Case 1:16-cv-02690-AT Document 121-12 Filed 08/05/16 Page 1 of 251
`Case 1:16-cv—02690-AT Document 121-12 Filed 08/05/16 Page 1 of 251
`
`E-2
`
`E-2
`
`

`

`Case 1:16-cv-02690-AT Document 121-12 Filed 08/05/16 Page 2 of 251
`Network Analysis Corporation
`
`. . y
`
`'<.,t' =•. / J v . .-
`- i j . i \ «•• / • / l-"> i '•.' ; ^ •■ ( ■' I
`
`IT ' : -i 1
`
`U»j> •• 'v ' - y
`(37)
`
`V.-.
`
`Y .. , (s,t;- - Y.. .... Ct) "(i,l) , (u,v)
`
`(38)
`
`(1,3! (tl
`
`"vu- In the previous section Y,. .. (t) - X.. .. (^ rsndoi.ü :• cd cv;.r
`
`:~.ece 1 or rr.o-.'o 2 distribution. The remaining quantity to ccrspuU:
`is:
`
`X (t) = *-f x vV(s,t) + r. . ,,U) (39)
`U/D' (u,v) in u'l] '(U'V; u'-"
`II(i,j) set
`These equations have been programmed for the grid of repeaters at
`the lattice points of the Euclidean plane, as earlier. The program is
`for the case where a repeater has a single fixed path to the origin or
`ground station. The program was run for ten time points using a single
`sample at each point. The numerical results are, therefore, subject to
`some variability. Some of the results of a single run are given in
`Tables 28 - 34. As is evident from the tables, saturation of the
`channel begins early for A = 5. In Table 31 for example the proba-
`bility that a message originating at d = 5 at time 1 gets to the ori-
`gin with a delay of less than four time units is only about .44. For
`X = 5 the situation deteriorates rapidly with time. To obtain a large
`set of representative data would require running the program for many
`time points, probably at least 15 or 20 for different values of A and
`slot size. This can be done using the available program.
`
`10.3. Delays and Average Delays as a Function of Distance
`We can extend the calculations and analyses described in the pre-
`vious two sections to include calculations of delay distributions and
`average delay. In addition to studying delays, we can develop equa-
`tions to study bottlenecks in a given network. These formulae have
`been programmed and numerical results can be obtained.
`Let D,. .»(t) be the random variable delay of a message which
`(i / j 1
`originates at (i,j) at time t. We assume that the probability that a
`message is delayed by k-units of time is given by the proportion of
`. 11.75
`
`

`

`Case 1:16-cv-02690-AT Document 121-12 Filed 08/05/16 Page 3 of 251
`Network Analysis Corporation
`
`Retransmissions at Point of Wipeout
`
`X=5, HflOO, Model
`
`0 12 3
`
`0
`
`0
`
`6
`
`6
`
`43
`
`0
`
`0
`
`6
`
`11
`
`26
`
`19
`
`16
`
`91
`
`67
`
`151
`
`0
`
`0
`
`0
`
`6
`
`18
`
`19
`
`29
`
`47
`
`67
`
`64
`
`0
`
`0
`
`0
`
`1
`
`1
`
`7
`
`13
`
`3
`
`1
`
`9
`
`0
`
`0
`
`6
`
`3
`
`1
`
`2
`
`0
`
`0
`
`0
`
`2
`
`0
`
`0
`
`0
`
`0
`
`0
`
`0
`
`0
`
`0
`
`0
`
`0
`
`>L
`
`1
`
`2
`
`3
`
`4
`
`5
`
`6
`
`7
`
`8
`
`9
`
`10
`
`TABLE 28
`
`X=5,
`
`m=100, Mode 2
`
`0
`
`0
`
`0
`
`7
`
`4
`
`34
`
`><
`
`1
`
`2
`
`3
`
`4
`
`5
`
`6
`
`7
`
`8
`
`9
`
`10
`
`1
`
`0
`
`0
`
`2
`
`10
`
`20
`
`37
`
`19
`
`28
`
`33
`
`2
`
`0
`
`0
`
`1
`
`1
`
`2
`
`2
`
`5
`
`6
`
`18
`
`4
`
`TABLE 29
`
`11.76
`
`3
`
`0
`
`0
`
`0
`
`0
`
`2
`
`1
`
`0
`
`4.
`
`1
`
`0
`
`4
`
`0
`
`0
`
`0
`
`2
`
`1
`
`1
`
`1
`
`1
`
`1
`
`1
`
`5
`
`0
`
`0
`
`0
`
`0
`
`0
`
`0
`
`0
`
`1
`
`0
`
`1
`
`

`

`Case 1:16-cv-02690-AT Document 121-12 Filed 08/05/16 Page 4 of 251
`Network analysis Corponitkm
`
`Delay Probability Tables for a Message Being
`
`Accepted d units from the Ground Station
`
`) t=5, m= =100, Mode 1
`
`?. Message Originating at d= =5 at time 0.
`
`4
`
`5
`
`1.000
`
`1.000
`
`->^dist
`delay^^
`
`0
`
`1
`
`2
`
`3
`
`4
`
`5
`
`6
`
`0
`
`.159
`
`.235
`
`.134
`
`.097
`
`.089
`
`1
`
`.271
`
`.247
`
`.1*7
`
`.076
`
`.045
`
`.041
`
`2
`
`.524
`
`.325
`
`.084
`
`.028
`
`.016
`
`.009
`
`.007
`
`TABLE 30
`
`3
`
`.933
`
`.059
`
`.004
`
`.002
`
`.001
`
`!
`
`^"v^ist
`delay"*-«^
`i °
`1
`
`2
`
`3
`
`4
`
`5
`
`6
`
`7
`
`A Messag« 2 Origination at d= =4 at time 0.
`
`3
`
`4
`
`1.000
`
`1.000
`
`0
`
`.249
`
`.251
`
`.196
`
`.083
`
`.054
`
`.046
`
`1
`
`.454
`
`.221
`
`.123
`
`.070
`
`.033
`
`.018
`
`.016
`
`2
`
`.727
`
`.153
`
`.083
`
`.021
`
`.007
`
`.004
`
`.002
`
`.002
`
`TABLE 31
`
`11. '7
`
`

