What Is the Steady-State Distribution in a Bernoulli Queuing Process?

In summary, to find the steady-state distribution of the number of jobs in a Bernoulli single-server queuing process, we can use the formula P(n) = (1-r/r)^n * r/r, where r is the arrival rate and r is the service rate. In this case, the steady-state distribution is P(0) = 1/2, P(1) = 1/4, and P(2) = 1/8.
  • #1
lina29
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Homework Statement


Consider Bernoulli single-server queuing process with an arrival rate of 2 jobs per minute, a service rate of 4 jobs per minute, frames of 0.2 minutes, and a capacity limited by 2 jobs. Compute the steady-state distribution of the number of jobs in the system.

Homework Equations


The Attempt at a Solution


For P I got
.6 .4 0
.48 .44 .08
0 .48 .44

and my system of equations became

.6a + .48b = a
.4a+ .44b + .48c =b
.08b + .44c = c

and then by simplifying I got

.48b = .4a
.08b = .56c

But I'm stuck right there and don't know where to go. Any help would be appreciated. thanks!
 
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  • #2


Hi there,

To solve this problem, we can use the steady-state distribution formula for a single-server queuing process:

P(n) = (1-r/r)^n * r/r

where P(n) is the probability of having n jobs in the system, r is the arrival rate, and r is the service rate.

In this case, r=2 and r=4, so we have:

P(0) = (1-2/4)^0 * 2/4 = 1/2
P(1) = (1-2/4)^1 * 2/4 = 1/4
P(2) = (1-2/4)^2 * 2/4 = 1/8

Therefore, the steady-state distribution of the number of jobs in the system is:

P(0) = 1/2
P(1) = 1/4
P(2) = 1/8

I hope this helps! Let me know if you have any further questions.
 

FAQ: What Is the Steady-State Distribution in a Bernoulli Queuing Process?

1. What is a Bernoulli process?

A Bernoulli process is a statistical model in which a sequence of independent events are observed, and each event can only have two possible outcomes, typically labeled as "success" and "failure". These events are also assumed to have a fixed probability of success, denoted by p. Examples of Bernoulli processes include coin flips, where the outcomes are either heads or tails, and medical trials, where the outcome can either be a successful treatment or not.

2. What is the Bernoulli distribution?

The Bernoulli distribution is a discrete probability distribution that describes the probability of observing a certain number of successes in a given number of independent Bernoulli trials. It is often represented as Ber(p), where p is the probability of success. The distribution has a single parameter, p, and its expected value is equal to p while its variance is equal to p(1-p).

3. How is the Bernoulli process related to binomial distribution?

The Bernoulli process is the basis for the binomial distribution, which is a discrete probability distribution that describes the number of successes in n independent Bernoulli trials. In other words, the binomial distribution is the probability distribution of the number of successes in a Bernoulli process. It is often represented as Bin(n,p), where n is the number of trials and p is the probability of success.

4. What are some real-life applications of Bernoulli process statistics?

Bernoulli process statistics has a wide range of applications in various fields such as finance, medicine, and engineering. In finance, it can be used to model stock prices and market movements. In medicine, it can be used to analyze the effectiveness of a treatment or medication. In engineering, it can be used to predict the probability of failure of a system or component. It is also commonly used in quality control and reliability analysis.

5. What are the assumptions of a Bernoulli process?

The main assumptions of a Bernoulli process are that the events are independent, each event can only have two possible outcomes, and the probability of success remains constant throughout the process. Additionally, the events should be mutually exclusive, meaning that the occurrence of one event does not affect the probability of another event. These assumptions are necessary for the Bernoulli process to accurately model real-life situations.

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