Case 1:16-cv-02690-AT Document 121-11 Filed 08/05/16 Page 1 of 251
`Case 1:16-cv—02690-AT Document 121-11 Filed 08/05/16 Page 1 of 251
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`E-1
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`E-l
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`Case 1:16-cv—02690-AT Document 121-11 Filed 08/05/16 Page 1 of 251
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`E-1
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`E-l
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`Case 1:16-cv-02690-AT Document 121-11 Filed 08/05/16 Page 2 of 251
`Network Analysis Corporation
`
`The repeaters at d=5 do not have messages arriving from
`other stations. They only receive their own traffic at Poisson
`rate. Repeaters at d< 5 which are on the "axes" (denoted by
`circles) have messages arriving from the three neighbors at d+1,
`as well as their own Poisson traffic.
`Repeaters off the axes at distance d < 5 have input at each
`time point from the two neighbors at d+1 as well as their own
`Poisson traffic.
`The network is activated at t=0 by having random Poisson
`arrivals with mean X at each of the 61 repeaters. This input
`traffic at each repeater is converted to received messages in
`each of the; two possible modes for different values of m by use
`of the "transfer functions".
`
`,m. min(k-j,m-j)
`
`m v =0
`
`(2) P* = (m) V (-1)" 0) '2ZJL) k . j=0, 1, 2,...,min(k,m)
`*J j ~Q v m
`
`R
`These calculations giv>? us P., .>(0) for all repeaters with
`coordinates (d,j), d=l, 2, 3, 4, 5, j=l,...,4d, an (0,0) the
`station at the origin.
`Vie can now determine message traffic at each repeater by using
`equations which describe me-dsage transmission in the direction of
`of the "origin".
`
`For Time t=l
`
`When d= 5; j = ly 2,...,20 , the repeaters at d=5 receive only
`their generated Poisson traffic. Thus, for time 1 we generate
`61 Poisson traffic numbers which describe direct (i.e., at the
`source) message input. When d <4, the repeater at coordinates
`(d,j) also receive traffic from its neighbors at further distance
`
`11.25
`
`
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`Network Analysis Corporation
`Case 1:16-cv-02690-AT Document 121-11 Filed 08/05/16 Page 3 of 251
`
`by one unit. The following equations describe messages arriving
`at each repeater fo:: arbitrary time t>l.
`
`On the Axis
`
`F(0,0, (t> ■ PU,1> <t-»+»U.2> (t-1,+PU,3) (t-l)^1>4/t-l»+Poi»son
`
`M_*i <P1,1»P1,2'P1.3'P1,4>
`
`*n.DM • ?u,l)(t-1,*pu.2)(t-1,*pu,8),t-1,+Polsson
`
`"(l^1' - PU,2)lt-1)+P(2,3)«t-1»+P^.4)(t-l»+PoiSSOn
`
`"(1,3) (t) " ^2,4)(t-1,+P(2.5)(t-1)+PU,6)(t-l)+PoisSOn
`
`P(1.4)(t> * PU,6)<t-1)+PU,7)(t-1»+P(2,8,'t-1»+PoiSSOn
`
`«1« P2,l' P2.3' P2,5' P2,7
`
`(t-l)+Poisson
`P(2,l)lt» =PU,l)ft-1)+P(3,2)(t-1)+PU,12)
`
`P(2,3)(t) " P(3,3,,t-1)+P(3,4)(t-1)+Pn,;,)(t-1)tPoiSSOn
`
`R
`(t-l)+Poisson
`P(2,5)(t) =PU,6)(t-1)+P(3,7)(t-1+P(3,8)
`
`P
`
`(t-l)+Pois£jn
`l2,7)<t) =P(3,9)(t-1)+P(3,10)(t-1)+P(3,ll)
`
`At_dÜ P3(1, P3f4, P3fV P3fl0
`
`' (3,1) (t) " P(4,D (t"1)+P(4,2) ^-1)+P(4,16) (t-D^Poisson
`
`P(3,4)(t) - P(4,4)(t-1)+P(4,5)(t-1)+P(4,6)(t-1)+PoiSSOn
`
`P(3,7) (t) " P(4,3) (t"1)+P(4,9) (t"1)+P(4,10) <t-l>+Pois.on
`
`(t-l)+Poisson
`P(3r10)(t)-P(4/12)(t-1)+P(4,13)(t-1)+P(4r14)
`
`11.26
`
`
`
`Network Analysis Corporation
`Case 1:16-cv-02690-AT Document 121-11 Filed 08/05/16 Page 4 of 251
`
`Ä^=l P(4,l)(t)' P(4,5){t)' P(4,9)(t)' P(4,13)(t)''
`
`P<4,1> (t) * P(5,D {t-1)+P(5f2) (t-1)+P(5r20) (t-D+Poisson
`
`P(4,5)(t) " P^5,6)(t-1)+P^5,5)(t'"1)+P?5,7)(t-:L)+Poi8Son
`
`P(4,9)(t) " P%,10)(t-1)+P^5rll)(t-1)+P%,12)(t-1)+Poi98On
`
`P(4,13) (t)" pR(5r15) (t-1)+P^5,16) (t-1)+Pkl7) <t-l)+PoiB.on
`
`Off the Axes:
`
`P(d,j)^
`
`P^d+l,j)(t-1+PJd+l,j+l)(t-1)+Poisson; ^=2'3 d;
`d=2,3,4.
`
`P(drj)(t)
`
`P^d+1 j+1) (t-1'+PW+l j+2) (t~1)+Poisson'' j=d+2,d+3,..2d;
`d=2,3,4.
`
`pRd+l,j+2)(t-1)+PU+l,j + 3)(t-1)+PoiSSOn; J = 2d+2,..3d;
`d=2,3,4.
`
`P(dfj)(fc)
`
`P^d+l/j+3)(t-1)+p!(d+l,j + 4)(t-1)+P0iss0n; J-3d+2,..4d;
`d=2,3,4.
`
`These equations relate arriving and received messages over
`neighboring time points and repeaters. Thus, the arriving number
`of messages can be computed in the grid at each point in time and
`each repeater.
`In terms of a flow diagram, the procedure for analyzing this
`and all finite grids follows."