`

`Case 1:16-cv-02690-AT Document 121-12 Filed 08/05/16 Page 5 of 251
`Network Analysis Corporation
`
`^dist
`dela^v^
`
`0
`
`1
`
`2
`
`3
`
`4
`
`5
`
`6
`
`7
`
`A Message Originating at d=3 at time 0
`
`G
`
`.916
`
`.036
`
`.023
`
`.013
`
`.004
`
`.003
`
`.002
`
`1 2 3
`
`.950 1.000 1.000
`
`.031
`
`.010
`
`.004
`
`.002
`
`.001
`
`.001
`
`.001
`
`TABLE 32
`
`A Message Originating at d=2 at time 0
`
`!
`
`1 2
`
`1.000 1.000
`
`"^N^dist
`dela^v^
`
`0
`
`1
`
`2
`
`3
`
`4
`
`0
`
`.823
`
`.170
`
`.004
`
`.002
`
`.001
`
`j
`
`TABLE 33
`
`A Message Originating at d=l, at time 0
`
`^^dist
`dela^—^
`
`0 •
`
`1
`
`1.000
`
`0
`
`1
`
`2
`
`3
`
`.778
`
`.183
`
`.038
`
`.«JOi
`
`TABLE 34
`
`11.78
`
`

`

`Case 1:16-cv-02690-AT Document 121-12 Filed 08/05/16 Page 6 of 251
`Network Analysis Corporation
`
`THE NUMBER OF MESSAGES ACCEPTED AT THE ORIGIN
`
`X = 5, m = 100, mode 2
`
`Originating at d = 5 at t = 0: 3 messages
`
`Time of Acceptance at Origin Number Delay
`
`t = 5 .814 0
`
`t = 6 1.130 1
`
`t = 7
`
`t = 8
`
`t = 9
`
`.502
`
`.253
`
`.161
`2.860
`
`2
`
`3
`
`4
`
`TABLE 35
`
`Originating at d = 5 at t = 1: 6 messages
`
`jptance at Or ig in
`Time of Ace«
`
`Number
`
`Delay
`
`t = 6 2.010 0
`
`t = 7 1.238 1
`
`t = 8 .988 2
`
`t = 9 .805 3
`5.061
`
`TABLE 36
`
`Originating at d= 5 at t - 2: 2 messages
`
`Time of Acceptance at Origin Number Del&y
`
`t = 7 .115 0
`
`t = 8 .249 1
`
`t = 9 .256 2
`.620
`
`TABLE 37
`
`11.79
`
`

`

`Network Analysis Corporation
`Case 1:16-cv-02690-AT Document 121-12 Filed 08/05/16 Page 7 of 251
`
`DELAY PROBABILITY TABLES FOR A MESSAGE
`
`BEING ACCEPTED d UNITS FROM THE GROUND STATION
`
`X = 5. m = 100, mode 2
`
`A Message Originating at d = 5 at time 0
`0
`1
`2
`3
`4
`
`5
`
`Dist
`Delay
`
`0
`
`1
`
`2
`
`3
`
`4
`
`5
`
`.271
`
`.496
`
`.854
`
`1.000
`
`1.000
`
`1,000
`
`.377
`
`.360
`
`.140
`
`.167
`
`.071
`
`.003
`
`.084
`
`.041
`
`.003
`
`.054
`
`.018
`
`.007
`
`-
`
`-
`
`-
`
`-
`
`-
`
`-
`
`-
`
`-
`
`-
`
`-
`
`-
`
`-
`
`-
`
`-
`
`-
`
`-
`
`TABLE 38
`
`A Message Originating at d = 4 at time 0
`4
`0
`1
`2
`3
`
`Dist
`Delay
`
`0
`
`1
`
`2
`
`3
`
`4
`
`5
`
`6
`
`.407
`
`.452
`
`.792
`
`1.000
`
`1.000
`
`.189
`
`.300
`
`.178
`
`.216
`
`.178
`
`.029
`
`.092
`
`.035
`
`.001
`
`.045
`
`.020
`
`.001
`
`.028
`
`.008
`
`-
`
`.003
`
`-
`
`-
`
`-
`
`-
`
`-
`
`-
`
`-
`
`-
`
`-
`
`-
`
`-
`
`TABLE 39
`
`11.80
`
`"—""lidMin'Maiiit «s_w ii ■)«Jai..iif,iiiliii j^_^j^aiijiBBg^
`
`

`

`Case 1:16-cv-02690-AT Document 121-12 Filed 08/05/16 Page 8 of 251
`Network Analysis Corporation
`
`A Mess acre Oriainatina at d = 3 at time 0
`0
`2
`1
`3
`
`Dist
`Delav
`
`0
`
`1
`
`2
`
`3
`
`4
`
`5
`
`6
`
`.511
`
`.425
`
`.031
`
`.020
`
`.007
`
`.003
`
`.002
`
`.962
`
`.022
`
`.009
`
`.005
`
`.001
`
`.001
`
`-
`
`I. 000
`
`1.000
`
`-
`
`-
`
`-
`
`-
`
`-
`
`-
`
`-
`
`-
`
`-
`
`-
`
`TABLE 4 0
`
`A Messacre Oricrinatinq at d = 2 at time 0
`3
`1
`2
`0
`
`Dist
`Delav
`
`0
`
`1
`
`2
`
`3
`
`4
`
`5
`
`.883
`
`.061
`
`.050
`
`.003
`
`i
`i
`
`.001
`
`.923
`
`.074
`
`.002
`
`-
`
`-
`
`1.000
`
`-
`
`-
`
`-
`
`-
`
`-
`
`TABLE 41
`
`11.81
`
`

