1 00:00:00,530 --> 00:00:03,500 This segment is probably the most critical one 2 00:00:03,500 --> 00:00:07,040 for the purpose of understanding what the Poisson process really 3 00:00:07,040 --> 00:00:09,260 is and how it behaves. 4 00:00:09,260 --> 00:00:12,050 There will be almost no mathematical formulas. 5 00:00:12,050 --> 00:00:14,560 But the segment will be quite dense 6 00:00:14,560 --> 00:00:16,960 in terms of conceptual reasoning. 7 00:00:16,960 --> 00:00:20,100 So pay a lot of attention. 8 00:00:20,100 --> 00:00:24,360 In a nutshell, we will argue that the Poisson process has 9 00:00:24,360 --> 00:00:28,470 memorylessness properties that are entirely similar to those 10 00:00:28,470 --> 00:00:30,862 that we have seen for the Bernoulli process. 11 00:00:30,862 --> 00:00:34,190 This should not be surprising, since the Poisson process can 12 00:00:34,190 --> 00:00:37,710 be thought of as a limiting case of the Bernoulli process. 13 00:00:37,710 --> 00:00:39,660 We will reason through these properties, 14 00:00:39,660 --> 00:00:42,610 not in the style of a formal mathematical proof, 15 00:00:42,610 --> 00:00:44,960 but with an intuitive argument. 16 00:00:44,960 --> 00:00:48,240 But I would like to assure you that the intuitive argument can 17 00:00:48,240 --> 00:00:52,690 be translated into a rigorous proof. 18 00:00:52,690 --> 00:00:55,050 The first property is the following. 19 00:00:55,050 --> 00:00:57,970 The process starts at time 0. 20 00:00:57,970 --> 00:01:03,130 You come in and start watching at let's say time 7. 21 00:01:03,130 --> 00:01:05,519 Or more generally, instead of time 7, 22 00:01:05,519 --> 00:01:07,500 suppose that you come in and start 23 00:01:07,500 --> 00:01:10,250 watching at some time, little t. 24 00:01:10,250 --> 00:01:13,900 The important thing here is that little t is a constant. 25 00:01:13,900 --> 00:01:16,500 It's a deterministic number. 26 00:01:16,500 --> 00:01:22,590 Starting at that time, what will you see? 27 00:01:22,590 --> 00:01:26,210 Well, the original process was Poisson. 28 00:01:26,210 --> 00:01:29,140 This means that disjoint intervals 29 00:01:29,140 --> 00:01:33,229 in the original process are independent. 30 00:01:33,229 --> 00:01:36,450 Therefore, disjoint intervals in the process 31 00:01:36,450 --> 00:01:41,360 that you will be seeing will also be independent. 32 00:01:41,360 --> 00:01:45,100 Furthermore, during any little interval of length 33 00:01:45,100 --> 00:01:47,509 delta in the process that you see 34 00:01:47,509 --> 00:01:49,880 will still have probability lambda times 35 00:01:49,880 --> 00:01:53,330 delta, approximately, of seeing and arrival. 36 00:01:53,330 --> 00:01:56,650 Therefore, what you see also satisfies the properties 37 00:01:56,650 --> 00:02:01,220 of a Poisson process, and is itself a Poisson process. 38 00:02:01,220 --> 00:02:05,860 Second, the original process was Poisson. 39 00:02:05,860 --> 00:02:08,509 So different intervals are independent. 40 00:02:08,509 --> 00:02:10,520 So whatever happens in this interval 41 00:02:10,520 --> 00:02:13,180 is independent from whatever happens in that interval. 42 00:02:13,180 --> 00:02:16,550 But that interval corresponds to the future of the process, 43 00:02:16,550 --> 00:02:21,079 and therefore, the future of the process, what you get to see, 44 00:02:21,079 --> 00:02:23,990 is independent from the past history. 45 00:02:23,990 --> 00:02:27,900 And so the conclusion is that the process that you get to see 46 00:02:27,900 --> 00:02:32,180 is a Poisson process, which is independent of the history 47 00:02:32,180 --> 00:02:35,150 until the time that you started watching. 