Question

Customers arrive at a single-server queue in accordance with a Poisson process having rate \(\lambda .\) However, an arrival that finds \(n\) customers already in the system will only join the system with probability \(1 /(n+1) .\) That is, with probability \(n /(n+1)\) such an arrival will not join the system. Show that the limiting distribution of the number of customers in the system is Poisson with mean \(\lambda / \mu .\)

Short answer

In summary, we analyzed the limiting distribution of the number of customers in a single-server queue following a Poisson process. By calculating the stationary probabilities for each state, we concluded that as the time approaches infinity, the limiting distribution is Poisson with mean \( \frac{\lambda}{\mu} \).

Step-by-step solution

Establish the limiting distribution as time approaches infinity

As t approaches infinity, the limiting distribution will be given by: \(P_n = \lim_{t \to \infty} P_n(t) = \lim_{t \to \infty} \frac{e^{-\lambda t} (\lambda t)^n}{n!}\)

Find the probabilities of each state n in the stationary distribution

Let's denote the stationary probabilities with π, so \(π_0 = 1\) \(π_n = \frac{π_{n-1}}{n}\), for n = 1, 2, 3, ... Using this recursive relation, we will calculate the stationary probabilities for each state: \(π_1 = \frac{π_0}{1} = 1\) \(π_2 = \frac{π_1}{2} = \frac{1}{2}\) \(π_3 = \frac{π_2}{3} = \frac{1}{2} \cdot \frac{1}{3} = \frac{1}{6}\) And so on, which means: \(π_n = \frac{1}{1!} \cdot \frac{1}{2!} \cdots \frac{1}{n!} = \frac{1}{n!} \)

Show that the limiting distribution is Poisson

Now we can show that the limiting distribution is Poisson with mean λ/μ. The Poisson distribution with mean λ/μ is given by: \(P_n = \frac{e^{-(\lambda / \mu)} (\lambda / \mu)^n}{n!} \) We need to demonstrate that the limiting distribution of customers in the system (π_n) is equal to the Poisson distribution with mean λ/μ. Note that when n = 0: \(P_0 = e^{-(\lambda / \mu)} \) If we take the ratio of the stationary probabilities: \(\frac{π_n}{π_{n-1}} = \frac{1}{n} \) Now consider the derivative: \(\frac{dP_{n}}{dn} = \frac{e^{-(\lambda / \mu)}}{n!}\left( -\frac{n\lambda}{\mu} + (\lambda/\mu) \right) \left(\frac {\lambda}{\mu}\right)^{n}\) \( \lim_{n\to\infty} \frac{dP_{n}}{dn} = 0 \) Therefore, as n approaches infinity, the limiting distribution will be Poisson with mean λ/μ, which is the result we wanted to prove. In conclusion, we have shown that the limiting distribution of the number of customers in the system is Poisson with mean λ/μ.

Key concepts

Poisson Process

If you've heard of random events occurring independently over a certain period of time or space, then you're on your way to understanding the Poisson process. It's a mathematical model used extensively in queueing theory to describe situations like the arrival of customers at a service center or calls coming into a call center. The key characteristic of a Poisson process is that the number of events occurring in disjoint time intervals is independent of each other, and the probability of a single event occurring within a very small time interval is proportional to the length of that interval.

For instance, if customers arrive at a bank according to a Poisson process with a rate \(\lambda\), this means that over a small interval \(dt\), the probability of one customer arriving is \(\lambda \cdot dt\), and the probability of two or more arrivals is negligible. As the intervals get extremely small, these probabilities define a continuous-time stochastic process, which is what we call the Poisson process.

Stationary Probabilities

Delving into the realm of queueing theory, we come across the set of stationary probabilities that describe the long-term behavior of the system. These probabilities, denoted as \(\pi_n\), do not change over time, which is why we call them 'stationary'. They give us the likelihood of finding the system in a particular state—say, with \(n\) customers in the queue—after a long period of operation, regardless of the initial state.

Think of it as the system's 'equilibrium' distribution. Calculating these values is essential because once we have them, we can determine various performance metrics of the queue, such as the average number of customers waiting or the probability that an arriving customer has to wait before being served. The exercise provided showcases a fundamental recursive relationship that simplifies the calculation of these probabilities for each state of the system.

Single-Server Queue

The single-server queue is a cornerstone example of a queueing system. Picture a local grocery store with a single checkout line. Customers form a line (the queue) to be served by the only available cashier (the server). In a single-server queue framework, we are often interested in answering questions like 'How long on average does a customer spend in the system?' or 'What's the probability that a new arrival will need to wait?'.

Such a queue can operate under various disciplines like 'First-Come-First-Served' (FCFS) or 'Last-In-First-Out' (LIFO), but the basic principle remains the same: there is only one service mechanism available, and it can handle one customer at a time. The exercise tweak with join probabilities (where customers may decide not to join the queue) adds an interesting twist to the classic model and affects the stationary probabilities.

Poisson Distribution

The Poisson distribution is the trusty sidekick of the Poisson process—it's the discrete probability distribution that tells us the probability of a given number of events occurring in a fixed interval of time or space, assuming these events occur with a known constant mean rate and independently of the time since the last event.

The classic example is the number of emails one might receive in an hour. If you know on average you receive 5 emails per hour, the Poisson distribution can be used to calculate the probability of receiving exactly 3 emails in the upcoming hour. In mathematical terms, the probability of observing \(n\) events is given by the formula \(P_n = \frac{e^{-\lambda} (\lambda)^n}{n!}\), where \(\lambda\) is the average rate of occurrence. When it comes to queues, understanding and utilizing the Poisson distribution is indispensable for planning and analysis, as it provides a fundamental link between the abstract theory and real-world applications.