`

`Case 1:16-cv-02690-AT Document 121-12 Filed 08/05/16 Page 9 of 251
`Network Analysis Corporation
`
`E.St
`
`A Message Originating at d = 1 at time 0
`0
`
`1
`
`0
`
`1
`
`2
`
`3
`
`.800
`
`1.000
`
`.191
`
`.005
`
`.004
`
`-
`
`-
`
`-
`
`TABLE 42
`
`A Message Originating at d = 0 at time 0
`
`Dist
`Delay
`
`0
`
`1
`
`1.000
`
`.000
`
`TABLE 43
`
`A Messao re Originating at d = 5 at time 1
`
`0
`
`1
`
`2
`
`3
`
`4
`
`5
`
`Dist
`Delay
`
`0
`
`1
`
`2
`
`3
`
`4
`
`5
`
`.335 .520
`
`.718
`
`.752
`
`,846
`
`1.000
`
`.206 .159
`
`.116
`
`.207
`
`.132
`
`.165 .161
`
`.125
`
`.039
`
`.020
`
`.134 .081
`
`.024
`
`.002
`
`.002
`
`-
`
`-
`
`.036
`
`.012
`
`.003
`
`-
`
`.003
`
`-
`
`-
`
`-
`
`-
`
`TABLE 44
`
`11.82
`
`■ '■•■ ■•■ • -
`
`

`

`Case 1:16-cv-02690-AT Document 121-12 Filed 08/05/16 Page 10 of 251
`Network Analysis Corporation
`
`PROBABILITY OF ZERO DELAY VS. DISTANCE OF ORIGINATION
`
`X 0
`
`MODE 1
`
`1
`
`2
`
`3
`
`4
`
`5
`
`0
`
`l
`
`2
`
`3
`
`4
`
`5
`
`6
`
`7
`
`8
`
`9
`
`1.000
`
`.778
`
`.823
`
`.916
`
`.346
`
`.329
`
`.458
`
`.125
`
`.242
`
`.229
`
`,045
`
`.085
`
`.207
`
`.043
`
`.013
`
`.058
`
`.349
`
`.091
`
`.053
`
`.021.
`
`.177
`
`.033
`
`.025
`
`.021
`
`.013
`
`.092
`
`.019
`
`.011
`
`.006
`
`.015
`
`.159
`
`.123
`
`.047
`
`.024
`
`.037
`
`.021
`
`.005
`
`.002
`
`.008
`
`.029
`
`.002
`
`.001
`
`.031
`
`.001
`
`.047
`
`TABLE ^5
`
`.,ivi:-i.->,/;j ...
`
`11.83
`
`

`

`Network Analysis Corporation
`Case 1:16-cv-02690-AT Document 121-12 Filed 08/05/16 Page 11 of 251
`
`X 0
`
`MODE 2
`
`1
`
`2
`
`3
`
`4
`
`5
`
`0
`
`1
`
`2
`
`3
`
`4
`
`5
`
`6
`
`7
`
`8
`
`9
`
`1.000
`
`.800
`
`.883
`
`.511
`
`.407
`
`.373
`
`.136
`
`.118
`
`.356
`
`.170
`
`.333
`
`.118
`
`.102
`
`.030
`
`.180
`
`.271
`
`.335
`
`.115
`
`.100
`
`.044
`
`.033
`
`.154
`
`.066
`
`.218
`
`.112
`
`.026
`
`.061
`
`.045
`
`.109
`
`.161
`
`.Co2
`
`.032
`
`.005
`
`.100
`
`.073
`
`.021
`
`.011
`
`.022
`
`.041
`
`o045
`
`.022
`
`.019
`
`.005
`
`.018
`
`.021
`
`TABLE 46
`
`11.84
`
`rtitiiHiitfiftiiih^ilnifctT^i;, Vm n „ iA^i^^^^^^-.^^^^,,^,.,.
`
` —-. .... ,-......„ _.
`
`

`

`Case 1:16-cv-02690-AT Document 121-12 Filed 08/05/16 Page 12 of 251
`Network Analysis Corporation
`
`messages which originate at time t which are delayed by k-units. If
`
`enough random samples are taken, this estimate becomes quite good. In
`
`notation, define Y, Ä» ,. .. . .. as the number of messages which
`(0,0) (1,3) (s,t)
`are accepted at the ground station at time s, which originated at
`
`(i,j) at time t. Then,
`
`iy(irj)<t>=*; -
`
`' c ft : , "> -•■ i 3 y \±2 "-'
`0{: •-. (t)
`
`t)
`
`Ihcv. coLvo ccr:putcr trials to dot..; :;ine 40
`Cc. ** a V.r*V
`ever aij -ri r,.xütc:-t' at: instance i <T.:K; oci:.rute tils, e» • < y d;.
`tribut:ion vs :. -.ruction oniv oi di£tc::cc tc the crcur.d '-La^i
`
`in.. (t)->;
`.1.
`
`i %— HD,. .. (t)=-ki, where
`vi,D)
`3-x
`
`(40)
`
`(41)
`
`D.. (t) is the randan variable delay of a message originating at
`
`eistcr.ee 5. at time t.
`
`To obtain a time invariant measure, we can average 41 OVOJ
`
`time and obtain the probability distribution of the ra.ndcn vari-
`
`able D. and its expectation, given by:
`
`E(D.) = L-t k P{D.=k)
`k=o
`
`(42)
`
`The sasie kind of analysis can be used to study "bottlenecks."
`Let D,. .. (v: t) be the random variable "delay of a message" ori-
`li / 31
`ginating at (i,j) at time t in getting w units from the ground
`station for w=0f 1, 2,..., i. As earlier we have:
`Y,, „, ,• .. (t+k+i-w, t)
`P{D.. ,, (w,t)=k} = -±^LLL1LLA} . k= o, 1, 2
`°(i,j)(t/
`
`(43)
`
`where (w,u) is the unique repeater on the path from (i,j) to (o,c)
`at distance w. Similarly as in 41 and 42 :
`
`P{D. (w,t)=k) =4 L P^DM -n(w,t)=k] anc
`OD J — i.
`E JD^ , t )1 = V^KP {Di (w, t) =k } .
`
`(44)
`
`C.5)
`
`_____
`
`11.85
`
`

