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.