# Queueing system with exponential arrival, service

• bjr_jyd15
In summary, to determine the probability of the next arrival before 1:45pm at a queueing station with two servers and exponential inter-arrival and service times, the exponential distribution can be used with a lambda value of 0.5 and x value of 1.75.
bjr_jyd15
I'm having troubles with the basics here--do I use exponential process or poisson?

Suppose a queueing station has two servers, an exponential inter-arrival time for customers with a mean of 2 hours and an exponential service time for service with mean of 2 hours. Furthermore a customer has just arrived at 12 noon.

1. What is the probability that the next arrival will be before 1:45pm?

I have tried using exponential distribution with lambda = 0.5...but can't figure out what I'm doing wrong.

Thanks.

The probability that the next arrival will be before 1:45pm can be calculated using the exponential distribution. The probability is given by the equation: P(x<1.45) = 1 - e^(-lambda*x), where lambda = 0.5 and x = 1.75 (1.45pm - 12 noon). Therefore, the probability of the next arrival being before 1:45pm is 0.38292492.

## 1. What is a queueing system with exponential arrival and service?

A queueing system with exponential arrival and service is a mathematical model that is used to analyze the behavior of waiting lines or queues. It assumes that arrivals and service times follow an exponential distribution, which allows for the calculation of key performance metrics such as waiting time, queue length, and utilization.

## 2. How is the arrival rate defined in a queueing system with exponential arrival and service?

The arrival rate in a queueing system with exponential arrival and service is defined as the average number of arrivals per unit time. It is often denoted by the symbol λ (lambda) and is a key parameter in determining the behavior of the system. The higher the arrival rate, the longer the queues and waiting times will be.

## 3. What is the difference between arrival rate and service rate in a queueing system with exponential arrival and service?

The arrival rate refers to the rate at which customers enter the system, while the service rate refers to the rate at which customers are served and exit the system. In an exponential queueing system, the arrival rate is typically denoted by λ (lambda) and the service rate by μ (mu). The ratio of arrival rate to service rate, λ/μ, is known as the traffic intensity and is a key determinant of system performance.

## 4. How is the waiting time calculated in a queueing system with exponential arrival and service?

The waiting time in a queueing system with exponential arrival and service is calculated using Little's Law, which states that the average waiting time is equal to the average number of customers in the system (including those in the queue and being served) divided by the arrival rate. This means that as the arrival rate or queue length increases, the waiting time also increases.

## 5. What are some real-world applications of a queueing system with exponential arrival and service?

Queueing systems with exponential arrival and service are used in a wide range of real-world applications, including telecommunications networks, transportation systems, healthcare facilities, and call centers. They can help organizations optimize resources, reduce waiting times, and improve customer satisfaction by providing insights into customer behavior and system performance.

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