48 00:02:35,150 --> 00:02:37,970 And we say, therefore, that what you see 49 00:02:37,970 --> 00:02:42,520 is a fresh starting process. 50 00:02:42,520 --> 00:02:46,280 The Poisson process starts fresh at time t. 51 00:02:46,280 --> 00:02:48,640 We have the fresh start property. 52 00:02:48,640 --> 00:02:51,360 And similar to the language we use for the Bernoulli process, 53 00:02:51,360 --> 00:02:55,340 the fresh start property means that you see a process that's 54 00:02:55,340 --> 00:02:57,630 independent of the past and which 55 00:02:57,630 --> 00:03:01,890 has the same statistical properties as if this was time 56 00:03:01,890 --> 00:03:06,930 0, as if the process was just starting right now. 57 00:03:06,930 --> 00:03:09,810 One consequence of this fresh start property 58 00:03:09,810 --> 00:03:11,230 is the following. 59 00:03:11,230 --> 00:03:13,600 You start watching at time t. 60 00:03:13,600 --> 00:03:15,720 And you're interested in the time 61 00:03:15,720 --> 00:03:19,820 it takes until the next arrival. 62 00:03:19,820 --> 00:03:23,640 What are the properties of this random variable? 63 00:03:23,640 --> 00:03:26,490 Well, since you have a fresh starting Poisson process 64 00:03:26,490 --> 00:03:30,410 at this time, this is the time until the first arrival 65 00:03:30,410 --> 00:03:34,270 in this fresh starting Poisson process. 66 00:03:34,270 --> 00:03:36,940 And the time until the first arrival 67 00:03:36,940 --> 00:03:39,079 in a process that is just starting, 68 00:03:39,079 --> 00:03:42,760 we know that it has an exponential distribution. 69 00:03:42,760 --> 00:03:46,090 So this is going to be an exponential random variable 70 00:03:46,090 --> 00:03:48,579 with the same parameter, lambda. 71 00:03:48,579 --> 00:03:51,329 Furthermore, because the process starts fresh, 72 00:03:51,329 --> 00:03:55,120 whatever happens in the future is independent from the past. 73 00:03:55,120 --> 00:03:58,310 And so this random variable, the remaining time, 74 00:03:58,310 --> 00:04:03,010 is independent of whatever happened in the past until time 75 00:04:03,010 --> 00:04:03,510 t. 76 00:04:09,810 --> 00:04:13,480 Now let us look at a somewhat different situation. 77 00:04:13,480 --> 00:04:19,000 You start watching the process at time T1. 78 00:04:19,000 --> 00:04:23,600 Time T1 is the time of the first arrival. 79 00:04:23,600 --> 00:04:27,130 And you start watching from here on. 80 00:04:27,130 --> 00:04:29,530 What is it that you're going to see? 81 00:04:29,530 --> 00:04:31,860 Suppose that the first arrival happens, 82 00:04:31,860 --> 00:04:34,760 let's say, at time equal to 3. 83 00:04:34,760 --> 00:04:37,659 So we're conditioning on this event. 84 00:04:37,659 --> 00:04:41,290 In that case, you start watching the process at time 3. 85 00:04:41,290 --> 00:04:45,540 And you also know that the first arrival happened at time 3. 86 00:04:45,540 --> 00:04:49,220 But this fact about the first arrival happening at time 3 87 00:04:49,220 --> 00:04:53,710 belongs to the history of the process until time 3. 88 00:04:53,710 --> 00:04:55,850 This is information about the past, 89 00:04:55,850 --> 00:05:00,400 and does not affect what is going to happen after time 3. 90 00:05:00,400 --> 00:05:04,710 The process after time 3 will be independent from the history 91 00:05:04,710 --> 00:05:08,120 until time 3 and whatever happened until that time. 92 00:05:08,120 --> 00:05:11,840 So starting at that particular time 3, what you see 93 00:05:11,840 --> 00:05:15,860 is a Poisson process that is independent from the past. 94 00:05:15,860 --> 00:05:18,830 Now, this argument is valid even if I 95 00:05:18,830 --> 00:05:24,260 were to use here a 3.5 or 3.4 or 3.7. 