Question

Consider a queueing system, where customers arrive according to a Poisson process with intensity \(\lambda\) customers per minute. Let \(X(t)\) be the total number of customers that arrive during \((0, t]\). Compute the correlation coefficient of \(X(t)\) and \(X(t+s)\).

Short answer

The correlation coefficient is \( \sqrt{\frac{t}{t+s}} \).

Step-by-step solution

Understand the Problem

In this exercise, we have a queueing system where customers arrive according to a Poisson process with rate \( \lambda \). We are asked to find the correlation coefficient between the number of customers arriving in one interval \( X(t) \) and a shifted interval \( X(t+s) \).

Define the Random Variables

Let \( X(t) \) be the number of arrivals in the interval \( (0, t] \) and \( X(t+s) \) be the number of arrivals in the interval \( (0, t+s] \). Thus, \( X(t+s) = X(t) + Y(s) \), where \( Y(s) \) is the number of arrivals in interval \( (t, t+s] \).

Calculate Expectation and Variance

For a Poisson process, \( E[X(t)] = \lambda t \) and \( \text{Var}(X(t)) = \lambda t \). Similarly, \( E[Y(s)] = \lambda s \) and \( \text{Var}(Y(s)) = \lambda s \). Assuming Poisson independent increments, \( \text{Cov}(X(t), Y(s)) = 0 \).

Compute Covariance of \( X(t) \) and \( X(t+s) \)

The covariance, \( \text{Cov}(X(t), X(t+s)) \), is \( \text{Cov}(X(t), X(t) + Y(s)) \). Therefore, \( \text{Cov}(X(t), X(t+s)) = \text{Cov}(X(t), X(t)) + \text{Cov}(X(t), Y(s)) = \text{Var}(X(t)) = \lambda t \).

Determine the Correlation Coefficient

The correlation coefficient, \( \rho(X(t), X(t+s)) \), is given by the formula: \[ \rho = \frac{\text{Cov}(X(t), X(t+s))}{\sqrt{\text{Var}(X(t)) \cdot \text{Var}(X(t+s))} }. \] Substituting \( \text{Cov}(X(t), X(t+s)) = \lambda t \) and the variances, \( \text{Var}(X(t+s)) = \lambda (t+s) \), the correlation coefficient becomes: \[ \rho = \frac{\lambda t}{\sqrt{\lambda t \cdot \lambda (t+s)}} = \sqrt{\frac{t}{t+s}}. \]

Conclude the Result

The correlation coefficient between \( X(t) \) and \( X(t+s) \) is \( \sqrt{\frac{t}{t+s}} \). This indicates that the arrival in the initial period and the extended period are positively correlated.

Key concepts

Understanding the Correlation Coefficient

The correlation coefficient is a measure used to indicate the strength and direction of the relationship between two random variables. In the context of a Poisson process involving a queueing system, we examine intervals of time and the number of customer arrivals. Specifically, we review the relationship between customer arrivals in overlapping time periods.
For the random variables, such as the number of customers arriving at a queue in a given time, the correlation coefficient helps us understand how one event predicts or relates to another. A positive correlation indicates a direct relationship; as one number increases, so does the other.
The formula for the correlation coefficient, \( \rho(X, Y) \), is given by:
\[ \rho = \frac{\text{Cov}(X, Y)}{\sqrt{\text{Var}(X) \cdot \text{Var}(Y)}} \]
where \( \text{Cov}(X, Y) \) is the covariance between \( X \) and \( Y \), and \( \text{Var}(X) \), \( \text{Var}(Y) \) are the variances of \( X \) and \( Y \) respectively.

Insights into Queueing Systems

Queueing systems are models used to describe processes where "customers" wait for service. These customers could be people in line at a bank, or data packets waiting to be sent over a network. The arrival of customers is often random and can be described using probability models such as the Poisson process.
In a queueing system with a Poisson process, the number of arrivals is random and occurs continuously over time. This arrival pattern is particularly common in systems where events occur independently and at a constant average rate \( \lambda \). This implies that the expected number of events happening in disjoint time periods are independent.
Queueing theory helps us model these situations to optimize service times, reduce wait times, and ensure smooth operation of a service mechanism, all of which are crucial in various industries like telecommunications and customer support.

Covariance in Poisson Processes

Covariance is a measure of how much two random variables change together. If two variables have high covariance, they tend to change in the same direction together. In the context of our Poisson-based queueing system, covariance plays a role in understanding how two time intervals are related.
In Poisson processes, we calculate covariance based on the assumption of independent increments. This means that the events (customer arrivals) in non-overlapping time intervals are independent, resulting in a zero covariance for these periods.
Precisely, for a Poisson process with rate \( \lambda \), if we have two intervals such \( X(t) \) and \( Y(s) \) which are independent increments, their covariance is:
\[ \text{Cov}(X(t), Y(s)) = 0 \]
This zero covariance is the property that allows us to substitute directly when calculating the correlation coefficient between two intervals \( X(t) \) and \( X(t+s) \).

Role of Independent Increments

Independent increments are a core concept in Poisson and many other stochastic processes. They refer to the idea that events occurring in non-overlapping time intervals do not affect each other, meaning the arrival during one interval does not influence the number of arrivals in another.
For our queueing system, this principle simplifies analysis since it implies that arrivals in different intervals can be considered separately when calculating probabilities and statistics. It holds that:

• For any two intervals \( (0, t] \) and \( (t, t+s] \), the number of arrivals is independent.
• Covariance between independent increments is zero, i.e., \( \text{Cov}(X(t), Y(s)) = 0 \).

Understanding independent increments is pivotal as it impacts how we compute other measures like expectation, variance, and ultimately the correlation coefficient. The independence of increments is what makes the Poisson process both practical and powerful for modeling many real-life situations where phenomena occur randomly yet with a predictable average rate.