Applications of Stochastic Processes in Engineering: The Poisson Process and a Simulation Study of Vehicle Arrivals

Abstract

Stochastic processes are important to model systems that have components of randomness, and that develop over time; they are useful in engineering. Of those, arguably the most basic and indispensable is the Poisson process which is also used to describe the likelihood of random, independent events to occur in various applications such as telecommunication, reliability, transportation, manufacturing and energy systems. Drawing upon mathematical theory the Poisson process is defined by its rate parameter which is λ, and also explains that it possesses nice features such as independent and stationary increments which make it suitable for modelling real-life phenomena. Communication systems apply the Poisson process to model packet arrivals and enhance the performance of the application, and reliability engineering uses it to estimate the failure rate and plan for maintenance. In transport, the process is useful in modelling ways of managing traffic and even the schedule of a public transport system. Manufacturing systems use quality assurance and inventory control modeling while energy systems use the Poisson process for power outages and renewable energy systems planning. The article also discusses generalizations of non-Poisson processes for practical situations where the rate assumptions and events are correlated. To move from description to demonstration, the paper further reports a Monte Carlo experiment in which 500 one-minute intervals of vehicle arrivals are generated from exponential interarrival times at a mean rate of λ = 20 vehicles per minute. The simulated counts are compared with the theoretical Poisson distribution through a time series, a histogram, an interarrival plot and a quantile-quantile plot, and are tested formally by a chi-square goodness-of-fit test (χ² = 7.40, df = 6, p = 0.285); the interarrival times are tested against the exponential law (Kolmogorov–Smirnov D = 0.012, p = 0.122). Neither test finds evidence against the Poisson model, and the experiment quantifies how much minute-to-minute variation an engineer should expect around a constant mean rate — a result with direct consequences for signal timing and capacity margins.