Q.An ice-cream parlour receives a customer at an average rate of 4 per minute. If the number of customers received by the parlour follows a Poisson distribution, what is the approximate probability that 16 customers will be coming to the parlour in a particular 4-minute period on a given day?

Poisson Queueing Theory: A First Look

Imagine a coffee shop. Customers arrive at random times — sometimes three come in a minute, sometimes none for five minutes. The barista serves them one at a time, and each service takes a random duration. You want to answer questions like: How long will the next customer wait? How many customers will be in line on average? That is queueing theory.

Poisson queueing theory is the simplest, most powerful model for such systems. It rests on two key ideas: arrivals happen according to a Poisson process, and service times follow an exponential distribution. Together, they make the mathematics clean and the predictions sharp.

Intuition: What makes arrivals "Poisson"?

Think of a Poisson process as completely random arrivals — no memory, no pattern. If you watch the coffee shop door, the chance that a customer arrives in the next second is always the same, regardless of how long it has been since the last customer. This is called the memoryless property.

Concretely, if the average arrival rate is $\lambda$ customers per hour, then:

• The number of customers arriving in any fixed time interval (say, 10 minutes) follows a Poisson distribution.
• The time between consecutive arrivals follows an exponential distribution with mean $1/\lambda$.

The precise statement

A queueing system is described by Kendall's notation: $A/B/c$, where $A$ is the arrival process, $B$ is the service-time distribution, and $c$ is the number of servers. The classic Poisson queue is $M/M/1$:

• M = Markovian (Poisson) arrivals
• M = Markovian (exponential) service times
• 1 = one server

The system is stable only if $\lambda < \mu$ — otherwise the queue grows without bound.

Key results for $M/M/1$

Let $\rho = \lambda / \mu$ be the traffic intensity (the fraction of time the server is busy). Then: