interarrivalajat

Interarrivalajat, or interarrival times, are the durations between consecutive events in a point process. They describe the spacing of events such as arrivals, failures, or Occurrences over time and are treated as random variables that help characterize the underlying process.

In the classic Poisson process with rate λ, the interarrivalajat are independent and identically distributed exponential random variables with mean 1/λ and variance 1/λ^2. The exponential distribution’s memoryless property means that the waiting time to the next event does not depend on the time already elapsed. The time to the k-th event is the sum of k interarrivalajat and follows a Gamma (or Erlang) distribution with shape parameter k and rate λ.

Beyond the Poisson model, interarrivalajat can follow a wide range of distributions in renewal processes, where the times between events are generally distributed and may be dependent on the modeling assumptions. The sum of the first n interarrivalajat gives the time of the n-th event, which is a fundamental relation in renewal theory.

Estimation and usage: interarrivalajat are typically inferred from observed event timestamps. Methods include nonparametric estimation of the empirical distribution or density, as well as fitting parametric families to data. They are used in queueing theory, computer networks, reliability engineering, finance, seismology, and other fields to analyze traffic, performance, and risk. Key considerations include the independence and identically distributed assumptions, potential heavy tails, and censoring in observed data.