`

`Case 1:16-cv-02690-AT Document 121-12 Filed 08/05/16 Page 13 of 251
`Network Analysis Corporation
`
`11. MESSAGES OUTWARD FROM THE ORIGIN
`
`11.1. A Single Message Originates at the Ground Station.
`
`The next major part of our study is to model the situation
`
`when message flow is outward from the origin. We begin our
`
`study with the dynamics of the simple model where the repeaters
`
`are at the lattice points of the plane. A single message ori-
`
`ginates at the ground station at time t=0. Every message re-
`
`ceived by a repeater is accepted and perfectly retransmitted to
`
`each of its four nearest neighbors. We determine:
`
`a) The number of repeaters which receive the message
`
`for the first time at time t: t=0, 1,2,....
`
`b) Th/e number of repeaters which have seen the single
`
`message by time t.
`
`c) The number of copies of the single message received
`
`by any repeater at time t.
`
`The assumptions are:
`
`1) A single message arrives at a given node at time t = 0,
`
`no other messages are introduced into the network. We assume
`
`the message originated at the origin , (Cartesian coordinates
`
`(0,0)).
`
`2) Message transmission is perfect, i.e., after one time
`
`unit each of the four neighbors to any repeater receive
`
`all messages transmitted by the repeater at the previous
`
`time point.
`
`11.86
`
` <...; ,^ .■.■...,-
`
`: ''■■''' ■.......■;■.«...;; ............
`
`

`

`Case 1:16-cv-02690-AT Document 121-12 Filed 08/05/16 Page 14 of 251
`Network Analysis Corporation
`
`Some diagrams and numbers are helpful to fix ideas.
`0
`10
`AQ
`0
`
`t = 0,
`
`0
`
`1
`
`0
`
`. o
`
`o
`
`t = 1,
`
`ln J 11 i in
`>°—< l
`
`1
`
`TT
`
`J
`i
`
`(one message at origin, no
`messages elsewhere)
`
`(a single message at each of the
`four neighbors, no messages
`
`elsewhere)
`
`/ i K
`o/ 2 0 2\0
`yf ii ii li V
`
`/ 1)40
`0^^ " h
`
`yt—H! ,Q ,_*
`
`\l 1 /
`
`(4 messages at the origin,
`0 Message at all repeaters
`1 step from origin, 1 message
`at each of four repeaters
`2 steps in either the horizontal
`or vertical directions, 2 messages
`at repeaters, 1 unit in horizontal
`direction and 1 unit in vertical
`
`direction.
`
`By examining the diagrams we are led to introduce a
`coordinate system based on distance as measured in steps to
`reach a repeater and horizontal distance of the repeater
`from the origin. The quadrant symmetry of the model also
`indicates use of these coordinates.
`
`11.87
`
`iWi'nituri'rin
`
`H&UHA-m--. ' ■ ■ ■■■•-■■• ■■ -.'- ■'»■-■■J-.-
`
`Ü*4ft*i»ÖB&1f«^
`
`

`

`Case 1:16-cv-02690-AT Document 121-12 Filed 08/05/16 Page 15 of 251
`
`The coordinates of a repeater are denoted by (d,j)
`
`where d is the distance of the repeater from the origin
`
`measured in minimum time units a message needs to arrive at
`
`the repeater from the origin; the second coordinate j is
`
`the horizontal distance of the repeater from the origin
`
`again measured in time units but only in the horizontal direction.
`
`For example, we give some coordinates:
`
`f(3,0)
`
`(2,0)
`
`.(4,2)
`
`, (1,0) .(2,1).(3,2)
`
`(3,3) (2,,!) g.,1) JUlSj (V1} (.2'2) (3,3)
`
`(3,2) (2,1)4(1'Ü) •(2<1)
`
`(2,0) .(3,1)
`
`(3,0)
`Some further notation which is necessary;
`
`B(t) = the number of repeaters which receive the message
`
`for the first time at time t.
`
`Clearly B (t) is the number of repeaters whose first
`
`coordinate is t. i.e. that are at distance t from the origin.
`
`A(t) = the number of repeaters which have seen the
`t
`me ssage by time t. Clearly A(t) =,C"B(-;)
`j=0
`
`N. (t) = number of copies of the message received by a given
`
`repeater at coordinates (d,j) at time t. Clearly
`
`a) Nd(t) = 0 for d >t
`
`b) Nt?(d+2k+l) = 0 for k = 1, 2,
`
`Thus it is necessary to compute N.(d+2k) k = 0, 1, 2,....
`
`11.88
`
`

