LINE Solver
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map_block.m
1function MAP=map_block(E1,E2,E3,G2,OPT)
2% map_block constructs a MAP(2) or MAP(1) according to the given moments
3% and autocorrelation parameters if no MMPP(2) can be constructed using the
4% same set of moment and autocorrelation parameters
5%
6% MAP=map_block(E1,E2,E3,G2,OPT) returns a MAP(2) with prescribed or
7% approximated prescribed moments and gamma2
8%
9% Input:
10% E1 = the first moment
11% E2 = the second moment
12% G2 = rou(i) / rou(i-1), where rou(i) is the autocorrelation of lag
13% i of a map
14% OPT = optional parameter, if OPT is 'scv', it means the second
15% input parameter is actually scv, squared cv
16%
17% Output:
18% MAP = a Hmap(2) or map(1) with the given moments and G2, but
19% sometimes we can't get the map with exact moments and G2, because
20% the given moments may not be feasible, so we give a Hmap(2) or
21% map(1) with the closest moments and gamma2
22%
23% Comments:
24% Sometimes we can't get the map with exact moments and G2, because
25% the given moments and gamma2 may not be feasible, so we try to find
26% a MAP(2) that gives the moments and autocorrelation as close as
27% possible to given moments and autocorrelations
28%
29
30if nargin==5
31 if strcmpi(OPT,'scv')
32 E2=(1+E2)*E1^2;
33 end
34else
35 OPT='';
36end
37
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E3^2+24*E1^5*E3^3+648*E1^4*E2^5-567*E1^2*E2^6+216*E1^7*E2^2*E3+144*E1^6*E3^2*E2-756*E1^5*E2^3*E3-270*E1^4*E3^2*E2^2-12*E1^3*E3^3*E2+810*E1^3*E3*E2^4+108*E1^2*E3^2*E2^3-270*E1*E3*E2^5+E3^4*E1^2)^(1/2))/(-3*E3^2*E2^2+48*E3^2*E1^4+81*E2^4*E1^2-36*E3^2*E1^2*E2+90*E2^3*E3*E1-126*E2^2*E1^3*E3-54*E2^5+2*E1*E3^3)*E1^3*E2^2-18*(24*E1^4*E3-27*E1^3*E2^2-18*E1^2*E2*E3+18*E1*E2^3+E3^2*E1+(-243*E1^6*E2^4-6*E1*E3^3*E2^2+162*E2^7+9*E2^4*E3^2+24*E1^5*E3^3+648*E1^4*E2^5-567*E1^2*E2^6+216*E1^7*E2^2*E3+144*E1^6*E3^2*E2-756*E1^5*E2^3*E3-270*E1^4*E3^2*E2^2-12*E1^3*E3^3*E2+810*E1^3*E3*E2^4+108*E1^2*E3^2*E2^3-270*E1*E3*E2^5+E3^4*E1^2)^(1/2))/(-3*E3^2*E2^2+48*E3^2*E1^4+81*E2^4*E1^2-36*E3^2*E1^2*E2+90*E2^3*E3*E1-126*E2^2*E1^3*E3-54*E2^5+2*E1*E3^3)*E1^2*E2*E3+12*E2*E1^2+2*(24*E1^4*E3-27*E1^3*E2^2-18*E1^2*E2*E3+18*E1*E2^3+E3^2*E1+(-243*E1^6*E2^4-6*E1*E3^3*E2^2+162*E2^7+9*E2^4*E3^2+24*E1^5*E3^3+648*E1^4*E2^5-567*E1