the
system will enter service with the idle server. An arrival finding
two others will wait in line for the first free server. The
capacity of the system is 3. All service times are exponential with
rate µ=3, and once a customer is served by either server, he...
Customers arrive to the checkout counter of a convenience store according to a Poisson process at...
Customers arrive to the checkout counter of a convenience store
according to a Poisson process at a rate of two per minute. Find
the mean, variance, and the probability density function of the
waiting time between the opening of the counter and the following
events:
a. The arrival of the second customer.
b. The arrival of the third customer.
c. What is the probability that the third customer arrives
within 6 minutes? You can use a computer if you’d like...
Customers arrive at a department store according to a Poisson process with an average of 12...
Customers arrive at a department store according to a Poisson
process with an average of 12 per hour.
a. What is the probability that 3 customers arrive between
12:00pm and 12:15pm?
b. What is the probability that 3 customers arrive between
12:00pm and 12:15pm and 6 customers arrive between 12:30pm and
1:00pm?
c. What is the probability that 3 customers arrive between
12:00pm and 12:15pm or 6 customers arrive between 12:30pm and
1:00pm?
d. What is the probability that a...
Customers arrive at a hair salon according to a Poisson process with an average of 16...
Customers arrive at a hair salon according to a Poisson process
with an average of 16 customers per hour. The salon has just one
worker due to covied-19 restriction. Therefore, the salon must
close whenever the worker leaves. assume that customers who arrive
while the salon is closed leave immediately and don’t wait until
the worker returns. The salon is closed on weekends.
a.
What is the probability that at most (less than) four customers
arrive in the hour before...
Customers arrive in a certain shop according to an approximate Poisson process on the average of...
Customers arrive in a certain shop according to an approximate
Poisson process on the average of two every 6 minutes.
(a) Using the Poisson distribution calculate the probability of
two or more customers arrive in a 2-minute period.
(b) Consider X denote number of customers and X follows binomial
distribution with parameters n= 100. Using the binomial
distribution calculate the probability oftwo or more customers
arrive in a 2-minute period.
(c) Let Y denote the waiting time in minutes until...
Customers arrive at an establishment according to a Poisson process of frequency = 5 per...
Customers arrive at an establishment according to a Poisson
process of frequency = 5 per hour. Since the establishment opens
at 9:00 am:
a) What is the probability that exactly a client arrived by 9:45
am?
b) What is the probability of maximum five clients by 11:45
am?
Customers arrive at a two pump system at Poisson rate two per hour. An arrival finding...
Customers arrive at a two pump system at Poisson rate two per
hour. An arrival finding the system empty is equally likely to
enter service with either pump. An arrival finding one customer in
the system will enter service with the idle pump. An arrival
finding two others in the system will wait in line for the first
free pump. An arrival finding three in the system will not enter.
Two service times are exponential with rates one per hour...
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