`

`Case 1:16-cv-02690-AT Document 121-12 Filed 08/05/16 Page 16 of 251
`A) Calculation Of B(t)t Network Analysis Corporation
`
`The quantity B(t), (the number of repeaters at distance
`
`t from the origin is the number of repeaters which receive
`
`the message for the first time t, is easy to compute. This
`
`quantity is given by the number of integer solutions to
`
`|i| + jjl = t. Since a repeater is at distance d from the
`
`origin if and only if its* Cartesian coordinates(i,j) satisfy
`
`|i | + |j| = t,we Can solve this equation and count solutions. Note that
`
`i=0, j=t or -t 2 solutions
`
`i-1, j=t-l or -t+1 2 solutions
`
`i=-l, j=t-l or -t+1 2 solutions
`
`i=2, j=t-2 or -t+2 2 solutions
`
`i=-2 i=t-2 or -t+2 2 solutions
`
`i=t-l/ j=l or j=-l , .
`
`i=-t+l# j=l or j=-l . . ,
`
`i=t j=0
`
`i=-t j=0
`mmummi^mmmmmmmmmmmmmmmmmmammaBmmnmmmmammmmmmmmmmmmmmmmmmmmmmmmmmmMmimmmmmmmm^ummmmB
`
`The number of solutions is,B(0) = 1
`
`B(t) = 2 + 4(t-l) +2 = 4t for t£l.
`
`To compute Aft) we sum B(t) and obtain
`t_ ' t t
`A(t) =ZT B(j) =1 +^J 4(j) = 1 + 4^T j = 1 + 4 t(t+l)
`j=0 j=l j=l 2
`
`= 1 + 2t2 + 2t = 2t2 + 2t + 1
`
`The rate at which A(t) the number of repeaters which receive
`the message by time t grows as A' (t) = 4t + 2 which is linear in t.
`
`11.89
`
`

`

`Case 1:16-cv-02690-AT Document 121-12 Filed 08/05/16 Page 17 of 251
`Network Analysis Corporation
`
`The quantity H. (d + 2k),
`
`K — U , X i <£,....
`
`B»
`
`To compute the number of copies of the message received
`
`at a repeater with coordinates (d,j) at time d + 2k, k = 0, 1, 2,..
`
`we first draw some diagrams. Due to symmetry it suffices to
`
`ex-unine only the first quadrant.
`
`t = 0 L
`
`t = 1
`
`t = 2,
`
`t = 3,
`
`9 K
`n ° X1
`0 •—.— i 1 9
`
`t = 4,
`
`0
`
`0 HXk4
`fV
`?n
`o k i i 24 o\.4
`36 [ L_ 30 lo\
`
`It seems clear from the diagram that a repeater with
`
`coordinates(d,j) will receive at t=d the number of messages
`
`which is given by the binomial coefficient (.). From the
`
`diagram we note the relationship of the outer edge to the
`
`d now of a Pascal triangle:
`
`1
`1 1
`12 1
`13 3 1
`14 6 4 1
`1 5 1010 5 1
`
`This result is also apparent from an argument based on the number of
`
`paths of a message from (0,0) to (d,j). The number of messages
`
`received at a repeater with coordinates (d,j) is given by the
`
`number of paths from (0,0) to (d,j) which is obviously (.). To
`
`determine a general formula for N.(d + 2k) for k> 1, we can
`
`write and solve the appropriate difference equation.
`
`-:!-""-' *-■•■" -•.■..■•..■■..■--
`
`11.90
`
`

`

`Network Analysis Corporation
`Case 1:16-cv-02690-AT Document 121-12 Filed 08/05/16 Page 18 of 251
`
`Nd(d + 2k) = Nd~*(d + 2k-l) + Nd*J (d + 2k-l)
`
`+ Ndt1(d + 2k-l) + NdT1(d + 2k-l)
`
`for k = 1, 2,...,,d = 0, 1, 2,...,;j = 0, 1, 2,...,d.
`
`The initial conditions are
`
`Nd(t) = 0 if t<d,
`
`Nd(d) =(d).
`
`The solution to this equation is given by Nd(d + 2k) =(d +j.2k) (d
`
`k+-2k)
`
`To check its validity note that the initial conditions are
`
`satisfied, and apply the well known definition of binomial coefficients.
`
`Kd"j; (d-1 +2k) + Nd^ (d+l+2(k-l)) + N*3"!"1 (d+1+2 (k-1)) + ^T1 (d-l+2k)
`
`_,d+2k-lwd+2k-l. , ,d+2k-lx ,d+2k-l^,d+2k-l. ,d+2k-l.,,d4 2k-lwd+2k-l.
`-( k )(k+j-l ' +( k-1 } ( k+j M k-1 } (k+j-l M k Mk+j ,
`
`,d+2k~l. r ,d+2k-lw/d+2k-l. ,. ,d+2k-l. , ,d+2k-l. , ,d+2k-l. ,
`= ( k )[(k+j-l) + ( k+j )] + ( k-1 )[(k+j )+(k+j-l )]
`
`_,d+2k-lwd+2k. , ,d+2k-lwd+2k.
`1 k M k+j' + [ k-1 M k+j'
`
`_,d+2k . r,d+2k-l. ,d+2k-l,, ,d+2k -d+2k
`1 k+j' u k ' ' k-1 n ^ k+j' * k '
`
`For k very large with respect to d we can use a Stirling
`
`approximation to note the N. (d+2k)">^-2 i.e. grows as 2 .
`
`11.91
`
`

