1 00:00:00,500 --> 00:00:02,380 We will now study what happens when 2 00:00:02,380 --> 00:00:05,520 we merge independent Poisson processes. 3 00:00:05,520 --> 00:00:07,920 The story as well as the final conclusion 4 00:00:07,920 --> 00:00:10,750 is going to be similar to what happened for the case 5 00:00:10,750 --> 00:00:13,720 where we merged independent Bernoulli processes. 6 00:00:13,720 --> 00:00:16,880 In particular, we will see that the merged process will also 7 00:00:16,880 --> 00:00:19,310 be Poisson. 8 00:00:19,310 --> 00:00:21,100 What is the story? 9 00:00:21,100 --> 00:00:23,690 Suppose that you have two light bulbs. 10 00:00:23,690 --> 00:00:27,830 One of them is red and flashes at random times that 11 00:00:27,830 --> 00:00:30,920 are described according to a Poisson process 12 00:00:30,920 --> 00:00:33,260 with a certain rate, lambda1. 13 00:00:33,260 --> 00:00:36,960 The other light bulb is green, flashes also as a Poisson 14 00:00:36,960 --> 00:00:39,040 process with a certain rate. 15 00:00:39,040 --> 00:00:41,700 Furthermore, we assume that the two light bulbs 16 00:00:41,700 --> 00:00:44,410 are independent of each other. 17 00:00:44,410 --> 00:00:47,600 If you're color blind so that the only thing that you see 18 00:00:47,600 --> 00:00:51,540 is flashes but without being able to tell the color, 19 00:00:51,540 --> 00:00:54,740 what kind of process are you going to see? 20 00:00:54,740 --> 00:00:58,530 Well, you will see a Poisson process also, 21 00:00:58,530 --> 00:01:00,140 and at this point, you can probably 22 00:01:00,140 --> 00:01:02,410 guess what the arrival rate is going 23 00:01:02,410 --> 00:01:04,239 to be for this Poisson process. 24 00:01:04,239 --> 00:01:07,790 It should be the sum of these two arrival 25 00:01:07,790 --> 00:01:10,950 rates of the processes that you started with. 26 00:01:10,950 --> 00:01:13,120 So this will be our final conclusion, 27 00:01:13,120 --> 00:01:16,880 but we want to verify that this is indeed 28 00:01:16,880 --> 00:01:18,789 the correct conclusion. 29 00:01:18,789 --> 00:01:22,000 So let us look at the situation in some more detail. 30 00:01:22,000 --> 00:01:26,130 We have the two processes, two arrival processes-- the red one 31 00:01:26,130 --> 00:01:29,200 and the green one-- and the merged process 32 00:01:29,200 --> 00:01:35,080 is formed by recording an arrival at any time where 33 00:01:35,080 --> 00:01:36,970 either of the two processes that you 34 00:01:36,970 --> 00:01:40,990 started with records an arrival. 35 00:01:40,990 --> 00:01:45,530 Let us now look at the time interval 36 00:01:45,530 --> 00:01:49,180 and think about the number of arrivals in the merged process 37 00:01:49,180 --> 00:01:51,220 during this time interval. 38 00:01:51,220 --> 00:01:52,850 What is that the number? 39 00:01:52,850 --> 00:01:56,870 That number is equal to the number of arrivals 40 00:01:56,870 --> 00:02:02,060 that you have in the first process plus 41 00:02:02,060 --> 00:02:08,590 the number of arrivals that you have in the second process. 42 00:02:08,590 --> 00:02:13,260 Let's call those numbers N1 and N2 so that what we have here 43 00:02:13,260 --> 00:02:15,960 is N1 plus N2. 44 00:02:15,960 --> 00:02:18,880 Now, N1 is a Poisson random variable 45 00:02:18,880 --> 00:02:20,930 because this is a Poisson process. 46 00:02:20,930 --> 00:02:23,380 Similarly, N2 is a Poisson random variable. 47 00:02:23,380 --> 00:02:26,270 We assume that these two processes are independent. 48 00:02:26,270 --> 00:02:29,800 Therefore, N1 plus N2 is the sum of independent Poisson 49 00:02:29,800 --> 00:02:33,010 random variables, and therefore, N1 plus N2 50 00:02:33,010 --> 00:02:36,190 is also a Poisson random variable. 51 00:02:36,190 --> 00:02:37,380 This is reassuring. 52 00:02:37,380 --> 00:02:39,960 It's a good piece of evidence that the blue process 53 00:02:39,960 --> 00:02:43,400 is a Poisson process, but this is not enough. 