`
`11.27
`
`
`
`Network Analysis Corporation
`Case 1:16-cv-02690-AT Document 121-11 Filed 08/05/16 Page 5 of 251
`Initial Stage
`
`Compute
`Poisson
`Input at 61 pt
`
`at each point in
`time
`
`Compute Tables
`of
`Transfer Functions
`
`Mode 1
`
`Mode 2
`
`I
`
`Compute
`randomize 1 P,-, (o)
`4 : r- (dii>
`over distributions JJ
`
`P(d..)(o)
`D3
`
`Obtain
`
`P'V(0)
`
`L
`
`\ randomize over
`transfer functions
`in each mode
`
`Compute
`
`P(d..)(t)
`31
`
`Compute
`
`L_
`
`Repeat for each point in time
`
`FIGURF 2
`
`11.28
`
`
`
`»e.work Analysis Corporation
`Case 1:16-cv-02690-AT Document 121-11 Filed 08/05/16 Page 6 of 251
`
`The parameters are, X = mean Poisson arrival at each time
`
`point, at each repeater, m the number of slots in each mode one
`
`and two.
`
`The output of the computer analysis is processed and pre-
`
`sented in two forms, tabular and graphical. The tabular format is
`
`for each m and X mode,
`
`1 0
`
`1
`
`2
`
`3
`
`4
`
`5
`
`6
`
`7
`
`• • •
`
`• * •
`
`• • •
`
`. . .
`
`P(0,0)(t)
`p*o,o)(t)
`
`p(l,l)(t)
`
`p(l,l){t)
`P(l,2)(t>
`
`P(l,2)^
`•
`
`•
`•
`
`•
`
`•
`•
`•
`
`•
`
`•
`•
`
`Various graphic analyses are also obtained.
`
`A. A graphic arrived and received messages at the origin
`
`as a function of time for various values of m and A.
`
`B. A frequency histogram of arrivals off the axis. There are
`
`24 points of the axis at distance 2, 3, 4,...
`
`We take for each time t;
`
`f(x) = n imber of stations with x arrivals at time t .
`24
`
`This is plotted for each time point.
`
`11.29
`
`
`
`Case 1:16-cv-02690-AT Document 121-11 Filed 08/05/16 Page 7 of 251
`Network Analysis Corporation
`
`C. The same histograms as in b except on the axis. There
`are 16 points on the axes at distances 1, 2, 3, 4.
`
`D. The mean number of arrived and received messages x(t)
`and A (t) as a function of time on and off the axis. These
`are given by,
`
`, 5(t) - £ xf(x), $R(t) = £ xfR(x)
`x=l y=l
`
`where f(x) if the frequency of arrivals and f (x) is the
`frequency of received messages.
`
`R(x),
`^A(t) = £ xf (x), ^AR(t) = £ xfA
`
`A x=l A A x=l A
`
`where f»(x) and f. (x) are frequencies on the axis of
`arriving and received messages. Some numerical results
`follow.
`
`8.1 Summary of Initial Computer Analysis
`Attached, are two curves which represent a summary of data
`compiled from a preliminary computer investigation of a closed
`grid network. The grid selected for initial analysis is the
`closed boundary grid at distance five. We combined computer
`runs with the closed form theoretical analyses of sections 4
`and 5 of this report to obtain some observations of network
`behaviour.
`The first six curves represent a study of messages arriving
`and being received at the origin (fixed ground station) as a
`function of time. We used 20 computer runs for each of the first
`fifty time units. In this initial study the number of slots was
`kept fixed at 100, but X(the mean number of messages originating
`at a given repeater) was set at 10, 20 and 30. All calculations
`were carried out for mode 1 and mode 2.
`
`11.30
`
`
`
`Network Analysis Corporation
`Case 1:16-cv-02690-AT Document 121-11 Filed 08/05/16 Page 8 of 251
`
`The message flow and reception at the origin settle down at
`about t=4 and remained relatively constant. For X=10 the number
`of arriving messages seemed to have a mean at about 155 and the
`number of received messages averaged to about 31. Since the sys-
`tem behaviour for A=10, m=100 settled down so quickly it seems
`reasonable to combine all time point data past t=10 to estimate
`the probability density function of arrivals and receptions at the
`origin in each of modes 1 and 2 when X=10. The curves would seem
`to indicate asymptotic Poisson behaviour with means about 31, 155
`in mode 1 and about 100, 300, in mode 2 respectively. Saturation
`occurs quic.-iy in mode 2 for X=10 or more. These results are
`summarized in the last four curves of probability density functions,
`
`11.31
`
`
`
`Case 1:16-cv-02690-AT Document 121-11 Filed 08/05/16 Page 9 of 251
`Network Analysis Corporation
`
`The number of arriving and received messages at
`
`the origin as a function of time.
`
`Mode l
`
`* - 10
`m x 100 slots
`
`CD
`ao-,
`
`D
`CM-
`
`o
`CD-
`cn
`
`o
`to-
`co
`
`;C\J
`
`O
`ao-
`
`o
`OJ-
`
`OH
`(D
`
`T
`Q
`0. 8. 16.r 24. 32. 40.
`TIME
`
`"1
`
`FIGURE 3
`
`
`
`Case 1:16-cv-02690-AT Document 121-11 Filed 08/05/16 Page 10 of 251
`Network Analysis Corporation
`
`The number of arriving and received messages at
`
`the origin as a function of time.
`
`Mode 1
`
`A
`m
`
`20
`100 slots
`
`o
`CD-i
`
`O
`CM- a*
`
`o
`CD-
`CO
`
`O
`CO-
`CO
`
`?CM
`
`o
`CO-
`
`o
`OJ-
`
`o-
`
`P(0f0)^)
`
`cb,
`
`8.
`
`16, 2*4.
`TIME
`
`32.
`
`HO.
`
`FIGURE 4
`
`11.33
`
`
`
`Nttwork Analysis Corporation
`Case 1:16-cv-02690-AT Document 121-11 Filed 08/05/16 Page 11 of 251
`
`The number of arriving and received messages at
`
`the origin as a function of time.
