Question
In: Statistics and Probability
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
departs the system.
a) Define the states and draw the rate diagram.
b) Write down the balance equations
c) Find the expected number of customers in the system.
Solutions
orchestra answered 1 year agoRelated Solutions
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 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...
During lunchtime, customers arrive at Bob's Drugs according to a Poisson distribution with λ = 5...
During lunchtime, customers arrive at Bob's Drugs according to a
Poisson distribution with λ = 5 per minute. Show your answers to 3
decimal places.
What is the probability of one customer arriving?
What is the probability of more than two customers arriving?
What is the probability of at most three customers arriving?
What is the probability of at least four customers arriving?
What is the probability of fewer than two customers
arriving?
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...
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 service facility according to a Poisson process of rate 5 /hour Let...
Customers arrive at a service facility according to a Poisson
process of rate 5 /hour Let N(t) be the number of customers that
have arrived up to time t hours).
a.
What is the probability that there is at least 2 customer walked
in 30 mins?
b.If there was no customer in the first30 minutes, what is the
probability that you have to wait in total of more than 1 hours for
the 1 st customer to show up?
c.For...
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....
2. Consider an N = 1 server queue with arrival rate λ > 0 and service...
2. Consider an N = 1 server queue with arrival rate λ > 0 and
service rate µ = 1.
(a) Under what conditions will the process be (i) transient,
(ii) positive recurrent, and (iii) null recurrent?
(b) If the process is positive recurrent, find the stationary
distribution, say π(x). What is the name of this distribution?
(c) If the process is transient, find ρx0 for x ≥ 1.
Consider a two-server queue with Exponential arrival rate λ. Suppose servers 1 and 2 have exponential...
Consider a two-server queue with Exponential arrival rate λ.
Suppose servers 1 and 2 have exponential rates μ1 and μ2, with μ1
> μ2. If server 1 becomes idle, then the customer being served
by server 2 switches to server 1.
a) Identify a condition on λ,μ1,μ2 for this system to be stable,
i.e., the queue does not grow indefinitely long.
b) Under that condition, and the long-run proportion of time
that server 2 is busy.
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