54 00:02:43,400 --> 00:02:45,280 To argue that it is a Poisson process, 55 00:02:45,280 --> 00:02:48,030 we need to check the defining properties of a Poisson 56 00:02:48,030 --> 00:02:49,220 process. 57 00:02:49,220 --> 00:02:52,430 One defining property is the independence property. 58 00:02:52,430 --> 00:02:55,950 If we take disjoint intervals, the number of arrivals 59 00:02:55,950 --> 00:02:58,810 here is independent, or should be independent, 60 00:02:58,810 --> 00:03:01,240 from the number of arrivals there. 61 00:03:01,240 --> 00:03:04,700 The argument here is exactly the same as for the Bernoulli case, 62 00:03:04,700 --> 00:03:08,970 so we will not go through it in any detail. 63 00:03:08,970 --> 00:03:12,980 We just notice that whatever happens during that time 64 00:03:12,980 --> 00:03:17,270 has to do with whatever happens during those times in the two 65 00:03:17,270 --> 00:03:19,260 processes that we started with. 66 00:03:19,260 --> 00:03:21,900 And similarly, what happens in these times 67 00:03:21,900 --> 00:03:24,530 has to do with what happens in these two processes 68 00:03:24,530 --> 00:03:25,930 during those times. 69 00:03:25,930 --> 00:03:29,890 Because for each one of the two processes that we start with, 70 00:03:29,890 --> 00:03:33,560 we have the Poisson assumption so that this interval 71 00:03:33,560 --> 00:03:36,480 is independent from that interval in the sense 72 00:03:36,480 --> 00:03:39,610 that arrivals here and arrivals there are independent. 73 00:03:39,610 --> 00:03:42,880 So because of this, whatever happens during those 74 00:03:42,880 --> 00:03:44,790 times has nothing to do with whatever 75 00:03:44,790 --> 00:03:47,340 happens in those times, so number of arrivals 76 00:03:47,340 --> 00:03:51,480 here is independent from the number of arrivals there. 77 00:03:51,480 --> 00:03:53,340 The other property that we need to check 78 00:03:53,340 --> 00:03:56,530 is that the probability of recording an arrival 79 00:03:56,530 --> 00:03:59,880 during a small time interval of length delta, 80 00:03:59,880 --> 00:04:03,800 that this probability has the right scaling 81 00:04:03,800 --> 00:04:07,560 properties, that it is linear in delta, 82 00:04:07,560 --> 00:04:10,630 in the length of this interval, and that the probability of two 83 00:04:10,630 --> 00:04:13,700 or more arrivals here is negligible. 84 00:04:13,700 --> 00:04:17,370 To see what happens during a typical interval in the merged 85 00:04:17,370 --> 00:04:20,040 process, we need to consider what 86 00:04:20,040 --> 00:04:23,160 is going to happen during that typical interval 87 00:04:23,160 --> 00:04:26,050 in the other two processes and consider 88 00:04:26,050 --> 00:04:28,520 all the possible combinations. 89 00:04:28,520 --> 00:04:31,370 During a little interval, the red process 90 00:04:31,370 --> 00:04:35,530 is going to have zero arrivals with this probability, 91 00:04:35,530 --> 00:04:37,380 one arrival with this probability, 92 00:04:37,380 --> 00:04:40,290 and two or more arrivals with this probability, which 93 00:04:40,290 --> 00:04:41,650 is negligible. 94 00:04:41,650 --> 00:04:46,159 Actually here, we're ignoring terms of order delta squared. 95 00:04:46,159 --> 00:04:47,920 These are the correct expressions 96 00:04:47,920 --> 00:04:52,180 if we only focus on terms that are either constants or linear 97 00:04:52,180 --> 00:04:53,000 in delta. 98 00:04:53,000 --> 00:04:55,290 We are ignoring terms that are of order delta 99 00:04:55,290 --> 00:04:56,900 square or higher. 100 00:04:56,900 --> 00:04:58,760 And similarly for the green process, 101 00:04:58,760 --> 00:05:01,510 we have these probabilities for the number of arrivals 102 00:05:01,510 --> 00:05:04,570 that may happen during a small interval. 103 00:05:04,570 --> 00:05:09,750 For the merged process, we will have zero arrivals if and only 104 00:05:09,750 --> 00:05:14,180 if we have zero arrivals in the red process 105 00:05:14,180 --> 00:05:16,190 and zero arrivals in the green process. 