`
`Mode 1
`
`X - 30
`m ■ 100 slots
`
`O
`CD-,
`zr
`
`o
`CM-
`3«
`
`O
`CD- en
`
`o
`CD-\
`CO
`
`FIGURE 5
`
`11.34
`
`/
`
`
`
`O
`00-
`
`O
`CM-
`
`O
`
`Network Analysis Corporation
`Case 1:16-cv-02690-AT Document 121-11 Filed 08/05/16 Page 12 of 251
`
`The number of arriving and received messages at
`
`the origin as a function of time.
`
`Mode 2
`
`X - 10
`m « 100 slots
`
`P<0,0)(t)
`
`16. 24.
`TIME
`
`FIGURE 6
`
`11.35
`
`
`
`Case 1:16-cv-02690-AT Document 121-11 Filed 08/05/16 Page 13 of 251
`Netwoik Analysis Corporation
`
`The number of arriving and received messages at
`
`the origin as a function of time.
`Mode-« 2
`X « 20
`m - 100 slots
`
`P(0,0)«t>
`
`(0,0) (t)
`
`O
`CD
`
`o
`CM-
`
`O
`en
`
`o
`OH
`CO
`
`;<M
`
`o
`OOH
`
`o
`rsi-
`
`o-|
`CD
`
`U
`
`8,
`
`16. 24.
`TIME
`
`32.
`
`40.
`
`FIGURE 7
`
`11.36
`
`
`
`O
`
`o
`rvj-
`3*
`
`Network Analysis Corporation
`Case 1:16-cv-02690-AT Document 121-11 Filed 08/05/16 Page 14 of 251
`
`The number of arriving and received messages at
`
`the origin as a function of time.
`Mode 2
`X - 30 P (t)
`m - 100 slots tu'U/
`
`8.
`
`16. 24.
`TIME
`
`40.
`
`FIGURE 8
`
`11.37
`
`
`
`Network Analysis Corporation
`Case 1:16-cv-02690-AT Document 121-11 Filed 08/05/16 Page 15 of 251
`
`O
`GO
`O
`
`O
`CD
`
`rr^K^i<-Y Density Functions
`
`Mode t
`A-IO
`
`80. 160. _ 240. 320. 400.
`Messages Received UNIT Messages Arriving
`
`■ICURE 9
`11.38
`
`
`
`Case 1:16-cv-02690-AT Document 121-11 Filed 08/05/16 Page 16 of 251
`
`Network Analysis Corporation
`
`3«
`o
`
`rsi
`o
`
`CD
`• ■ o
`
`\n
`•-•
`•• o
`
`ftp
`
`o
`
`(O o
`1«- o
`
`o
`o
`
`cb.
`
`Probability Density Functions
`
`Mode 2
`A«|0
`
`J
`
`80.
`
`160. 240.
`UNIT
`
`320. 1100.
`
`Messages Received
`
`Messages Arriving
`
`^
`
`FIGURE 10
`
`11.39
`
`
`
`Case 1:16-cv-02690-AT Document 121-11 Filed 08/05/16 Page 17 of 251
`Network Analysis Corporation
`
`9. DYNAMICS OF A SINGLE MESSAGE ON ROUTE
`
`In this section, we will develop the theoretical basis for
`
`a compute . analysis of the dynamics of a single message originating
`
`at a repeater in the net and attempting to reach the ground station
`
`at the origin. The equations derived are directed towards a computer
`
`analysis. Let us assume that the given message originates at a
`
`repeater with coordinates (i,j) at time t. If the incoming and ac-
`
`ceptance numbers at (k,j) at time t are respectively X (t) and
`A (i,j)
`X (t), we assume the given message is one of the X (t)
`(i,j) (i,j)
`messages. Furthermore, we assume that each of the X (t) messages
`(i,j)
`is equally likely to be one of the accepted messages. Under these
`
`assumptions, it follows that at (i,j), there are two types of
`
`messages which have arrived. The first type is one message (the
`
`given one), the second type are X (t)-l messages. The proba-
`(i,j)
`bility of acceptance at (i,j) is given by the hypergeometric pro-
`
`bability density function:
`
`x(ij)(t)-i
`
`kX^. (t)-l
`^ ' (22,
`x(ij)(t)
`
`X,. .. (t)
`
`At each repeater on every path to the ground station the
`
`same analysis applies. At any given repeater, on the path, say
`
`with coordinates (k,e) there may be several copies of the original
`
`message which arrives.
`
`Suppose (k,e) is on a path from (i,j) to (0,0) and the number
`
`11.40
`
`
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`Case 1:16-cv-02690-AT Document 121-11 Filed 08/05/16 Page 18 of 251
`Network Analysis Corporation
`
`of paths from (i,j) to (k,e) is w. Then at (k,e), at time t
`
`plus the distance from (i,j) to (k,e), either 0, 1, 2,..., up to
`
`w copies of the messaqe may arrive. If d is the distance from
`A
`(i,j) to (k,e) and at time (t + d), X (t+d) and X (t+d)
`(k,e) (k,e)
`messages respectively arrive and are accepted then we can compute
`
`the probability that exactly Z copies of the original messages
`
`are accepted. The computation of the required probabilities is
`
`a direct extension of
`
`P{ex3ctly Z copies of original message is accepted at
`(k,e) at time t + d/v copies are amongst the arrivals}
`
`vwx(kfe)(t+d)V
`
`X"(k,e)(t+d)-Z
`
`X(k,e)(t+d)
`
`X(k,e)(t+d)
`
`Z = 0, 1, 2,
`
`Equation(25) is valid at every repeater along every path from (i,j) to
`
`(0,0), and in particular at ne origin. The only ingredient needed to apply
`
`the equations to a computer analysis and generate numerical values is a for-
`
`mula for the probability that exactly v copies of the message arrive at each
`
`repeater. This formula can be obtained recursively using the idea of isodesic
`
`line and wedge joint density functions as developed in Section 7.
`
`If a single copy of the given message is accepted at its origination re-
`
`peater, it is:
`
`a) repeated to each of two repeaters one unit closer to the origin if
`
`it is not on an axis;
`
`b) repeated to the one repeater one unit closer to the origin if it is
`
`on an axis.