96 00:05:24,260 --> 00:05:28,130 No matter when this first arrival occurred, 97 00:05:28,130 --> 00:05:30,820 what I see starting from this time 98 00:05:30,820 --> 00:05:36,860 is a Poisson process which is independent from the past. 99 00:05:36,860 --> 00:05:38,850 At the time of the first arrival, 100 00:05:38,850 --> 00:05:43,230 the process just starts fresh. 101 00:05:43,230 --> 00:05:47,070 As a consequence of this, and by repeating the argument that we 102 00:05:47,070 --> 00:05:50,260 carried out for the remaining time until the next arrival 103 00:05:50,260 --> 00:05:52,900 up here, we can repeat this argument 104 00:05:52,900 --> 00:05:56,590 and argue that the time until the next arrival 105 00:05:56,590 --> 00:06:00,440 in this fresh starting process, this 106 00:06:00,440 --> 00:06:05,490 will also be an exponential random variable. 107 00:06:05,490 --> 00:06:07,780 Now, this time until the next arrival 108 00:06:07,780 --> 00:06:10,890 is the difference between the second arrival 109 00:06:10,890 --> 00:06:13,380 time and the first arrival time. 110 00:06:13,380 --> 00:06:16,310 And we denote it by T2. 111 00:06:16,310 --> 00:06:21,280 What we just argued is that this time until the next arrival 112 00:06:21,280 --> 00:06:25,230 is going to be an exponential random variable. 113 00:06:25,230 --> 00:06:29,620 And also, it is independent from the past. 114 00:06:29,620 --> 00:06:35,620 And in particular, it is independent from T1. 115 00:06:35,620 --> 00:06:38,270 So the time until the second arrival, 116 00:06:38,270 --> 00:06:42,500 starting from the first arrival, the second inter-arrival time 117 00:06:42,500 --> 00:06:46,350 is a random variable that has an exponential distribution that 118 00:06:46,350 --> 00:06:49,190 is the same distribution as that of T1, 119 00:06:49,190 --> 00:06:52,030 and is independent from T1. 120 00:06:52,030 --> 00:06:54,870 Now we can extend this argument and look at the kth 121 00:06:54,870 --> 00:06:56,550 inter-arrival time. 122 00:06:56,550 --> 00:07:00,410 For example, if the arrival numbered k minus 1 123 00:07:00,410 --> 00:07:03,960 occurred here, and the k arrival occurs here, 124 00:07:03,960 --> 00:07:08,650 this difference, here we denote it by Tk, 125 00:07:08,650 --> 00:07:12,270 and by arguing in a similar way that the process starts 126 00:07:12,270 --> 00:07:16,490 fresh at this particular time, the time until the next arrival 127 00:07:16,490 --> 00:07:19,400 will also be an exponential random variable 128 00:07:19,400 --> 00:07:21,510 with the same distribution. 129 00:07:21,510 --> 00:07:23,660 And furthermore, will be independent 130 00:07:23,660 --> 00:07:26,540 from the past history, and therefore, 131 00:07:26,540 --> 00:07:31,000 independent from the earlier inter-arrival times. 132 00:07:31,000 --> 00:07:34,730 And this has lots of important implications. 133 00:07:34,730 --> 00:07:38,840 For example, the time until the kth arrival, 134 00:07:38,840 --> 00:07:43,260 which is the sum of the first k inter-arrival times, 135 00:07:43,260 --> 00:07:47,850 is the sum of independent, identically distributed, 136 00:07:47,850 --> 00:07:50,920 exponential random variables. 137 00:07:50,920 --> 00:07:55,170 In particular, this means that we can find the PDF of Yk 138 00:07:55,170 --> 00:07:58,370 by convolving the exponential PDF 139 00:07:58,370 --> 00:08:03,350 of these inter-arrival times, convolving this exponential PDF 140 00:08:03,350 --> 00:08:05,570 with itself k times. 