# Bernoulli single-server queueing process

• lowball
In summary: XZlIHRoZSBzYW1wbGUgaXMgbW9kZWx5IGJldHdlZW4gdG8gcGxhY2UgbW9kZWx5IGJ5IHRoZSBzaW5nbGUtc2VydmVyIEJlcm5vdWxsaSBxdWVlcmluZyBwcm9wZXJ0eSB3aXRoIDItbWludXRlIGZyYW1lcy4gQ2FycyBhcnJpdmUgZXJlY3RlZCBldmVuIGEgZmVhcm1lIHJ
lowball

## Homework Statement

Performance of a car wash center is modeled by the single-server Bernoulli queueing process with 2-minute frames. Cars arrive every 10 minutes, on the average. The average service time is 6 minutes. Capacity is unlimited. If there are no cars at the center at 10 am, compute the probability that one car is being washed and another car is waiting at 10:04 am.

## Homework Equations

$Δ = 2$ min
$λ_A = .1$ min$^{-1}$
$λ_S = .167$ min$^{-1}$
$p_A = λ_AΔ = .2$
$p_S = λ_SΔ = .333$
$p_{00} = 1-p_A = .8$
$p_{01} = p_A = .2$
$(1-p_A)p_S = .267$
$(1-p_S)p_A = .133$
$1 - .267 - .133 = .6$

## The Attempt at a Solution

Using the above calculations I formed this transition probability matrix:

$P = \begin{pmatrix} .8 & .2 & 0 & 0 & \dots\\ .267 & .133 & .6 & 0 & \dots\\ 0 & .267 & .133 & .6 & \dots\\ 0 & 0 & .267 & .133 & \dots\\ \vdots & \vdots & \vdots & \vdots & \ddots \end{pmatrix}$

With no cars in the system, the initial distribution is:
$P_0 = \begin{pmatrix}1&0&0&0\end{pmatrix}$

With a frame size of 2 minutes, 10:04 is 2 frames away from 10:00, thus the distribution after 2 frames is:
$P_2 = P_0P^6 = \begin{pmatrix}1&0&0&0\end{pmatrix}\cdot P^2$
$= \begin{pmatrix}.6934&.1866&.12&0\end{pmatrix}$

And the probability for two cars to be in the system after 2 frames is $P_2(2) = .12$

But that's not accepted as the right answer. The answer in the back of the book says $\frac{2}{75}$, but that's not even anywhere in the matrix of $P^2$. Any idea what I'm doing wrong?

Shouldn't the second row of the matrix read:
10: 0.8*1/3 = .267
11: 0.8*2/3 + .2*1/3 = 0.6
12: 0.2*2/3 = .133
?

Ah... now I see. The example in my book had it ordered somewhat oddly. After swapping those around, I do indeed get the correct answer of .0266. Thanks!

lowball said:
Ah... now I see. The example in my book had it ordered somewhat oddly. After swapping those around, I do indeed get the correct answer of .0266. Thanks!

You should not use such inaccurate numbers, especially not at the beginning.
$$\lambda_A = 1/10,\; \lambda_S = 1/6\\ p_A = 1/5,\; p_S = 1/3\\ p_{00} = 4/5, \; p_{01} = p_A = 1/5\\ (1-p_A)p_S = (4/5)(1/3) = 4/15, \; (1-p_S)p_A = (2/3)(1/5) = 2/15\\ 1 - 4/15 - 2/15 = 9/15 = 3/5$$
Later, you can round off, and to get *accurate* multi-step probabilities (via Pn) you should keep a lot more digits in P---full machine floating-point accuracy would be best. (Of course, maybe you did keep all those figures and just rounded off for presentation purposes, in which case you should say so.)

RGV

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

A Bernoulli single-server queueing process is a mathematical model that is used to analyze and predict the behavior of a single-server queueing system. It assumes that arrivals to the system follow a Bernoulli distribution and that the service time of each customer is exponentially distributed. This process is commonly used in fields such as operations research and computer science to study real-world queueing systems.

## 2. How does a Bernoulli single-server queueing process work?

In this process, customers arrive at a single server, which can only serve one customer at a time. The service time for each customer is independent and exponentially distributed. When a customer arrives, they join the queue and wait for the server to become available. Once the server is free, the customer at the front of the queue is served, and the process continues until all customers have been served.

## 3. What is the purpose of using a Bernoulli single-server queueing process?

The purpose of using this process is to analyze the performance of a single-server queueing system. By studying the behavior of this model, researchers can make predictions and optimize the system to improve its efficiency and customer satisfaction. This process is particularly useful in situations where there is limited capacity, such as in call centers, transportation systems, and computer networks.

## 4. What are the assumptions made in a Bernoulli single-server queueing process?

There are several assumptions made in this process, including the following:

• Arrivals to the system follow a Bernoulli distribution.
• The service time for each customer is exponentially distributed.
• The system has a single server that can only serve one customer at a time.
• The service times are independent and do not depend on previous service times.
• The queue has an infinite capacity.

## 5. How is the performance of a Bernoulli single-server queueing process evaluated?

The performance of this process is evaluated using various metrics, such as the average waiting time, the average number of customers in the system, and the utilization of the server. Additionally, the probability of a customer having to wait in the queue and the probability of the server being idle are also measured. These metrics can help researchers identify any bottlenecks in the system and make improvements to increase its efficiency.

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