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The Poisson process is fundamental in queueing theory and is used to model random events happening over time, such as customer arrivals in a queue. This process is characterized by a constant average rate, denoted as \( \lambda \), which represents the expected number of events (arrivals) per unit of time. An important feature of the Poisson process is that the number of events in non-overlapping intervals are independent of each other.
Poisson processes are applicable in various real-world scenarios beyond queueing, such as call arrivals at a call center or cars passing through a toll booth. In the context of an M/M/1 queue, it handles the arrival of customers efficiently since the system assumes a constant arrival rate, making it predictable.
The probability of observing exactly \( n \) events in a time interval \( t \) is given by the Poisson distribution formula:

• \( P(n; \lambda t) = \frac{e^{-\lambda t}(\lambda t)^n}{n!} \)

This property allows systems like the M/M/1 queue to model arrivals mathematically, estimate waiting times, and manage capacity planning effectively.

In the study of the M/M/1 queue, the service times and inter-arrival times are modeled using the exponential distribution. This distribution is essential due to its memoryless property, which means that the probability of an event occurring in the future is independent of how much time has already elapsed.
The exponential distribution is a continuous probability distribution, and it is characterized by its rate parameter \( \lambda \), which is the reciprocal of the mean, \( 1/\lambda \). In the case of service times, the rate parameter \( \mu \) represents the average number of services that can be completed per unit of time.
The probability density function (PDF) of an exponentially distributed random variable \( X \) is given by:

• \( f(x; \lambda) = \lambda e^{-\lambda x} \) for \( x \geq 0 \)

Furthermore, the cumulative distribution function (CDF), which shows the probability that the variable takes a value less than or equal to \( x \), is:

• \( F(x; \lambda) = 1 - e^{-\lambda x} \)

These mathematical functions are instrumental in determining system behaviors such as average waiting times and efficiency in queueing models.

The M/M/1 queue is a simple yet foundational model in queueing theory, representing a system with a single server and infinite calling population. The notation M/M/1 indicates a Markovian (memoryless) arrival process (first M), a Markovian service process (second M), and one server (1).
The M/M/1 model assumes that:

• Arrivals occur according to a Poisson process with rate \( \lambda \).
• Service times are exponentially distributed with rate \( \mu \).
• There is a single server handling arrivals.
• The system has infinite capacity for waiting customers.

A significant aspect of the M/M/1 queue is that it helps compute performance metrics like average number of customers in the system, average time a customer spends in the system, and the probability of having \( n \) customers in the system. Furthermore, because of its simplicity, it serves as a building block for more complex queuing models. In particular, the M/M/1 queue allows analysts to derive waiting time distributions and service efficiency, which are crucial in designing and managing systems effectively.

Probability distribution is a core concept in defining the likelihood of different outcomes in random experiments. In queueing theory, it determines how random variables, such as the number of customers arriving or the service times, behave. Understanding probability distributions helps model these uncertainties and make informed decisions in system management.
A Poisson distribution is used when modeling the number of events happening in a fixed interval of time, characterized by a single parameter \( \lambda \), the average rate. It is suitable for scenarios where events occur independently, like arrivals in a queue.
An exponential distribution, on the other hand, models the time between events in a Poisson process. It is key for understanding how long a customer might wait or how quickly a service might be completed.
In an M/M/1 queue, these distributions help derive key probabilities, such as:

• The likelihood that a customer's waiting time in queue will be precisely \( x \).
• The probability that no other customers are waiting when a new customer arrives.
• The expected number of customers in the system at any given time.

By applying these distributions, we are able to manage and optimize queueing systems effectively, ensuring that they operate smoothly and efficiently.