# Statistics Bernoulli single-server queuing process

• zzzzz
In summary: The correct matrix should be:5/6 1/6 05/12 1/2 1/12 = [x y z]7/12 5/12 0Solving the system of equations from this matrix gives the correct answers:x = 25/37y = 10/37z = 2/37Therefore, the proportion of time you spend on the telephone is 10/37 + 2/37 = 12/37.
zzzzz

## Homework Statement

[/B]Suppose your office telephone has two lines, allowing you to talk with someone and have at most one other person on hold. You receive 10 calls per hour and a conversation takes 2 minutes, on average. Use a Bernoulli single-server queuing process with limited capacity and 1-minute frames to compute the proportion of time you spend using the telephone.

## The Attempt at a Solution

[/B]
Found the transition probability matrix as:

5/6 1/6 0
5/12 1/2 1/12 = [x y z]
0 5/12 7/12

From this matrix, I found the following system of equations
5/6x + 5/12y = x
1/6x +1/2y + 5/12z = y
1/12y + 7/12z = z
Solving the system of equations from this matrix I got
x=25/81
y = 10/27
z = 26/81

I thought that the proportion of time you would spend on the telephone is 56/81, which would be the steady state probabilities of y ( One customer on the phone) and z (One customer on the phone and another one on hold), but that answer is wrong.
I also tried the steady state probability of y = 10/27, but that is also wrong.

Can you please explain what I am doing wrong?

Thank you so much.

I realized I made an algebra mistake while computing the system of equations..
The correct answers to the system of equations were
x = 25/37 y = 10/37 and z = 2/37

zzzzz said:

## Homework Statement

[/B]Suppose your office telephone has two lines, allowing you to talk with someone and have at most one other person on hold. You receive 10 calls per hour and a conversation takes 2 minutes, on average. Use a Bernoulli single-server queuing process with limited capacity and 1-minute frames to compute the proportion of time you spend using the telephone.

## The Attempt at a Solution

[/B]
Found the transition probability matrix as:

5/6 1/6 0
5/12 1/2 1/12 = [x y z]
0 5/12 7/12

From this matrix, I found the following system of equations
5/6x + 5/12y = x
1/6x +1/2y + 5/12z = y
1/12y + 7/12z = z
Solving the system of equations from this matrix I got
x=25/81
y = 10/27
z = 26/81

I thought that the proportion of time you would spend on the telephone is 56/81, which would be the steady state probabilities of y ( One customer on the phone) and z (One customer on the phone and another one on hold), but that answer is wrong.
I also tried the steady state probability of y = 10/27, but that is also wrong.

Can you please explain what I am doing wrong?

Thank you so much.
Your transition probability matrix is incorrect: its second row adds up to less than 1.

## 1. What is a Bernoulli single-server queuing process?

A Bernoulli single-server queuing process is a mathematical model used in statistics to analyze and predict the behavior of a single-server queue. It is named after Swiss mathematician Jacob Bernoulli and is often used to study systems where customers arrive randomly and are served one at a time by a single server.

## 2. What are the assumptions of the Bernoulli single-server queuing process?

The Bernoulli single-server queuing process assumes that arrivals and service times are independent and follow a Bernoulli distribution, meaning they have only two possible outcomes (success or failure) with a fixed probability. It also assumes that customers arrive randomly and are served on a first-come, first-served basis.

## 3. How is the performance of a single-server queue measured using the Bernoulli single-server queuing process?

The performance of a single-server queue is measured using several key metrics, including the average waiting time, average queue length, and average waiting time in the queue. These metrics can be calculated using the formulas and equations derived from the Bernoulli single-server queuing process.

## 4. What are some real-world applications of the Bernoulli single-server queuing process?

The Bernoulli single-server queuing process has many practical applications in different industries. For example, it can be used to analyze customer wait times in a call center, predict traffic flow at a toll booth, or optimize service levels at a bank. It is also commonly used in computer simulations to test the efficiency of different queuing strategies.

## 5. What are some limitations of the Bernoulli single-server queuing process?

While the Bernoulli single-server queuing process is a useful tool for analyzing queues, it has some limitations. It assumes that arrivals and service times are independent and follow a Bernoulli distribution, which may not always hold true in real-world scenarios. Additionally, it does not take into account factors such as customer impatience or server breakdowns, which can affect the performance of a queuing system.

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