`
`11 /M
`
`
`
`Case 1:16-cv-02690-AT Document 121-11 Filed 08/05/16 Page 19 of 251
`Network Analysis Corporation
`
`We will focus only on (a) since (b) is essentially identical
`
`as far as the analysis is concerned. The message, when accepted
`
`at its origination, is then repeated to repeaters at (i-l,j-l)
`
`and (i-l,j). Acceptances at (i-l,j-l) and (i-l,j) are determined
`
`according to Equation (2 3) The isodesic line joint density of
`
`receptions and acceptances are computed at (i-l,j-l) and (i-l,j).
`
`This joint density then determines arrivals and acceptances at
`
`(i-2,j-2), (i-2,j-l) and (i-2,j). The process then continues
`
`recursively until all computations are carried out at the origin.
`
`9.1 Outline of Computer Analysis
`
`In our computer analysis we used the above results to compute
`
`the probability distributions and mean value of the number of
`
`copies accepted at the origin of a single message which originates
`
`at distance of 5, 4, 3, 2, 1, 0 units from the ground station. For
`
`convenience and realism of the numerical results, we selected each
`
`originating repeater to have the maximum number of paths to the
`
`origin. The coordinate system we used for these calculations is
`
`given in Figureil, below:
`
`j/l \
`5\ s\
`
`S8 S6
`A /7 / 5
`
`K
`/ 1p/ 8/ 6/ 4 V 2V2\
`
`, 3\
`
`/\\/\9/ 7 / 5 / 3 V\ r\
`\x U / '•r /
`
`9
`
`V
`
`'K
`
`'*■
`
`FIGURE 11
`
`11.42
`
`
`
`Case 1:16-cv-02690-AT Document 121-11 Filed 08/05/16 Page 20 of 251
`Network Analysis Corporation
`
`The repeaters selected for originating messages at distances 5,4,3,2,1,0
`
`are respectively at (5,4), (4,3), (3,3), (2,2), (1,1), (0,0). The routes arc
`
`designated in Figure 12, and the maximum number of copies of an originating mes-
`
`sage which can be received along each repeater on the route is given in Table 1
`
`below. Note that the maximum number of possible copies is given by the number
`
`of paths from an originating repeater to the receiving repeater. Note in Table
`
`1 that no copies can be received at a repeater further from the origin than the
`
`originator.
`
`(3,2)
`
`(5,4)
`
`0,0)
`
`(2,3)
`
`Fig.12 Routing From (5,4) to (0,0)
`
`(0,0) (1,1) (1,2) (2,1) (2,2) (2,3) (3,2) (3,., (3,4) (4,3) (4,4) (5,4)
`/
`10
`1
`1
`2
`1
`3
`6
`1
`1
`
`3 ! 1
`
`1
`
`0
`
`1
`
`1
`
`0
`
`1
`
`1
`
`1
`
`0
`
`1
`
`0
`
`1
`
`0
`
`0
`
`1
`
`2
`
`1
`
`1
`
`1
`
`0
`
`1
`
`1
`
`2
`
`0
`
`1
`
`1
`
`0
`
`1
`
`0
`
`1
`
`0
`
`0
`
`1
`
`0
`
`0
`
`1
`
`3
`
`3
`
`1.
`2
`
`1
`
`1
`
`1
`
`0
`
`1
`
`1
`
`3
`
`0
`
`1
`
`2
`
`0
`
`1
`
`1
`
`0
`
`I
`
`4
`
`6
`
`1
`
`3
`
`3
`
`1
`
`2
`
`1
`
`1
`
`1
`
`1
`
`(0,0)
`
`(1,1)
`
`(1,2)
`
`(2,1)
`
`(2,2)
`
`(2,3)
`
`(3,2)
`
`(3,3)
`
`(3,4)
`
`(4,3)
`
`(4,4)
`
`(5,4)
`
`Table 1 Maximum Number of Copies - Between Two Repeaters
`
`11.43
`
`
`
`Case 1:16-cv-02690-AT Document 121-11 Filed 08/05/16 Page 21 of 251
`Network Analysis Corporation
`
`A. The Equation for ZQ(j;t)
`
`Clearly Z (j;t) is simply given by
`
`».a.« - Vs* (l,j;t)
`
`J0XJ'- A (0,0;t)
`'(0,0)
`
`; j = 0, 1; t = 0, 1, 2, ..., 40.
`
`(24)
`
`B.
`
`The Equation for Z.(j;t)
`
`1 A.„ „%(u.j»t)
`
`for j = 0, 1; t = 1, 2, ..., 40; where
`
`1 A(l i)(1'3;t)
`1 f^j.-t) = ; ln —. ? J = 0, 1; t = 0, 1, 2,
`A(lfl)(0,0;t) '
`
`., 39.
`
`C. The Equation for Z„(j;t)
`
`2 n=o v=o 2 D A(o,o)lu'u't;
`
`for j = 0, 1, 2; t = 2, 3, ..., 40; where
`
`2r • H Y f2r -I- n ^ ^ A(1,D CM>i,t) . A(l,2)(p'j;t)
`f2(x,D;t) = I f (y,t-i) (.) (,.__—. ———
`f
`y=0 (1,1) (1,2)
`
`for t = 1, 2, ..., 39; i = 0, 1; j = 0, 1; where
`
`2 A(2 2)(1'j;t)
`f H-t) =
`V3'*' A ^ -0,0;t)
`(2,2)
`
`; t = 0, 1, 2, ... , 38; j = 0, 1.