^2*E2^6+216*E1^7*E2^2*E3+144*E1^6*E3^2*E2-756*E1^5*E2^3*E3-270*E1^4*E3^2*E2^2-12*E1^3*E3^3*E2+810*E1^3*E3*E2^4+108*E1^2*E3^2*E2^3-270*E1*E3*E2^5+E3^4*E1^2)^(1/2))/(-3*E3^2*E2^2+48*E3^2*E1^4+81*E2^4*E1^2-36*E3^2*E1^2*E2+90*E2^3*E3*E1-126*E2^2*E1^3*E3-54*E2^5+2*E1*E3^3)*E3^2*E1+27*(24*E1^4*E3-27*E1^3*E2^2-18*E1^2*E2*E3+18*E1*E2^3+E3^2*E1+(-243*E1^6*E2^4-6*E1*E3^3*E2^2+162*E2^7+9*E2^4*E3^2+24*E1^5*E3^3+648*E1^4*E2^5-567*E1^2*E2^6+216*E1^7*E2^2*E3+144*E1^6*E3^2*E2-756*E1^5*E2^3*E3-270*E1^4*E3^2*E2^2-12*E1^3*E3^3*E2+810*E1^3*E3*E2^4+108*E1^2*E3^2*E2^3-270*E1*E3*E2^5+E3^4*E1^2)^(1/2))/(-3*E3^2*E2^2+48*E3^2*E1^4+81*E2^4*E1^2-36*E3^2*E1^2*E2+90*E2^3*E3*E1-126*E2^2*E1^3*E3-54*E2^5+2*E1*E3^3)*E1*E2^3-3*(24*E1^4*E3-27*E1^3*E2^2-18*E1^2*E2*E3+18*E1*E2^3+E3^2*E1+(-243*E1^6*E2^4-6*E1*E3^3*E2^2+162*E2^7+9*E2^4*E3^2+24*E1^5*E3^3+648*E1^4*E2^5-567*E1^2*E2^6+216*E1^7*E2^2*E3+144*E1^6*E3^2*E2-756*E1^5*E2^3*E3-270*E1^4*E3^2*E2^2-12*E1^3*E3^3*E2+810*E1^3*E3*E2^4+108*E1^2*E3^2*E2^3-270*E1*E3*E2^5+E3^4*E1^2)^(1/2))/(-3*E3^2*E2^2+48*E3^2*E1^4+81*E2^4*E1^2-36*E3^2*E1^2*E2+90*E2^3*E3*E1-126*E2^2*E1^3*E3-54*E2^5+2*E1*E3^3)*E3*E2^2-3*E2^2);
39mu10=3*(24*E1^4*E3-27*E1^3*E2^2-18*E1^2*E2*E3+18*E1*E2^3+E3^2*E1+(-243*E1^6*E2^4-6*E1*E3^3*E2^2+162*E2^7+9*E2^4*E3^2+24*E1^5*E3^3+648*E1^4*E2^5-567*E1^2*E2^6+216*E1^7*E2^2*E3+144*E1^6*E3^2*E2-756*E1^5*E2^3*E3-270*E1^4*E3^2*E2^2-12*E1^3*E3^3*E2+810*E1^3*E3*E2^4+108*E1^2*E3^2*E2^3-270*E1*E3*E2^5+E3^4*E1^2)^(1/2))/(-3*E3^2*E2^2+48*E3^2*E1^4+81*E2^4*E1^2-36*E3^2*E1^2*E2+90*E2^3*E3*E1-126*E2^2*E1^3*E3-54*E2^5+2*E1*E3^3)*(2*E1^2-E2)*(G2-1);
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E2^5+E3^4*E1^2)^(1/2))/(-3*E3^2*E2^2+48*E3^2*E1^4+81*E2^4*E1^2-36*E3^2*E1^2*E2+90*E2^3*E3*E1-126*E2^2*E1^3*E3-54*E2^5+2*E1*E3^3)*E2^3);
42
43D0=[ -mu00-mu01, 0;
44 0, -mu10-mu11];
45
46D1=[ mu00, mu01;
47 mu10, mu11];
48
49MAP={D0,D1};
50
51if any(imag(MAP{1})>10^-4, 'all') || any(imag(MAP{2})>10^-4, 'all')
52 mu00 = -1; % just to be sure the check below fails
53end
54
55% note: if the third moment is infeasible, ignore the third moment
56if ~issym(mu00) && (mu00<0 || mu11<0 || mu01<0 || mu10<0 || (map_isfeasible(MAP) == 0))
57 % if any element in D1 is negative, it indicates that this is an
58 % infeasible map, then we will find a map that matches the first and
