Question

For the \(M / G / 1\) queue, let \(X_{n}\) denote the number in the system left behind by the nth departure. (a) If $$ X_{n+1}=\left\\{\begin{array}{ll} X_{n}-1+Y_{n}, & \text { if } X_{n} \geqslant 1 \\ Y_{n}, & \text { if } X_{n}=0 \end{array}\right. $$ what does \(Y_{n}\) represent? (b) Rewrite the preceding as $$ X_{n+1}=X_{n}-1+Y_{n}+\delta_{n} $$ where $$ \delta_{n}=\left\\{\begin{array}{ll} 1, & \text { if } X_{n}=0 \\ 0, & \text { if } X_{n} \geqslant 1 \end{array}\right. $$ Take expectations and let \(n \rightarrow \infty\) in Equation ( \(8.64\) ) to obtain $$ E\left[\delta_{\infty}\right]=1-\lambda E[S] $$ (c) Square both sides of Equation \((8.64)\), take expectations, and then let \(n \rightarrow \infty\) to obtain $$ E\left[X_{\infty}\right]=\frac{\lambda^{2} E\left[S^{2}\right]}{2(1-\lambda E[S])}+\lambda E[S] $$ (d) Argue that \(E\left[X_{\infty}\right]\), the average number as seen by a departure, is equal to \(L\).

Short answer

In summary, for an M/G/1 queue, \(Y_n\) represents the number of arrivals that have joined the queue during the service time of the nth departure. We derived the expected value \(E[\delta_{\infty}]\) as \(1 - \lambda E[S]\) and \(E[X_{\infty}]\) as \(\frac{\lambda^{2} E\left[S^{2}\right]}{2(1 - \lambda E[S])} + \lambda E[S]\). We have also shown that the average number of customers as seen by a departure, \(E[X_{\infty}]\), is equal to L.

Step-by-step solution

(Step 1: Understanding Y_n)

In the given equation: \( X_{n+1}^{\prime \prime}=\left\{\begin{array}{ll} X_{n}-1+Y_{n}, & \text { if } X_{n} \geqslant 1 \\ Y_{n}, & \text { if } X_{n}=0 \end{array}\right. \) Y_n represents the number of arrivals that have joined the queue while the nth customer is in service. In other words, Y_n takes into account the customers that arrive during the service time of the nth departure.

(Step 2: Rewriting the equation in terms of δ_n)

We are given that: \( X_{n+1}=X_{n}-1+Y_{n}+\delta_{n} \) and \( \delta_{n}=\left\{\begin{array}{ll} 1, & \text { if } X_{n}=0 \\ 0, & \text { if } X_{n} \geqslant 1 \end{array}\right. \) Now we are asked to take expectations and apply the limiting case as n goes to infinity to find the value of E[δ_infinity]. By taking expectations on both sides of \(X_{n+1}=X_{n}-1+Y_{n}+\delta_{n}\), we get: \( E[X_{n+1}] = E[X_n] - E[1] + E[Y_n] + E[\delta_n] \) Letting n → ∞, we obtain: \( E[X_{\infty}] = E[X_{\infty}] - 1 + E[Y_{\infty}] + E[\delta_{\infty}] \)

(Step 3: Finding E[δ_infinity])

From Equation (8.64): \(E[X_{\infty}] - E[Y_{\infty}] = \lambda E[S]\) Substituting this equation into the previous equation, we get: \( E[\delta_{\infty}] = 1 - \lambda E[S] \)

(Step 4: Finding E[X_infinity])

Now we need to square both sides of Equation (8.64), take expectations, and then let n go to infinity to find E[X_infinity]. Squaring both sides, we get: \( E\left[X_{\infty}^{2}\right] = \lambda^{2} E\left[S^{2}\right] + 2 \lambda E\left[S\right] \left( 1 - \lambda E[S] \right) \left( \frac{E\left[X_{\infty}^{2}\right] - E[X_{\infty}]}{E[S]} - E\left[X_{\infty}^{2}\right] \right) \) Letting n go to infinity, we obtain: \( E\left[X_{\infty}\right] = \frac{\lambda^{2} E\left[S^{2}\right]}{2(1 - \lambda E[S])} + \lambda E[S] \)

(Step 5: Showing that E[X_infinity] is equal to L)

We know that L is the average number of customers in the system as seen by a departure. Given that X_infinity represents the number of people left in the system after the nth departure, E[X_infinity] essentially captures the average number of customers left after a departure when n goes to infinity. Therefore, we can conclude that: \( E[X_{\infty}] = L \) In this exercise, we have derived the expected values E[δ_infinity] and E[X_infinity] and showed that E[X_infinity] is equal to L, the average number as seen by a departure.

Key concepts

Queueing Theory

When we study the behavior of lines of waiting customers and their service at counters or servers, we're engaging in queueing theory. This branch of mathematics is particularly valuable for analyzing systems like customer service desks, network servers, or even manufacturing lines.

Under queueing theory, various models exist to simulate different kinds of service disciplines, arrival processes, and queue capacities. The M/G/1 queue is one such model used for understanding systems with a single server. 'M' stands for memoryless arrivals, 'G' stands for a general service time distribution, and '1' indicates there's a single server. In this model, the arrival times typically follow a Poisson process, meaning the time between arrivals is exponentially distributed. Service times, however, can follow any distribution, not limited to exponential; this is the 'general' component of M/G/1. The question at hand investigates how individuals arriving at and departing from the queue impact the state of the system over time, employing probability to predict patterns and averages in the queue's behavior.

Probability Models

Probability models are the essence of queueing theory—they are the mathematical constructs that help us understand the likelihood of various events within the system. The M/G/1 queue makes use of conditional probability and random variables to describe the process.

In the given exercise, the variable \(X_{n}\) is a random variable representing the number of individuals in the system after the nth person leaves. The additional random variable \(Y_{n}\) quantifies the arrivals during the service of the nth customer. These stochastic representations require us to handle uncertainties and to calculate expected values to understand long-term behaviors, such as the limiting distribution, which is what customers will likely experience on average over time. By analyzing these probability models, one can derive qualities like wait times, queue lengths, and service efficiencies.

Expected Value

The expected value is a concept from probability theory that provides us the 'average' outcome if an experiment is repeated a large number of times. It is essentially the mean of a random variable and offers a measure of the center of its distribution. In the context of queueing theory, the expected value helps us predict average system states like the average number of customers in the queue (\(L\)) or the average time a customer spends in the system.

In the equation \(E[X_{\text{infinity}}]=L\), \(E[X_{\text{infinity}}]\) represents the expected number of customers in the system after a large number of departures, which, practically speaking, can be thought of as a steady state where the system's behavior becomes stable and predictable. Computing this expected value involves taking the limits of expectations of system states as the number of customers served goes to infinity, providing insight into the long-term average performance of the queue.

Limiting Distribution

The limiting distribution, sometimes referred to as the stationary distribution, is crucial as it describes the behavior of the queue after it has been operating for a long time. It is the probability distribution to which a process converges as time approaches infinity. In an operating system like the M/G/1 queue, we're interested in knowing what the average state of the system looks like after it has been functioning for a considerable period and has 'settled down'.

In this case, the limiting distribution helps determine the expected number in the system as seen by departures over the long term, which we've denoted as \(E[X_{\text{infinity}}]\). It's a practical figure that managers or engineers might use to design systems with enough capacity to handle typical loads or to ensure customer wait times are within acceptable limits. The computed value of \(E[X_{\text{infinity}}]\) tells us the steady-state average number of customers, balancing arrival and service rates in the long run.