`
`D. The Equation for Z3(j;t)
`
`(25)
`
`(26)
`
`«,(„« - I I fj«,,.„t-i, ,»> W!?I^
`„»0v=0 - j '"A(0.0)t0'0,tl
`
`(27)
`
`for t = 3, ..., 40; j = 0, 1, 2, 3; where
`
`!,<i.j»t> - I I f>,V;t-l) (J) (T)-,(1,1)mn.„ ' ,(1'2' "'
`1 j ' A(1/1)(0,0;t) A(1/2)(0,0;t)
`P-:Q V-0
`
`11.44
`
`
`
`Case 1:16-cv-02690-AT Document 121-11 Filed 08/05/16 Page 22 of 251
`Network Analysis Corporation
`
`for t = 2, ..., 39; i = 0, 1; j = 0, 1, 2; where
`
`».0 <i> «j» A(2i2)(0.0,t) Ä(22) 10,0,« '
`
`for t = 1, 2, ..., 38; i = 0, 1; j = 0, 1; where
`
`3 *(3 3,(i'3;t)
`«i«'« - A ' (oto,tT '' t = °' l* 2 37; j = °' l'
`
`E. The Equation for Z4(j;t)
`
`VJlt,.f f ^,,.,-»,7,.>■»»irr,
`
`4 p=0v=0 4 D A(0f0)l0'°'t}
`
`for t = 4, 5, ..., 40; j = 0, 1, 2, ..., 16; where
`
`. 12 1. A.. (y-h>,i;t)
`
`V=0 y=0 p=0 J (1,1)
`
`A(l,2)(y + P>j;t)
`A(lf2)(0,0;t)
`
`for t = 3, 4, ..., 39; i = 0, 1, 2, 3; j = 0, 1, 2, 3; where
`
`11. . A. (vi,i;t) A . (p+v,j;t)
`f li,i*tl - 1 1 f <U.v.-t-l, $ .7, fr. "' ' -a • ,0,0(t-
`U=0 V=0 (2,1) (2,2)
`
`A(2,3)(v'k;t)
`A(2f3J(0.0,t> '
`
`for t = 2, 3, ..., 38; i = 0, 1; j = 0, 1, 2; k = 0, 1; where
`
`f4(i 1-t) - y f4(u-t 1» I»«. «»», . A(3,2)(^i?t) A(3,3)(^j;t)
`r (i,];t) - l r (U;t-1) { ; l.) • . ■ ■ — • . ■ •
`2 vi0 1 13 A(3,2)(°'0?t) A(3,3)(°'0;t)
`
`for t = 1, 2, ..., 37; : = 0, 1; j = 0, 1; where
`
`4 A{4 3)(1'3;t)
`f (-i-t) _ \ti^i (28)
`1 A(4 3)(0,0;t) ; t = 0, 1, ..., 36; j = 0, 1.
`
`11.45
`
`
`
`Case 1:16-cv-02690-AT Document 121-11 Filed 08/05/16 Page 23 of 251
`Network Analysis Corporation
`
`The numbers in Table 1 give the upper limits of the summation for the pos-
`
`sible copies of messages which can be received at each repeater of a single
`
`message originating at a repeater further from the origin but within the net of
`
`Figure 12. With the selected net and the numbers of Table 1, we can use the re-
`
`sults of section 12 to obtain numerical data.
`
`At time zero a random number of messages has arrived at each repeater. To
`
`compute the distribution of copies arriving at (0,0) from (5,4) we assume one
`
`of the messages arriving at (5,4) is singled out and followed along the route
`
`using the hypergeometric analysis of section 12. The procedure was used for
`
`t -- 0, 1, 2, ..., 40 in conjuction with the random Poisson number generator de-
`
`veloped and discussed earlier.
`
`Specifically we seek to compute the five numbers:
`
`z0(j;t)
`
`~ 0, 1;
`
`t = 0, 1, 2, ..., 40;
`
`= 0, 1;
`
`Z2(j;t)
`
`= 0, I.- 2;
`
`t = 1, 2, ..., 40;
`
`t = 2, 3, ..., 40;
`
`Z3(j;t)
`
`= 0, 1, 2, 3;
`
`t = 3, 4, ..., 40;
`
`z4(j;t)
`
`Z5(j;t)
`
`= 0, 1, 2, 3, 4, 5, 6; t = 4, 5, 6, ..., 40;
`
`= 0, 1, 2, 3, ..., 10; t = 5, 6, ..., 40;
`
`where Z (j;t) is the probability that exactly j copies of a message originating
`
`at a repeater at distance k at time t-k, are accepted at the origin at time t.
`
`For the computer analysis we considered one repeater at each of the distances,
`
`as in Figure 11. The maximum j values are given by the first row of Table 1.
`
`Using the hypergeometric analyses the following equation can be used to compute
`
`each of the Z, (j;t); as a function of:
`k
`
`1) X = mean number of originations at each repeater.
`
`2) m = number of slots fixed at 100.
`
`3) Each of two capture nodes 1 and 2.
`
`For ease of notation we denote:
`
`A.. ., (w,X;t) =
`(13)
`
`X,. ., (t)-w
`(13)
`
`X,. .,(t)-X
`(13)
`
`11.46
`
`
`
`Case 1:16-cv-02690-AT Document 121-11 Filed 08/05/16 Page 24 of 251
`Network Analysis Corporation
`
`F. The Equation for Z5(j;t)
`
`(6.6)
`
`Zb <j;t) = I I f (p,v;t-l) (7> .A
`
`(°'0)
`(00,t) ;
`
`for-t = 5, 6, ..., 40; j = 0, 1, 2,
`
`, 10; where
`
`13 3
`I I I
`f^(i,j;t) = l_ ^ l_ fj(v.i..p,t-l) O <p+p)
`v=0 y=0 p=0
`
`A(l 1) ^+v,i;t)
`A(lfl)(0,0;t)
`
`for t = 4, 5, 6, ..., 39; i = 0, 1, 2, 3, 4; j = 0, 1, 2, ..., 6;
`
`A(l,2)(ti+Pfj;t)
`* A(12)(0,0;t) '•
`
`where
`
`1 2 l
`v+y. ,y+p,
`E?(i,j,k;t> =111 f'(v,y,p;t-l) (Y) (*») ("£>)
`v=0 y=0 p=0 J
`
`A(2,l)(v>i;t)
`A(2fl)(0.0,t)
`
`A(2>2)(v+y,j;t)
`A(2f2JC0,0,t)
`
`A(2t3)(y+P,k;t)
`A(2>3)(0,0;t)
`
`for t = 3, ..., 38; i = 0, 1; j = 0, 1, 2, 3; k = 0, 1, 2, 3;
`
`where
`
`f*(i,j,k;t) = I I f^(y,v;t-l) (\) <^V) (J)
`v=0 y=0
`
`A(3,2)(ti>i;t)
`A(3/2)(0,0;t)
`
`A(3>3)(y+v,j;t)
`* A(33)(0,0;t)
`
`A(3,4)(V'k;t)
`* A(3f4)(0,0,t) '
`
`for t = 2, 3, ..., 37; i = 0, 1; j = 0, 1, 2; k = 0, 1; where
`
`f5n i ti I f5fu t n r"i fvi • A(4>3)(M'1,t} . A(4,4)(u'j;t)
`f (i,j;t) = I f (y,t-l) (.) ( ) , . . . ;
`2 y=Q 1 13 A(4f3)(0,0,t) A(4f4)(0.0;t)
`
`11.47
`
`
`
`Case 1:16-cv-02690-AT Document 121-11 Filed 08/05/16 Page 25 of 251
`Network Analysis Corporation
`
`for t « 1, 2, 3, ..., 36» i ■ 0, 1» j ■ 0, 1» where
`
`U.jit)
`(0,0; t) I t - 0, I, 2, ..., 35; j - 0, 1.