`

`Network Analysis Corporation
`Case 1:16-cv-02690-AT Document 121-12 Filed 08/05/16 Page 19 of 251
`
`To summarize:
`
`a) B(t) * 4t , Ol B(0) = 1,
`
`b) A(t) = 2 t + 2t+l t^O,
`
`c) N*(d+2k) =(dk+jXdk2k)~ 24k for large k ,
`
`11.2. A Fixed Number of Messages Originate at the Ground Station
`and Subject to Non-Capture
`
`The model of message flow from the ground station out to re-
`
`peaters can be extended to alle«' the possibility of erasure ox
`
`non-capture of messages.
`
`We assume that at each point of time t, x messages are being
`
`generated outward from the origin. Of messages accepced at each
`
`repeater, a fixed proportion k are a^'lressed to that, repeater and
`o
`hence are not repeated. We study the distributions of the number
`
`of messages received ?.:ia accepted at each repeater at each point
`
`ir* time, assuming an infinite net.
`(1)
`Recall X (t) = number of messages arriving at a repeater
`(i,j)
`with coordinates (i,j) at time t with Mode 1 capture.
`A, (1)
`X (t) = number of messages accepted at a repeater with
`(i,j)
`coordinates (i,j) at time t in Mode 1 capture. In Mode 2 capture,
`
`we use the same notation except that (1) as a superscript is re-
`
`place by a (2).
`
`A. In Mode One;
`
`In Mode 1 capture, the relationship between arriving and
`
`accepted messages is described by the transfer function:
`
`11.92
`
`WmMUtm m . mm^M.,, m maä ' ■• ■ ■ . —_
`
` :— .._ ' _,^_
`
`

`

`Case 1:16-cv-02690-AT Document 121-12 Filed 08/05/16 Page 20 of 251
`Network Analysis Corporation
`
`(") min(k,m)-j .
`
`when j < min(k,m), and zero otherwise.
`
`We can study x;...(t) and XA!:. (t) recursively.
`(ij) (13)
`
`At the origin for each t = 0, 1, 2, ...; X. ^.(t) = T, a fixed constant.
`
`'.•"urthermore:
`
`O min(T,m)-w
`Hx^V) =w} =P = ^ I
`(-1) ( .. ) ; ; , (m-w-V)
`v ' (T-W-V)!
`(0C; T'W mT vi0
`
`(47)
`
`If we assume T < m;
`
`m
`(. ) T-W
`— ) (-1 ( ~, 7-r (m-w-v) if w < T
`T L_ \) (T-V7-V) ! —
`<:> -•'-'■'v=o
`
`if w >
`
`(48)
`
`At coordinates (1,1), (1,0) the distribution of X. . (t) and X ,»(t)
`(1,0/ (1,1)
`
`are identical, hence we write only X. .(t).
`(1,0)
`
`We have:
`
`xKx?(fe-»
`
`XU) (t) =
`x(i,o)(t)
`
`t ~~ X / *L 1 • • • jj
`
`(49)
`
`if t = 0.
`
`For the acceptance at t = 1, 2, ...;
`
`T
`P{X(I,li!(t) = j} = J, Pkj * P{Xa!o)(t) " k} D = 0, 1, 2, ...;
`
`or recursively:
`
`■ofclilw-j).
`
`J. p« • p(x!o.w ,t-1' "k} if Ü'.
`
`\ 0
`
`otherwise.
`
`(50)
`
`11.93
`
`

`

`Network Analysis Corporation
`Case 1:16-cv-02690-AT Document 121-12 Filed 08/05/16 Page 21 of 251
`
`(1) £
`Equation(50) is recursively solvable since PfX ~ ' (t-1) = k} is given by
`
`(48)and P.. is given by (46).
`
`Now more generally,at a repeater at distance d with coordinates (d,j);
`
`j + 0,d. We have for d = 2, 3, ...;
`
`x((do)(t> - "-V [x((d-M-i,(t-1> + CM)^1']-
`
`The acceptances are given by:
`
`(2m)
`PO$!»Cti-r>- I »k.r •*«{?,„<«-W,
`I
`k=r
`
`(51)
`
`(52)
`
`where P. is given by (1) and P{X,J .. (t) *- k} can be computed recursively from
`kr " (d,j)
`(51) using the notion of isodesic line jcint densities,
`
`equations for mode 2 analysis are identical except that P... is replaced by P ..
`
`When j » 0 or d, i.e. the repeater is on the axis at distance d a simpler
`
`analysis unfolds. Since the random variables X., .,(t) and X,. ,.(t) have the
`(d,0) (d,d)
`
`Larr.e probability distribution.we write equations only for X,, .. (t); d > 2 since
`(d,0) —
`
`X. . (t) and X. . (t) have already been determined.
`
`x((dlo)(t) - (i-v x((d-;?o) (t-i} <
`
`t = d, d + 1,...;
`
`(53)
`
`For the acceptances;
`
`,(l),Ai
`
`,(D
`
`- 0, 1, 2, ..., m (54)
`
`The equations (4g)through (54)can be used with computer generated data to
`
`study messaqe arrivals and acceptances at each repeater.
`
`11.94
`
`c^nim HI r naiBifcja .,.,- urtfrnuTMai'ii , g^m^nyj
`
`■ ^'"■■" ■ ■ ■
`
`

`

`Network Analysis Corporuiion
`Case 1:16-cv-02690-AT Document 121-12 Filed 08/05/16 Page 22 of 251
`
`In particular, we can write equations such as 46-54 for the closed
`
`net under consideration and obtain numerical data for flow from
`
`the ground station.
`
`Let X,. -A(t) be the number of messages received at (i,j)
`at time t and Y,. . »(t) be the number of messages accepted at
`(i,j) at tine t. We assume X» . (t)=J a fixed constant for all
`lo ,o;
`tine Doints. As in the inward model, Y,. .. (t) is obtained from
`X-- .. (t) by randomizing over either mode 1 or mode 2 slotting.
`I1 r J 1
`When performing the calculations en the computer, we assumed a
`finite grid of 61 repeaters as earlier. However, now a repeater
`repeats messages to those repeaters which are one unit of
`distance further from the origin or ground station. Thus for
`example^ a repeater in a quadrant repeats to its two farther
`neighbors, while a repeater on the axis repeats to the one re-
`peater which is one unit further.
`
`The specific equations used for the first quadrant calcu-
`lations follcv/, we assume that -rj of those messages accepted
`are addressed to each repeater and hence not repeated. The
`calculations were carried out in each of the two slotting modes.
`
`Step 1. Set X, . (t) - J = 80, t = I, 2, ... ,35.
`(o,o) '
`Step 2. ComuuLe Y, > (t) by randomizing over the transfer
`£ - (o,o)
`distribution in each of two modes.
`
`Step 3. Compute X,, -^ (t) and X.. ?s(t) from:
`
`Xd,i)(t) = Xa,2)(t) - IT Y<o,o>(t-1>-
`Stop 4. Compute Y,, ,, (t) and Y ,, 2\ (t) in each of two modes.
`
`11.9b
`
`i in« ii *-—-
`
`■ '■-' ' :-.- .. . ■>. - , <• .. .
`
`

