# Statistics Bernoulli single-server queuing process with ATMS

• zzzzz
In summary: Here's a correct one:\begin{pmatrix}0.8 & 0.2 & 0 & 0 & 0 \\0.4 & 0.5 & 0.1 & 0 & 0 \\ 0 & 0.4 & 0.5 & 0.1 & 0 \\0 & 0 & 0.4 & 0.5 & 0.1 \\0 & 0 & 0 & 0.4 & 0.6\end{pmatrix}To find the steady-state probabilities, you need to find the eigenvector corresponding to the eigenvalue 1. This can be done by repeatedly multiplying the matrix by itself until
zzzzz

## Homework Statement

Customers arrive at an ATM at a rate of 12 per hour and spend 2 minutes using it, on average. Model this system using a Bernoulli single-server queuing process with 1-minute frames.

a. Compute the transition probability matrix for the system.

b. If the ATM is idle now, find the probability distribution of states in 3 minutes. (That is, compute the probabilities that, in 3 minutes, the ATM will be: (i) idle, (ii) serving one customer with no one in queue, (iii) serving one customer with one customer waiting, and (iv) serving one customer with two customers waiting.

c. Use your answer in part b to compute the expected number of customers in the system 3 minutes after the ATM was idle.

d. Use your answers in parts b and c to compute the expected number customers waiting in queue 3 minutes after the ATM was idle.

## Homework Equations

3. The Attempt at a Solution [/B]
I got the correct matrix for part a.
\begin{pmatrix}
0.8 & 0.2 & 0& 0 \\ 0.4 & 0.5 & 0.1 & 0 \\ 0 & 0 & 0.4 & 0.5
\end{pmatrix}

But I am unable to compute the steady state probabilities for part b, and therefore, my answers for part c and d are also wrong.
My attempt at b is as follows.
I found the probability distribution in 3 minutes which would be the transition probability matrix 3,
\begin{pmatrix}
0.7200000000000002 & 0.26 & 0.020000000000000004 & 0 \\ 0.52& 0.37& 0.1& 0.010000000000000002\\
0.16000000000000003 & 0.4 & 0.33000000000000007& 0.1\\ 0& 0.16000000000000003& 0.44 & 0.4
\end{pmatrix}

and solving it I got pi = \left< \frac 32 53 \, \frac 16 53, \frac 4 53, \frac 1 53 right>

Steady State Probabilities I got were
32/53 , 16/53, 4/53, 1/53

Realized my mistake!
My matrix was incorrect, but to solve the problem you should still find the matrix to the 3rd power.

zzzzz said:

## Homework Statement

Customers arrive at an ATM at a rate of 12 per hour and spend 2 minutes using it, on average. Model this system using a Bernoulli single-server queuing process with 1-minute frames.

a. Compute the transition probability matrix for the system.

b. If the ATM is idle now, find the probability distribution of states in 3 minutes. (That is, compute the probabilities that, in 3 minutes, the ATM will be: (i) idle, (ii) serving one customer with no one in queue, (iii) serving one customer with one customer waiting, and (iv) serving one customer with two customers waiting.

c. Use your answer in part b to compute the expected number of customers in the system 3 minutes after the ATM was idle.

d. Use your answers in parts b and c to compute the expected number customers waiting in queue 3 minutes after the ATM was idle.

## Homework Equations

3. The Attempt at a Solution [/B]
I got the correct matrix for part a.
\begin{pmatrix}
0.8 & 0.2 & 0& 0 \\ 0.4 & 0.5 & 0.1 & 0 \\ 0 & 0 & 0.4 & 0.5
\end{pmatrix}

But I am unable to compute the steady state probabilities for part b, and therefore, my answers for part c and d are also wrong.
My attempt at b is as follows.
I found the probability distribution in 3 minutes which would be the transition probability matrix 3,
\begin{pmatrix}
0.7200000000000002 & 0.26 & 0.020000000000000004 & 0 \\ 0.52& 0.37& 0.1& 0.010000000000000002\\
0.16000000000000003 & 0.4 & 0.33000000000000007& 0.1\\ 0& 0.16000000000000003& 0.44 & 0.4
\end{pmatrix}

and solving it I got pi = \left< \frac 32 53 \, \frac 16 53, \frac 4 53, \frac 1 53 right>

Your transition probability matrix is wrong: it should have 4 rows, not three, and all its row-sums should be 1.

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

A Bernoulli single-server queuing process with ATMS (Automatic Teller Machine System) is a statistical model used to analyze the waiting times of customers in a single-server queuing system, where the service times follow a Bernoulli distribution. This type of queuing process is commonly used in banks and other service industries where customers line up and wait for a service from a single server, such as an ATM machine.

## 2. How is the Bernoulli single-server queuing process with ATMS different from other queuing models?

The Bernoulli single-server queuing process with ATMS differs from other queuing models in that it assumes a constant service time for each customer, which follows a Bernoulli distribution. This means that the service time for each customer is independent of the service times of other customers in the queue, and the probability of being served is the same for each customer.

## 3. What are the key components of a Bernoulli single-server queuing process with ATMS?

The key components of a Bernoulli single-server queuing process with ATMS are the arrival rate of customers, the service rate of the server, and the number of customers in the queue. These components are used to calculate important metrics such as the average waiting time, the average queue length, and the probability of a customer having to wait for service.

## 4. How is the Bernoulli single-server queuing process with ATMS used in real-life applications?

The Bernoulli single-server queuing process with ATMS is commonly used in banks, retail stores, and other service industries to optimize customer service and reduce waiting times. By using this statistical model, businesses can make data-driven decisions about staffing levels, queue management, and service times to improve overall customer satisfaction.

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

One limitation of the Bernoulli single-server queuing process with ATMS is that it assumes a constant service time for each customer, which may not be realistic in real-life situations where service times can vary. Additionally, this model does not take into account external factors such as unexpected events or customer behavior, which can affect waiting times in a queuing system.

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