Implementation of Poisson Process Algorithm in Stochastic Processes

In stochastic processes, the Poisson algorithm serves as one of the fundamental methodologies. The MATLAB implementation of the Poisson process algorithm involves the following sequential steps: 1. Configure the parameter λ (lambda), representing the average number of events occurring per unit time. This parameter defines the rate of the Poisson process and is typically implemented as a scalar input variable in MATLAB code. 2. Generate event time sequences using random number generators that produce Poisson-distributed random variables. MATLAB's built-in functions like poissrnd() or alternative approaches using exponential distributions can be employed to simulate event occurrences. 3. Sort the generated time series chronologically to facilitate subsequent analysis. This sorting operation ensures proper sequencing of events and can be achieved using MATLAB's sort() function. 4. Calculate inter-event time intervals by computing differences between consecutive event occurrence times. These intervals follow an exponential distribution with mean 1/λ, which can be verified through statistical analysis. 5. Plot frequency distribution histograms of the time intervals to visually examine event occurrence patterns. MATLAB's histogram() function can be utilized to display the distribution characteristics and validate the exponential nature of inter-arrival times. By implementing these steps, researchers can gain deeper insights into the application of Poisson algorithms in stochastic processes and master their practical implementation in MATLAB. The complete routine typically involves approximately 10-15 lines of code, incorporating random number generation, mathematical operations, and visualization commands.