`

`Network Analysis Corporation
`Case 1:16-cv-02690-AT Document 121-12 Filed 08/05/16 Page 23 of 251
`
`Step 5. ?or t = 2, 3, 4,...,37; Compute X(2fl)(t)f X{2/2)(tl
`
`and X(2#3) from:
`
`X(2,l)(t> -§1^(1,1)«*-»
`X(2,2>(t> "I? <Y(1,1> (t-D + V(lf2)(t-D)
`
`X(2,3) (t) = FI Y(l,2) (t-1) •
`
`Step 6. Compute Y,2 ■, \ (t) , Y,2 ?) ^^ an<* Y(2 31^ ^Y random-
`izing in each slotting mode.
`
`Step 7. Compute X(3 „. (t) , X(3 3) (fc) » X(3 A\ (fc) from:
`^n 1 T
`<« ='lr[Y(2,i)'t-1» +Y(2.2,(t-1!]
`X(3,2)
`
`«*> =lr[1(2.2)(t-1» +Y(2.3)(t-1)]
`X(3,3)
`
`X(3,4) *"" 61 *(2,3) lC ■L' *
`
`Step 8. Compute Y(3 2« (t) , Y(3 3> (t) , Y 4x (t) by randomizing
`in mode 1 and mode 2.
`
`Step 9. Compute X(. 3)^) and Y(4 4) (fc) bY randomizing.
`
`Step 10. Compute Xft- 4\ (t) from:
`
`X(5,4)(t> -fl[Y<4,3)(t-1) + Y(4f4)(t-«]'
`
`Step 11. Compute Y,- ..(t) by randomizing and print out X's
`and Y's for t = 1, 2,...,40. The numerical data is
`summarized in Table 26 and the accompanying Figure 10
`
`11.96
`
`a*a^ IM
`
`ö^it-^säiÄ-aajui-ui-,"" nrrr-mh'tilUmiär^Miia^m
`
`

`

`Network Analysis Corporation
`Case 1:16-cv-02690-AT Document 121-12 Filed 08/05/16 Page 24 of 251
`
`11. 3 "eosages Coming Outward i-'ro.-t the Orig-P.
`
`It is assumed that at each point in time eighty (80)
`messages originate at the origin (ground station). The messages
`are repeated outwards to the various repeaters. At each point in
`time each repeater sends all but a fixed proportion of its ac-
`cepted message to each neighboring repeater, one unit of distance
`further from the ground station. The fixed proportion not re-
`peated is 1/61 of the number of accepted messages, which are
`assumed to be addressed to the given repeater. The number of
`slots is fixed at 100.
`
`In table 47 we summarize the result of this calculation
`by giving the average number of messages accepted and arriving
`as a function of distance and mode. The numerical data is dis-
`played graphically in figure 22 .
`
`DISTANCE
`
`1
`MODE
`
`ARRIVING ACCEPTED
`
`MODE 2
`ARRIV:
`:NG ACCEPTED
`
`80
`
`32
`
`27
`
`32
`
`46
`
`58
`
`33
`
`21
`
`19
`
`22
`
`30
`
`31
`
`80
`
`51
`
`47
`
`57
`
`87
`
`114
`
`52
`
`36
`
`33
`
`41
`
`58
`
`66
`
`2
`
`3
`
`4
`
`5
`
`TABLE 47
`
`11.97
`
`•■■"•■■
`
`iMiMiiittai—____^___,
`
`mmmmmmmmmm
`
`

`

`Network Analysis Corporation
`Case 1:16-cv-02690-AT Document 121-12 Filed 08/05/16 Page 25 of 251
`
`1
`
`CM
`
`W w
`
`u I (0
`cu
`w
`u
`a
`
`>
`H
`OS
`§
`
`(0 w
`
`u 1
`
`W
`U
`
`CM
`w
`Q
`O
`as
`
`W
`Q
`Q
`S
`
`e>
`« o fM
`s CM
`s g
`w H u fr, z
`
`fa D
`O
`
`<
`Ei
`W
`
`—I o
`
`saovssaw &o naawnN
`
`11.98
`
`jj^jto^^M&g^
`
`^.■■VI,J .■:, ■-. ■ ■■■ -v.^U^i^ifc^^fif.--;- ■: ■
`
`.. , ,,.:,„-Ü;i. ;■ ■ ■■ •-■■ . .•',-,i,.,-..- ■■■■
`
`

