PStohastice
9
„Lanţul Markov timp discret” Scopul lucrării: Studierea metodelor de redare, descrierea, analiza a proprietăţilor de comportare ale lanţurilor „Markov timp discret” şi a caracteristicilor numerice de performanţă. Consideraţii teoretice: Lanţul aleatoriu de timp Markovian este un şir de variabile aleatoare care satisface conditia lui Markov şi anume: probabilitatea sistemului discret la moment (k+1) (deseori numită şi epocă sau perioadă), să se găsească în starea discretă (i k+1 ), condiţia de faptul ca sistema s-a găsit respectiv la momentele 1,2,…,k-1,k în stările i 1 , i 2 ,…, i k , nu depinde decât de ultima stare, adică: P r (X k+1 = i k+1 /X k =i k , X k-1 = i k-1 , …, X 1 =i 1 )= P r (X k+1 = i 1 /X k = i k ). Probabilitatea că sistemul să fie în starea i la momentul dat k, vom nota: i (k) = P r (x k =i) 0 i (k) 1, Probabilitatea ca sistemul să treacă în sistemul j la momentul (k+1), ştiind că in momentul precedent k el s- a aflat in starea i, adică probabilitatea conditionată P r (X k+1 = j/X k = i)= p ij (k+1) i, j=1,…n şi k=0,1,2… poarta numele de probabilitate de trecere.
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Lanul Markov timp discret
Lanul Markov timp discret
Scopul lucrrii:
Studierea metodelor de redare, descrierea, analiza a proprietilor de comportare ale lanurilor Markov timp discret i a caracteristicilor numerice de performan.
Consideraii teoretice:
Lanul aleatoriu de timp Markovian este un ir de variabile aleatoare care satisface conditia lui Markov i anume: probabilitatea sistemului discret la moment (k+1) (deseori numit i epoc sau perioad), s se gseasc n starea discret (ik+1), condiia de faptul ca sistema s-a gsit respectiv la momentele 1,2,,k-1,k n strile i1, i2,, ik, nu depinde dect de ultima stare, adic:
Pr(Xk+1= ik+1/Xk=ik, Xk-1= ik-1, , X1=i1)= Pr(Xk+1 = i1/Xk = ik).
Probabilitatea c sistemul s fie n starea i la momentul dat k, vom nota:
(i(k) = Pr(xk=i)
0((i(k) (1,
Probabilitatea ca sistemul s treac n sistemul j la momentul (k+1), tiind c in momentul precedent k el s-a aflat in starea i, adic probabilitatea conditionat
Pr(Xk+1 = j/Xk = i)= pij (k+1) i, j=1,n i k=0,1,2poarta numele de probabilitate de trecere.
Un lan Markov este complet determinat daca cunoatem: mulimea strilor discrete S={si,i=1,n}, vectorul linie al probabilitilor de stare iniial ((0) i matricea stohastic a probabilitilor de trecere:
(Pi,j) (i,j=1,,n), 0( Pi,j (1,
Relaia prin care determin probabilitile de stare la momentul (k+1) cu ajutorul probabilitilor de trecere i a vectorului de stare corespunztor momentului k, este descris de ecuaia Kolmogorov [9]:
(j(k+1)= , j=1,,n; k=0,1,2,
Dac la fiecare stare j se va ataa o funcie cost cj(k) de aflare a lanului DLM n aceasta stare, atunci costul mediu c(k) de funcionare a lanului este:
C(k)=
n figura de mai jos este prezentat lanul Markov discret ergotic.
Lan Neergodic:
Pentru Lanul Markov dat trebuie de calculat cu ajutorul pachetului de programe QM probabilitatea (Sb(k) i costul mediu CSb(k) de aflare a lanului DLM la momentul k ntr-o submulime de stari Sb(S, astfel nct Sb(Sr, Sb(Sr=(n acest caz:
(i(k)= , iar CSb(k)=
Unde k este numrtorul perioadelor, ( probabilitatea ca sistemul la momentul de timp k se afl n starea SB, iar C este mediu de a lanului dat ntr-o submulime de stri.