`
`1 A(5,4)
`
`9.2 Probability of at Least One Message Getting Threrc'i
`
`The first set of curves, piguresl3-18# p3ot the probability
`of at least one message getting through as a function of the
`mean number of originations at each repeater. There is one sot oh
`curves "for each unit of distance d ranging from 0 to 5. Lach
`figure contains one curve for mode 1 and one curve for ir.ocio 2.
`The number of slots was fixed at 100. The data for the curves is
`summarised in Table 2 below.
`
`X=l
`distance Mode 1 Mode 2
`
`X=3
`Mode 1 Mode 2
`
`X=5
`Mode 1 Mode 2
`
`0
`
`1
`
`' 2
`
`3
`
`4
`
`5
`
`.398
`
`.243
`
`.355
`
`.428
`
`.613
`
`.740
`
`.589
`
`.434
`
`.614
`
`.695
`
`.874
`
`.967
`
`.285
`
`.192
`
`.166
`
`.164
`
`.250
`
`.341
`
`.421
`
`.273
`
`.341
`
`.358
`
`.534
`
`.692
`
`.264
`
`.1.17
`
`.129
`
`.119
`
`.159
`
`.157
`
`.431
`
`.321
`
`.326
`
`.285
`
`.271
`
`.198
`
`Table 2: Probability That at Least One Message Gets Through
`
`11.48
`
`
`
`Case 1:16-cv-02690-AT Document 121-11 Filed 08/05/16 Page 26 of 251
`Nctwtnk Anatyiii Conwmlion
`
`m
`** o
`
`o o o
`
`e «o
`
`• m
`
`. n
`
`' CM
`
`o
`N
`•0
`
`W
`
`H
`e» w
`H 06
`o e>
`b En
`O
`&
`W
`
`5 z
`
`d\
`
`00
`
`VO
`
`in
`
`ro
`
`CM
`
`HonoHHi sxao aovssaw
`3N0 XSV31 XV XYHX XXniHYHOHd
`
`11.49
`
`\
`
`
`
`Case 1:16-cv-02690-AT Document 121-11 Filed 08/05/16 Page 27 of 251
`Network Analysis Corporation
`
`n
`O -H
`H c
`« 3
`O iH o
`
`N II
`e -o
`
`.. in
`
`.. r>i
`
`ON
`
`00
`
`vo
`
`PI
`
`(N
`
`HonoHHJ. si3D aovssaw
`
`3N0 xsvai iv xvHi AimavaoHd
`
`11.50
`
`
`
`f.'
`
`Case 1:16-cv-02690-AT Document 121-11 Filed 08/05/16 Page 28 of 251
`Network Analysis Corporation
`
`m
`O
`
`o
`o
`
`II
`e
`
`•H
`C
`3
`
`II
`-0
`
`1
`
`w
`Q
`O
`s
`
`o
`«
`Q
`O
`S
`
`> in
`
`. CM
`
`•o
`
`CO z o
`
`H
`
`z
`H u
`H « o
`tu o
`« w
`
`2
`
`in
`
`w
`u
`H
`
`■■■
`
`CTl
`
`00
`
`\£>
`
`(N
`
`H9H0HHX SJ.39 3DYSS3W
`3N0 isvai xv xvHX AxnifiYeotfd
`
`11.51
`
`
`
`"
`
`Case 1:16-cv-02690-AT Document 121-11 Filed 08/05/16 Page 29 of 251
`Network Analysis Corporation
`
`0) Hi
`
`0 i
`
`(0 3
`
`O PI
`o
`
`II II
`6 -0
`
`HM
`
`.. in
`
`. > ro
`
`■ ■ <N
`
`H 1 I
`
`o a\ oo
`
`I I »-
`vo in
`
`m fM
`
`HOflOHHi SX39 30VSS3W
`3NO xsvsi xv XVHX ÄxniflvaoHd
`
`11.52
`
`
`
`Case 1:16-cv-02690-AT Document 121-11 Filed 08/05/16 Page 30 of 251
`Network Analyst* Corporation
`
`• • m
`
`II
`
`2
`O
`H
`
`2
`M
`U
`CO M
`05
`O
`
`W
`«
`D
`Ü
`
`CM
`O
`«
`W
`
`D
`2
`
`<N
`
`■4 1-
`
`Ch 00
`
`VO
`
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`
`CO
`
`fN
`
`HOnOHHl SI3D 30VSS3W
`
`3N0 ISV31 IV IVJU AJillieVSOHd
`
`11.53
`
`
`
`-^——•SSfc
`Case 1:16-cv-02690-AT Document 121-11 Filed 08/05/16 Page 31 of 251
`Setwork Analysis Corporation
`
`m u
`■P -P
`O -H
`H C
`0) 9
`
`o o in
`
`II II
`6 TJ
`
`u
`Q
`
`W z o
`u
`Q I
`
`in
`
`II
`•0
`
`Z o
`H oo
`z
`H W
`
`PS o
`O H
`tu
`O
`
`g D
`
`<N Z
`Z
`
`:
`
`:
`
`-I 8 1 I
`
`tfi 00
`
`VD
`
`< I
`in
`
`ro ts
`
`HDnoBHX sxao acvssaw
`3N0 I.SV31 I.V XVH1 AilliaVSOHd
`
`11.54
`
`
`
`Case 1:16-cv-02690-AT Document 121-11 Filed 08/05/16 Page 32 of 251
`
`Network Analysis Corporation
`
`9# 3 Distribution of Message Explosion as a Function of Slot
`Size and Mean Number of Originations
`
`The equations for message explosion derived earlier
`in the report were used to obtain numerical data
`for message explosion. The results of the numerical analysis
`follow in Tables 3 through 26 and Figures 1920 , and 21.