106 00:05:16,190 --> 00:05:18,570 The probability of these two events happening, 107 00:05:18,570 --> 00:05:21,760 because we assume that the two processes that we started with 108 00:05:21,760 --> 00:05:24,250 are independent, is going to be the product 109 00:05:24,250 --> 00:05:29,130 of the probabilities of zero arrivals in one process times 110 00:05:29,130 --> 00:05:33,480 zero arrivals in the other process. 111 00:05:33,480 --> 00:05:37,550 We multiply those two terms, and if we throw away 112 00:05:37,550 --> 00:05:40,500 the term delta squared, which is negligible, 113 00:05:40,500 --> 00:05:44,570 we see that this event is going to happen with probability 1 114 00:05:44,570 --> 00:05:49,170 minus lambda1 plus lambda2 times delta. 115 00:05:52,630 --> 00:05:57,180 What's the probability that we get one arrival? 116 00:05:57,180 --> 00:06:00,310 This is an event that can happen in two ways. 117 00:06:00,310 --> 00:06:03,910 We could have one arrival in the red process 118 00:06:03,910 --> 00:06:07,060 and zero arrivals in the green process, 119 00:06:07,060 --> 00:06:11,340 and this combination happens with this probability. 120 00:06:11,340 --> 00:06:16,040 Alternatively, we could have one arrival in the green process 121 00:06:16,040 --> 00:06:18,573 and zero arrivals in the red process. 122 00:06:18,573 --> 00:06:23,000 This is this event and it happens with this probability. 123 00:06:23,000 --> 00:06:25,600 Having one arrival in the blue process 124 00:06:25,600 --> 00:06:28,060 can happen either this way or that way, 125 00:06:28,060 --> 00:06:29,990 so the probability of one arrival 126 00:06:29,990 --> 00:06:32,860 will be the sum of these two probabilities. 127 00:06:32,860 --> 00:06:34,800 And if we throw away terms that are 128 00:06:34,800 --> 00:06:38,670 order of delta squared, what we're left with 129 00:06:38,670 --> 00:06:47,220 is just lambda1 plus lambda2 times delta. 130 00:06:50,630 --> 00:06:54,570 Finally, there's the possibility that the blue process 131 00:06:54,570 --> 00:06:58,820 is going to have two or more arrivals. 132 00:06:58,820 --> 00:07:03,720 This happens if we have one red and one green arrival, 133 00:07:03,720 --> 00:07:06,710 which happens with this probability, 134 00:07:06,710 --> 00:07:12,470 or if anyone of the processes has two or more arrivals, which 135 00:07:12,470 --> 00:07:17,170 would be terms here, here, and these would be the scenarios. 136 00:07:17,170 --> 00:07:19,470 But we notice that each one of these scenarios 137 00:07:19,470 --> 00:07:22,090 has probability that's order of delta squared. 138 00:07:22,090 --> 00:07:26,130 This term also has probability of order delta squared, 139 00:07:26,130 --> 00:07:29,870 so overall, the possibility that the blue process has 140 00:07:29,870 --> 00:07:33,750 two or more arrivals-- this is something that has probability 141 00:07:33,750 --> 00:07:37,530 that's of order delta squared. 142 00:07:37,530 --> 00:07:41,310 So during a typical small interval, 143 00:07:41,310 --> 00:07:44,150 there is probability proportional 144 00:07:44,150 --> 00:07:47,250 to the length of the interval of having one arrival, 145 00:07:47,250 --> 00:07:51,590 and lambda1 plus lambda2 is the factor of this proportionality, 146 00:07:51,590 --> 00:07:54,855 and the remaining probability is assigned to the event 147 00:07:54,855 --> 00:07:56,770 that there are zero arrivals. 148 00:07:56,770 --> 00:07:58,810 There's essentially negligible probability 149 00:07:58,810 --> 00:08:01,820 of having two or more arrivals, but this 150 00:08:01,820 --> 00:08:04,070 together with the independence assumption 151 00:08:04,070 --> 00:08:08,440 is exactly what comes in the definition of a Poisson process 152 00:08:08,440 --> 00:08:12,230 with an arrival rate equal to this number. 153 00:08:12,230 --> 00:08:15,080 And so we have established that the merged process 154 00:08:15,080 --> 00:08:17,540 is a Poisson process whose rate is 155 00:08:17,540 --> 00:08:22,040 the sum of the rates of the processes that we started from.