59 % second moment, still fit it with MMPP(2)
60 SCV=(E2-E1^2)/E1^2;
61 if SCV>=1
62 % if SCV >= 1 only use the first and second moments to match a
63 % HMAP in the form of D1 = [(1-p)/mu1,p/mu2;p/mu1,(1-p)/mu2]
64 % ignore the 3rd moment
65 % warning('map_block: SCV>=1\n')
66 mu1=E1-1/2*(-4*E1^2+2*E2)^(1/2);
67 mu2=E1+1/2*(-4*E1^2+2*E2)^(1/2);
68 p=1/2-1/2*G2;
69 mu1=real(mu1);
70 mu2=real(mu2);
71 D0=[-1/mu1,0;0,-1/mu2];
72 else
73 % if scv is less than 1, HMAP can't fit it, use a more general form
74 % map(2) to fit it, the map is in such a form: D1 = [(1-p)/mu1 +
75 % 1-p, p + (1-p)/mu2], D0 = [-1/mu1 -1,1;0,-1/mu2];
76 % warning('map_block: SCV<1\n')
77 % mu1=-1/4*(-4*E1+8*E1^2-3*E2-(-16*E1^2-32*E1^3+16*E1*E2+E2^2+8*E2)^(1/2))/(-E1+2*E1^2-E2+1);
78 % mu2=(1/4*(-4*E1+8*E1^2-3*E2-(-16*E1^2-32*E1^3+16*E1*E2+E2^2+8*E2)^(1/2))/(-E1+2*E1^2-E2+1)+2*E1-1/2*(-4*E1+8*E1^2-3*E2-(-16*E1^2-32*E1^3+16*E1*E2+E2^2+8*E2)^(1/2))/(-E1+2*E1^2-E2+1)*E1)/(1-1/2*(-4*E1+8*E1^2-3*E2-(-16*E1^2-32*E1^3+16*E1*E2+E2^2+8*E2)^(1/2))/(-E1+2*E1^2-E2+1));
79 % mu1=real(mu1);
80 % mu2=real(mu2);
81 % p=1/2-1/2*G2;
82 % D0=[-1/mu1-1,1;0,-1/mu2];
83 MAP={[-1/E1],[1/E1]};
84 return
85 end
86 P=[1-p,p;p,1-p];
87 MAP={D0,-D0*P};
88end
89% E1=map_moment(MAP,1);
90% SCV = (map_moment(MAP,2) - map_moment(MAP,1)^2)/(map_moment(MAP,1)^2);
91% E3=map_moment(MAP,3);
92% G2=map_gamma2(MAP);
93% eddy_question: we don't need to return G2, why do we need to compute
94% map_gamma2?
95
96 function G2=map_gamma2(MAP)
97 % function G2=map_gamma2(MAP)
98 %
99 % Input:
100 % MAP = a map in the form of {D0,D1}
101 %
102 % Output:
103 % G2 = rou(i) / rou(i-1), where rou(i) represents the i-th
104 % autocorrelations of interarrival times of the given map
105 %
106 % Comments:
107 % In a map, the ratio of two consecutive autocorrelations is always
108 % the same
109
110 if issym(MAP{1}(1,1))
111 % if map is symbolic
112 P=inv(-MAP{1})*MAP{2};
113 g=simplify(eig(P));
114 if g(1)==1
115 G2=g(2);
116 else
117 G2=g(1);
118 end
119 else
120 G2=at(sort(eig(inv(-MAP{1})*MAP{2}),'descend'),2);
121 end
122 end
123
124 function b=issym(x)
125 b=strcmpi(class(x),'sym');
126 end
127
128 function [a]=at(A,r,i)
129 a=A(r);
130 if nargin==3
131 a=A(r,i);
132 end
133 end
134
135
136
137end
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