`
`9.4 Distributions of Copies Getting Through as a Function of
`Slot Size and Mean Originations
`
`Tables 3-26 contain the probability distributions for the
`number of copies of a sirigle message which are received at the
`origin (ground stations) for each distance (d=0,2,3,4,5,) of
`origination of the message. The tables vary according to mode
`(each of two modes), mean number of messages originating at each
`repeater (A=l,3,5), and each of four slot sizes (m=25, 50, 75,
`100). This produces a total of 4 x 3 x 2 = 24 tables.
`
`In table 2 7, we summarize the results of the twenty-four
`tables by considering only the probability that at least one
`copy of the message gets through .s a function of distance and
`the three parameters; mode, mean, and ."lot size.
`
`Tie results of table 2 7 are presented pictorially in
`figures 19,20, and21 for distances of zero, two and four respec-
`tively of origination of the message.
`
`11.55
`
`
`
`Case 1:16-cv-02690-AT Document 121-11 Filed 08/05/16 Page 33 of 251
`Network Analysis Corporation
`
`Mode 1, X= 1, m = 25
`
`distance^
`0
`0
`.786
`1
`.921
`.900
`.397
`.831
`
`2
`3
`4
`5
`
`1
`.214
`
`.079
`.096
`.097
`
`.153
`
`2
`
`3
`
`.004
`.005
`
`,015
`
`1
`
`.001
`
`Table 3_
`
`Mode 2, X- 1, m 25
`
`distance
`0
`1
`2
`3
`4
`5
`
`0
`.689
`.834
`.759
`.716
`.554
`
`1
`.311
`.166
`.219
`.245
`.324
`
`2
`
`3
`
`4
`
`.021
`
`.037
`.103
`
`.002
`.018
`
`.002
`
`— ■ . ■ - . w
`
`Table 4
`
`11.56
`
`
`
`Case 1:16-cv-02690-AT Document 121-11 Filed 08/05/16 Page 34 of 251
`NsMörk Analysis Corporation
`
`I:of!e 1, X - 3,
`
`/!>
`
`dit:tancc\
`! o
`i" 1
`i 1
`1 2
`3
`
`4
`5
`
`.80!)
`
`.942
`
`.952
`.9GG
`
`.953
`
`.3 95
`
`.058
`.04 7
`.033
`
`.045
`
`.001
`
`.001
`.002 i
`
`1
`
`Table 5
`
`Kerle 2, A = 3, in
`
`25
`
`distance \
`0
`1
`
`2
`
`3
`4
`5
`
`.261
`.739
`.102
`.890
`.887 . .107
`.892
`.101
`
`.826
`
`.153
`
`.006
`.007
`.019
`
`
`
`.001
`
`"able §_
`
`11.57
`
`\
`
`
`
`Case 1:16-cv-02690-AT Document 121-11 Filed 08/05/16 Page 35 of 251
`Network Analysis Corporation
`
`Mode 1. X - 5, ra
`
`25
`
`distanced
`0
`1
`2
`3
`4
`5
`
`0
`.813
`.951
`.963
`.979
`.977
`
`1
`.1S7
`.049
`.036
`.021
`.022
`
`2
`
`.001
`.OCI
`.001
`
`Table 7
`
`node 2, X = 5, m = 25
`
`u
`\
`ir
`distance \
`0
`1
`2
`3
`4
`5
`
`0
`.753
`.914
`.916
`.930
`.900
`
`1
`.247
`.086
`.080
`.066
`.091
`
`2
`
`.004
`.004
`.008
`
`Table 8
`
`11.58
`
`
`
`Case 1:16-cv-02690-AT Document 121-11 Filed 08/05/16 Page 36 of 251
`Network Analysis Corporation
`
`i!od'> 1, X
`
`1, m -50
`
`di stance \ V
`0
`
`1
`2
`3
`
`4
`5
`
`0
`755
`.878
`
`. Si. 3
`
`. 796
`
`.661
`
`1
`.245
`.122
`.]6fi
`
`.185
`
`.272
`
`2
`
`3
`
`I
`
`l ~^
`
`.001
`.019
`.05!>v
`
`.00.1
`
`.007
`
`Table 9
`
`Mode 2, X - 1, n
`
`50
`
`distanceN.
`0
`
`1
`2
`3
`4
`5
`
`0
`.605
`.736
`.602
`.531
`.306
`
`1
`.395
`.264
`
`.338
`.356
`.360
`
`2
`
`3
`
`4
`
`5
`i !
`
`.060
`.103
`.232
`
`.010
`.083
`
`.017
`
`j
`.002
`
`Table 10
`
`11.59
`
`
`
`Case 1:16-cv-02690-AT Document 121-11 Filed 08/05/16 Page 37 of 251
`Network Analysis Corporation
`
`Mode 1, X = 3, m = 50
`
`distance\ 0
`.783
`0
`.926
`1
`2
`.921
`.926
`3
`.882
`4
`5
`
`1
`.212
`.074
`.076
`.070
`.108
`
`2
`
`.003
`.006
`.009
`
`Table 11
`
`Mode 2, X = 3, m = 50
`
`distonceX ' 0
`.709
`0
`.860
`1
`2 "
`.816
`.800
`.875
`
`3
`4
`
`5
`
`1
`.291
`
`.140
`.170
`
`.178
`.258
`
`2
`
`3
`
`4
`
`.014
`.021
`.058
`
`.001
`.008
`
`.001
`
`Table 12
`
`11.60
`
`
`
`Case 1:16-cv-02690-AT Document 121-11 Filed 08/05/16 Page 38 of 251
`Network Analysis Corporation
`
`V.QsZo 1, \ - 5, ui = 59
`
`oi :;i;<-r V
`0
`I .734
`1 .937
`. -1' ]
`. 9 5 4
`
`i
`
`?.
