1 00:00:00,260 --> 00:00:03,800 In this lecture, we introduce the Poisson process, which is 2 00:00:03,800 --> 00:00:07,320 a continuous time analog of the Bernoulli process. 3 00:00:07,320 --> 00:00:10,690 One way of thinking about it is as follows. 4 00:00:10,690 --> 00:00:14,610 Time is continuous, but conceptually we divide it into 5 00:00:14,610 --> 00:00:16,640 a very large number of slots. 6 00:00:16,640 --> 00:00:19,090 And during each slot, we have a tiny 7 00:00:19,090 --> 00:00:21,240 probability of an arrival. 8 00:00:21,240 --> 00:00:23,570 This probability is proportional to the 9 00:00:23,570 --> 00:00:25,610 length of the slot. 10 00:00:25,610 --> 00:00:28,150 Furthermore, we have an independence assumption for 11 00:00:28,150 --> 00:00:29,400 the different slots. 12 00:00:31,860 --> 00:00:35,560 The Poisson process is a very elegant model of arrival 13 00:00:35,560 --> 00:00:37,870 processes in continuous time. 14 00:00:37,870 --> 00:00:40,850 It models many real-world phenomena. 15 00:00:40,850 --> 00:00:44,390 And it also has a very clean mathematical structure that 16 00:00:44,390 --> 00:00:46,970 allows us to calculate practically 17 00:00:46,970 --> 00:00:50,300 every quantity of interest. 18 00:00:50,300 --> 00:00:53,820 Our development will parallel our analysis of 19 00:00:53,820 --> 00:00:55,640 the Bernoulli process. 20 00:00:55,640 --> 00:00:59,830 For example, we will find the PMF of the number of arrivals 21 00:00:59,830 --> 00:01:03,030 during a time interval and the PDF of the 22 00:01:03,030 --> 00:01:05,930 time of the kth arrival. 23 00:01:05,930 --> 00:01:08,950 We will discuss the memorylessness properties of 24 00:01:08,950 --> 00:01:10,900 the Poisson process. 25 00:01:10,900 --> 00:01:13,539 Similar to the case of the Bernoulli process, this is 26 00:01:13,539 --> 00:01:16,450 just a consequence of the independence assumptions that 27 00:01:16,450 --> 00:01:18,580 we are making. 28 00:01:18,580 --> 00:01:21,530 We will then exploit these independence properties to 29 00:01:21,530 --> 00:01:25,580 argue that the interarrival times are independent 30 00:01:25,580 --> 00:01:27,950 exponential random variables. 31 00:01:27,950 --> 00:01:30,480 And we will conclude with a comprehensive example.