Question
In: Statistics and Probability
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.
Solutions
orchestra answered 2 years agoRelated Solutions
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.
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 infinitely long.
b) Under that condition, find the long-run proportion of time
that server 2 is busy.
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...
A queue with one server without buffer, the probability of a customer’ arrival and departure in...
A queue with one server without buffer, the
probability of a customer’ arrival and departure in a time unit is
p and q respectively. Please try to 1) give the one step state
transition probability matrix. 2) give the balance equations. 3)
calculate the limiting probabilities for p=0.3 and q=0.5. (12
points)
A queuing system with a Poisson arrival rate and exponential service time has a single queue,...
A queuing system with a Poisson arrival rate and exponential
service time has a single queue, two servers, an average arrival
rate of 60 customers per hour, and an average service time of 1.5
minutes per customer.
Answer the following questions. Show
ALL formulas and calculations used in your
response.
The manager is thinking of implementing additional queues to
avoid an overloaded system. What is the minimum number of
additional queues required? Explain.
How many additional servers are required to...
A queuing system with a Poisson arrival rate and exponential service time has a single queue,...
A queuing system with a Poisson arrival rate and exponential
service time has a single queue, two servers, an average arrival
rate of 60 customers per hour, and an average service time of 1.5
minutes per customer.
The manager is thinking of implementing additional queues to
avoid an overloaded system. What is the minimum number of
additional queues required? Explain.
How many additional servers are required to ensure the
utilization is less than or equal to 50%? Explain.
If the...
Consider an M / M / 1 queueing system with capacity N = 1, arrival rate...
Consider an M / M / 1 queueing system with capacity N = 1,
arrival rate λ = 0.2 customers per minute, and service rate µ =0.3
customers per minute. Let X(t) be the number of customers present
at time t.
a) Suppose that X(0) =0. Calculate P{X(1) = 0} and P{X(10) =
0}
b) Suppose that X(0) =1. Calculate P{X(1) = 0} and P{X(10) =
0}
c) Could you have predicted the difference between the answers
in part a,...
Consider the following two-server one-queue system from time = 0 to time = 20 min. If...
Consider the following two-server one-queue system from time = 0
to time = 20 min. If both servers are available when a customer
arrives, the customer will choose server1. Customers waiting in the
queue enter service whenever any one of the two servers becomes
available (first come, first serve). Arrivals and service times
are: • Customer #1 arrives at t = 0 and requires 2 minutes of
service time • Customer #2 arrives at t = 1 and requires 5...
We define the Liouville function λ(n) by setting λ(1)=1. If n>1, we consider the prime power...
We define the Liouville function λ(n) by setting λ(1)=1. If
n>1, we consider the prime power factorization n=p_1^(a_1 )
p_2^(a_2 )…p_m^(a_m ) and define λ(n)=(-1)^(a_1+a_2+⋯+a_m ) Prove
that the summatory function of λ,Λ(n)=∑_d|n┤▒〖λ(d)〗 is
multiplicative.
(a) Let Λ = {λ ∈ R : 0 < λ < 1}. For each λ...
(a) Let Λ = {λ ∈ R : 0 < λ < 1}. For each λ ∈ Λ, let Aλ =
{x ∈ R : −λ < x < 1/λ}. Find U λ∈Λ Aλ and ~U λ∈Λ Aλ
respectively.
(b) Let Λ = \ {λ ∈ R : λ > 1}. For each λ ∈ Λ, let Aλ = {x ∈
R : −λ < x < 1/λ}. Find U λ∈Λ Aλ and ~U λ∈Λ Aλ
respectively.
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