`3
`/'
`
`5
`
`C.'C \
`
`j
`
`i
`i
`j
`i .93b
`
`*/
`
`1
`i .206 1
`. .063 '
`
`i i
`
`.002 !
`
`. 002 j
`
`.004
`
`! .057
`I .044
`! .0G2
`i
`
`!
`
`Table 13
`
`Mode 2, A = 5,
`
`50
`
`distance
`0
`1
`2
`3
`4
`5
`
`.733
`.890
`.868
`.870
`.791
`
`.267
`
`.110
`.124
`.120
`.180
`
`.008
`.010
`.027
`
`.002
`
`Table 14
`
`11.61
`
`
`
`Case 1:16-cv-02690-AT Document 121-11 Filed 08/05/16 Page 39 of 251
`Network Analysis Corporation
`
`Kode 1, \ = 1, n 75
`
`distancoX
`! o
`1
`2
`
`3
`4
`
`5
`
`0
`.711
`
`.831
`.742
`.863
`.512
`
`1
`
`.289
`.1Ü9
`.235
`.269
`.342
`
`2
`
`4
`3
`1 ;
`
`(
`i
`;
`
`.023 1
`.043
`.121
`
`.002
`
`i
`i
`.003
`.023
`! i
`
`Ta*i.!e 15
`
`Kode 2, X = 3 , :* =- 75
`
`distancc\
`0
`
`1
`2
`3
`4
`
`5
`
`0
`.526
`.655
`
`.486
`.392
`.134
`
`1
`.474
`.345
`.403
`.408
`.304
`
`5
`
`I
`
`1 1
`!
`! !
`i
`
`6
`
`"
`
`.049
`
`.00 3
`
`.001
`
`i
`
`■
`
`2
`
`3
`
`4
`
`.106
`
`.175
`.295
`
`.025
`.158
`
`Table 16
`
`11.62
`
`
`
`Case 1:16-cv-02690-AT Document 121-11 Filed 08/05/16 Page 40 of 251
`Neto/ork Analysis Corporation
`
`r:ou3 1, x --- 3,
`
`75
`
`65. :st-i IC6»
`0
`i .782
`i)
`.913
`
`r •
`
`1
`2
`
`i
`i
`•
`
`.895
`.891
`.818
`
`1
`i .218
`i .087
`i
`t .300
`i
`1 .103
`i
`i .161
`1
`
`2
`
`.005
`.006
`.019
`
`■?
`
`l !
`1 1
`
`i )
`.
`\
`
`i
`1 .001
`- j i
`i
`
`Table 17
`
`3
`4
`
`J n ,
`
`Mode 2, X=3,m=75
`
`distance \s
`0
`1
`
`2
`
`3
`
`5
`
`0
`.674
`.815
`.750
`.72?
`.561
`
`1
`.326
`.„85
`.225
`.233
`.313
`
`2
`
`3
`
`-I
`
`1 i
`i
`
`I
`
`i
`
`i
`
`.002
`.020
`.002:
`i i !
`
`.025
`.040
`.103
`
`Table 18
`
`11.63
`
`
`
`Case 1:16-cv-02690-AT Document 121-11 Filed 08/05/16 Page 41 of 251
`Network Analysis Corporation
`
`Ftoda 1. X = 5, n - 75
`
`uistanct\ 0
`.788
`0
`.929
`1
`.924
`2
`.932
`3
`.894
`4
`
`5
`
`1
`.212
`.071
`.074
`.065
`j .098
`
`1
`
`2
`
`! .003
`: .003
`! .008
`!
`
`Table 19
`
`Mode 2, X = i", ,-y_g__75
`
`2
`
`3
`
`1
`.289
`.137
`.108
`.16 3
`
`i
`
`0
`distanced.
`.711 |
`0
`.863 !
`1
`.819
`2
`i 3
`.818
`1 4 ! .703
`is! !
`
`1 !
`i
`
`.001.
`
`• 0 06'i
`i
`
`.013
`.019
`
`.239
`
`.051
`
`Table 20
`
`11.64
`
`
`
`Case 1:16-cv-02690-AT Document 121-11 Filed 08/05/16 Page 42 of 251
`Network Analysis Corporation
`
`V.OC.G
`
`«n - 3 00
`
`c. Lst.vic?', 0
`
`1
`
`2
`
`3
`
`4
`
`5
`
`0 ! .602
`3 | .737
`
`2 .645
`3 ! .57.?
`4 j .387
`5 .260
`
`■ • ~1 J *.'
`
`.21?
`i. :s02
`
`.325
`^ .369
`
`! .318
`
`1 1
`
`I
`I
`
`.053
`
`.093
`
`.3 84
`.250
`
`.012
`
`.050
`.122
`
`.003
`
`.039
`
`.008
`
`r,
`
`i
`
`i •
`
`!
`i
`*
`!
`.003
`
`Table 21
`
`Mode 2, X = 1, m = 100
`
`distance^ 0
`C
`.411
`
`1
`2
`
`3
`4
`5
`
`.566
`.386
`.305
`.126
`
`.003
`
`]
`
`.589
`.434
`
`A *' ^
`
`.399
`
`.244
`
`.110
`
`2
`
`3
`
`4
`
`5
`6
`7
`i ! ; i
`i !
`
`.181
`.240
`
`.308
`
`.200
`
`.056
`
`.215
`.248
`
`•
`
`.087
`.211
`
`.019
`.126
`
`.002
`.054
`
`! !
`1 1 !
`\
`
`i .03 4
`
`. ÜOi]
`
`Table 22
`
`11.65
`
`
`
`Case 1:16-cv-02690-AT Document 121-11 Filed 08/05/16 Page 43 of 251
`Network Anclysiu C-wF&rGtion
`
`Mode 1, X = 3f m = 100
`
`distance\ # 0
`.715
`0
`.808
`
`1
`
`.834
`836
`
`2
`3
`4 | .750
`.659
`
`5
`
`1
`.285
`.192
`.143
`.147
`! .211
`; .26"
`
`2
`
`3
`
`4
`
`.02 3
`.016
`• .035
`
`: .06
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