Python实现

exponential()使用，自动poisson过程实现。

python">import numpy as np
import matplotlib.pyplot as plt# Parameters
lambda_rate = 5  # rate parameter (events per time unit)
T = 10  # total time# Generate Poisson process
times = np.random.exponential(1/lambda_rate, int(lambda_rate * T * 1))
arrival_times = np.cumsum(times)
arrival_times = arrival_times[arrival_times < T]# Plot Poisson process
plt.step(np.concatenate([[0], arrival_times]), np.arange(len(arrival_times) + 1), where='post')
plt.xlabel('Time')
plt.xticks(np.arange(0, T + 1, 1))
plt.ylabel('Number of events')
plt.title('Poisson Process')
plt.show()

 def poisson函数使用。

python">#正常显示中文标签
plt.rcParams['font.sans-serif'] = ['SimHei']
plt.rcParams['axes.unicode_minus'] = False
#%%
import random
import numpy as np
import matplotlib.pyplot as pltdef poisson(T,lambdas):t = 0i = 0s = [t]while True:u = random.expovariate(lambdas)t = t + uif t <= T:i = i + 1#计数s.append(np.round(t,2))#记录发生事件的时间，保留两位小数else:breakprint(f'{T}时间内事件发生的次数:',i,'\n 各个事件发生的时刻:',s)#绘制样本轨迹n = i + 1x = np.arange(n)plt.step(s,x, where='post')#绘制阶梯图plt.xticks(s)#设置x轴刻度plt.xlabel('时间t')plt.ylabel('N(t)')plt.title('Poisson Process')plt.show()return s
#生成图像
poisson(4,1)#参数分别为T和lambda，T为时间长度，lambda为单位时间内事件发生的次数

 

R语言实现

#R语言实现poisson过程
# Parameters
lambda_rate <- 5  # rate parameter (events per time unit)
T <- 10  # total time# Generate Poisson process
set.seed(123)  # for reproducibility
times <- rexp(n = ceiling(lambda_rate * T * 1.5), rate = lambda_rate)
arrival_times <- cumsum(times)
arrival_times <- arrival_times[arrival_times < T]# Plot Poisson process
plot(stepfun(arrival_times, 0:length(arrival_times)), do.points = FALSE, xlab = "Time", ylab = "Number of events", main = "Poisson Process")