`

`Network Analysis Corpomtion
`Case 1:16-cv-02690-AT Document 121-12 Filed 08/05/16 Page 26 of 251
`
`12. OTHER MODELS
`
`A number of models other than the "basic model" were consi-
`
`dered. The results for quantities of interest were obtained in
`
`closed forms under the assumption of an infinite grid. One ex-
`
`ample of such a model was described in Section 3 for
`
`messages repeated only toward the ground station. Of course
`
`that was part of our "basic model". In the process of developing
`
`the results of Section 3 , we assumed that a single message
`
`originated at each repeater at each point in time and multiplied
`
`the resuJts by the mean, T, to obtain average flows. We will
`
`now "justify" that calculation and study uninhibited passage of
`
`messages in each direction in an infinite grid. Our new assump-
`
`tions are:
`
`1) At each point in time, starting at t=0/messages
`
`originate at each repeater according to a Poisson
`
`distribution with mean X•
`
`2) The arrivals (originations)at each repeater are
`
`independent over time and different repeaters.
`
`The probability that exactly j new messages originate at
`
`any time point at any repeater is
`
`V^- j j = 0, 1, 2,.
`j
`We compute formulas for ;
`
`a) N (t) = average number of messages which arrive at the
`
`origin at time t. Since all repeaters are statistically
`
`identical there is no loss in generality in studying message
`
`flow at the origin.
`
`^j^t^ =*^ : ■ -■ -—-.■. —- -^—^^
`
`

`

`Network Analysis Corporation
`Case 1:16-cv-02690-AT Document 121-12 Filed 08/05/16 Page 27 of 251
`
`b) N' (t) = Average number of distinct messages which arrive
`
`at the origin at time t for the fust time.
`
`c)Ieff(t)=Inefficiency of the network defined
`
`by;
`
`No(t)
`leff(t) = „. (t).
`
`average number of messages
`average number of new messages for the 1st tine,
`
`This is a measure of inefficiency since the larger Ieff,
`
`the more inefficient the system.
`
`The actual number of messages which arrive at the origin at
`
`time t is a random variable. In fact it is a sum of a large number
`
`(when t is fairly large) of independent random variables. The
`
`summand random variables can loosely be described as the con-
`
`tribution to message flow at the origin arising out of some
`
`number of messages originating at each repeater at each point
`
`in time.
`
`To compute N (t) we can sum up all the contributions» This
`
`is interesting but tedious. A simpler method is to compute
`
`X (t) which we define as the number of messages arriving at the
`
`origin at time t in a deterministic model obtained by assuming
`
`that at each point in time, at each repeater, exactly one new
`
`message originates. It will then follow that
`
`^> (t) = K 1;0 (t) .
`
`Similarly, if we define X^ (t) to be the number of distinct messages
`
`that arrive at the origin at time t in the deterministic model it
`
`will follow that
`
`N'0 (t) =Nx0 (t).
`
`We indicate an armwaving proof of the first assertion. The
`
`quantity No(t) is a sum of average contribution to the flow at
`
`11.100
`
`■MüMtaa. u^jdl mm
`
`

`

`Case 1:16-cv-02690-AT Document 121-12 Filed 08/05/16 Page 28 of 251
`Network Analysis Corporation
`
`the origin at time t, as a result of messages originating at
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`repeaters less than t units in distance and times earlier
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`than t. The average contribution from each repeater is
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`a constant (not with time) at each fixed time point and
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`repeater, multiplied by X^ the average generation rate,
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`(the constant is given by the calculations in section II
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`and depends on the coordinates (d,j) and time. Thus X factors
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`from the sum and N0(t) isXmuitiplied by the total flow resulting
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`from a single message originating at each repeater at each
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`point in time.
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`Ke now make the specific assumption.
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`At each point in time and at each repeater, a single new
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`message is originated.
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`Under this model to compute X^(t) and hence N (t) is trivial,
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`To fix ideas we depict the situation at three time points
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`t = 0
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`At time zero one message originates at each repeater, hence the
`message flow is X (0)=1.
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`t=l
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`5H—V
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`5 1 k
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`11.101
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`Network Analysis Corporation
`Case 1:16-cv-02690-AT Document 121-12 Filed 08/05/16 Page 29 of 251
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`At time 1 each repeater receives 5 messages,one from each of
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`four nearest neighbors and one new message.
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`21
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`21
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`21 k 21
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`t = 2,
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`At time w = 2 each repeater receives 5 messages from
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`each of its four nearest neighbors and one new message for a total
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`of 21.
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`To compute X (t) in general we note that each repeater
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`is statistically identical in terms of message flow. Hence,
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`XQ(t) = 4X0(t-l) +1 t^l
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`Xo(0) = 1.
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`This difference equation is trivial to solve and hence,
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`XQ(t) = 4t+1-l t>0 .
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`Thus
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`NQ(t) = ^ (4t+l_1}
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`To compute N ' (t) we consider the same deterministic model
`o
`and compute X ' (t), the number of distinct messages which arrive
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`at the origin for the first time at time t. It is easy to com-
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`pute X' (t) from the following table by summing contributions.
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`11.102
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`Network Analysis Corporation
`Case 1:16-cv-02690-AT Document 121-12 Filed 08/05/16 Page 30 of 251
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`Time of Origination Distance from Origin Number of Messages
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`0 t 4t
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`1 t-1 4(t-l)
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`2 t-2 4(t-2)
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`t-i i 4
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`t 0 1
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`The first column is the time the message first appeared in
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`the system if it is received by the origin for the first time
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`at time t. The second column indicates the distance from the
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`origin that the message originated. The third column indicates
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`the number of message originated at that time and distance which
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`are received at the origin at time t. Thus,
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`t
`X^(t) = 4t + 4 (t-1) + .. +4+1 = 27 4j+l
`2 j=1
`= 2t + 2t +1, O 0 .
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`Thus,
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`NJ (t) = \(2t2 + 2t +1)
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`The inefficiency of this "undamped" network is
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`leff (t) = J^t+l^j _ 4t+l-1 4t+l
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`X(2t2 +2t+l 3(2t2+2t+l) 6t2
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`The inefficiency grows rapidly with time for this undamped
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`system.
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`11.103
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`Case 1:16-cv-02690-AT Document 121-12 Filed 08/05/16 Page 31 o