141 00:08:05,570 --> 00:08:09,490 And this is indeed one way to find the PDF of Yk. 142 00:08:09,490 --> 00:08:12,340 But fortunately for us, we were able to find it 143 00:08:12,340 --> 00:08:13,970 with a much simpler argument. 144 00:08:13,970 --> 00:08:16,590 And we already know what it is. 145 00:08:16,590 --> 00:08:20,490 But this property here is also useful for finding 146 00:08:20,490 --> 00:08:22,790 the mean and the variance of Yk. 147 00:08:22,790 --> 00:08:25,850 The mean of the sum is the sum of the means. 148 00:08:25,850 --> 00:08:27,960 And since the random variables are independent, 149 00:08:27,960 --> 00:08:30,980 the variance of the sum is the sum of the variances. 150 00:08:30,980 --> 00:08:34,280 We know what is the mean and the variance of an exponential. 151 00:08:34,280 --> 00:08:37,308 And so by multiplying that by k, we 152 00:08:37,308 --> 00:08:40,429 obtain the mean of the kth arrival time 153 00:08:40,429 --> 00:08:43,630 and the variance of the kth arrival time. 154 00:08:43,630 --> 00:08:45,720 And so we now know the mean and variance 155 00:08:45,720 --> 00:08:49,820 of the Erlang PDF of order k. 156 00:08:49,820 --> 00:08:53,290 A second implication of this property 157 00:08:53,290 --> 00:08:56,150 is more theoretical, more conceptual. 158 00:08:56,150 --> 00:08:58,960 Recall that we defined the Poisson process 159 00:08:58,960 --> 00:09:01,320 in terms of an independence assumption 160 00:09:01,320 --> 00:09:04,540 and an assumption on the probability of arrivals 161 00:09:04,540 --> 00:09:06,340 during a small interval. 162 00:09:06,340 --> 00:09:11,780 But we could have defined the Poisson process as follows. 163 00:09:11,780 --> 00:09:14,600 Consider a sequence of independent, identically 164 00:09:14,600 --> 00:09:16,830 distributed exponentials. 165 00:09:16,830 --> 00:09:18,650 Call them Tk. 166 00:09:18,650 --> 00:09:23,370 And use these to define the arrival times. 167 00:09:23,370 --> 00:09:27,140 This is a way of constructing a process. 168 00:09:27,140 --> 00:09:29,880 What we argued in this segment is 169 00:09:29,880 --> 00:09:33,500 that a Poisson process under the original definition 170 00:09:33,500 --> 00:09:36,970 satisfies this new definition. 171 00:09:36,970 --> 00:09:38,830 One can complete the argument to show 172 00:09:38,830 --> 00:09:40,770 that the two definitions are equivalent. 173 00:09:40,770 --> 00:09:44,260 It is possible to argue that if we define an arrival 174 00:09:44,260 --> 00:09:48,570 process in this manner, this arrival process will also 175 00:09:48,570 --> 00:09:52,490 satisfy the basic properties of the Poisson process. 176 00:09:52,490 --> 00:09:54,720 This argument can indeed be carried out, 177 00:09:54,720 --> 00:09:57,810 but we will not go through it. 178 00:09:57,810 --> 00:10:01,720 A final implication, which is a little more practical. 179 00:10:01,720 --> 00:10:04,440 If you want to simulate the Poisson process, 180 00:10:04,440 --> 00:10:06,590 how would you do it? 181 00:10:06,590 --> 00:10:11,130 Given what we now know, the most natural way is the following. 182 00:10:11,130 --> 00:10:15,320 We generate independent, identically distributed, 183 00:10:15,320 --> 00:10:17,560 exponential random variables, using 184 00:10:17,560 --> 00:10:20,070 for example a random number generator. 185 00:10:20,070 --> 00:10:23,730 And then use these exponential random variables 186 00:10:23,730 --> 00:10:27,530 to construct the values of the inter arrival times. 187 00:10:27,530 --> 00:10:30,710 And this way, construct a complete time history 188 00:10:30,710 --> 00:10:33,050 of the Poisson process.