,
Prelucrarea datelor experimentale:
Tabelul de probabiliti pentru lanul Markovian ergodic:
012345678910111213C
10,300,130,080,140,120,140,140,150,150,150,150,150,15150
200,090,090,130,110,130,120,130,130,130,130,130,130,13160
30,30,120,230,210,210,20,190,180,180,170,170,170,170,17-155
400,270,170,180,160,150,140,030,130,130,130,130,130,13157
50,30,090,050,040,040,040,040,060,030,030,030,030,030,03-160
600,160,080,080,060,060,060,010,060,060,070,070,070,07-153
70,10,40,020,020,010,010,010,050,010,010,010,010,010,01168
8000,050,030,040,040,050,10,050,050,050,050,050,05170
900,120,040,090,080,10,10,060,10,110,110,110,110,11-171
1000,100,070,040,060,060,010,060,060,060,060,060,06173
11000,010,010,010,010,010,010,010,010,010,010,010,01180
1200,010,120,070,080,080,080,080,080,080,080,080,070,07179
Pi(SB)0,40,630,360,360,390,410,440,440,450,460,460,460,450,45
Pi(SR)0,60,730,630,650,590,590,560,420,540,530,540,540,540,54
C(SB)61,865,84629,73734,138,852,240,338,638,638,636,838,3
C(SR)-95-0,7-13-0,3-3,6-0,4-2,1-121,132,681,151,151,151,15
C(k)-3365,13329,433,433,736,740,541,441,339,739,737,939,4
L(Sb Sr)0,40,480,380,280,30,280,290,410,290,280,280,280,280,28
L(SR SB)00,210,190,230,220,230,230,230,240,250,250,250,250,25
EMBED Excel.Chart.8 \s
Tabelul de probabiliti pentru lanul Markovian ergodic(cu ntrri externe deplasate cu 3 stri):
012345678910111213C
100,020,040,070,10,120,130,140,140,140,150,150,150,15150
200,020,080,110,120,120,120,120,130,130,130,130,130,13160
30,30,210,230,260,250,230,210,20,190,180,180,170,170,17-155
400,240,260,210,190,170,160,150,140,140,130,130,130,13157
50,30,060,090,050,050,040,040,040,030,030,030,030,030,03-160
600,30,110,080,060,050,050,060,060,060,060,070,070,07-153
700,030,040,020,010,010,010,010,010,010,010,010,010,01168
80,300,020,020,030,040,040,040,050,050,050,050,050,05170
900,040,030,050,070,080,090,10,10,10,110,110,110,11-171
100,10,030,020,030,040,050,060,060,060,060,060,060,060,06173
110000,010,010,010,010,010,010,010,010,010,010,01180
1200,050,080,090,080,080,080,080,080,080,080,080,070,07179
Pi(SB)0,40,170,230,280,330,380,410,430,440,440,460,460,450,45
Pi(SR)0,60,830,770,720,680,620,590,580,560,550,540,540,540,54
C(SB)68,315,328,83031,135,837,337,138,838,838,638,636,838,3
C(SR)-95-47-13-8,2-5,1-2-0,5-21,152,71,131,151,151,15
C(k)-26-3215,521,82633,836,835,139,941,539,739,737,939,4
L(Sb Sr)0,580,310,390,330,30,30,290,280,290,290,280,280,280,28
L(SR SB)00,140,180,20,210,220,230,230,240,250,240,250,250,25
Tabelul de probabiliti pentru lanul Markovian neergodic:012345678910111213C
10,300,080,020,030,020,010,010,0100000150
200,090,10,140,120,120,120,120,120,120,120,120,120,12160
30,30,120,280,290,360,360,390,390,40,410,410,410,410,42-155
400,270,170,230,230,260,250,270,270,270,270,270,270,28157
500,090,050,050,050,050,050,050,050,050,050,050,050,05-160
60,30,160,080,070,050,040,040,030,030,030,030,020,020,02-153
700,0100,010000000000168
80,100,050,020,020,010,010,01000000170
900,120,040,040,030,020,010,010,0100000-171
1000,100,040,010,010,010,01000000173
1100,030,110,080,10,090,10,10,110,110,110,110,110,11180
1200,010,030,010,010,010,010000000179
Pi(SB)0,40,240,20,140,10,070,050,040,0200000
Pi(SR)0,60,760,790,860,910,920,950,960,980,990,990,980,981
C(SB)620,25199,956,294,85,013,22-0,200000
C(SR)-924,71-1,29,251,866,31,886,556,85,255,256,786,786,8
C(k)-304,9617,919,28,1511,16,899,776,595,255,256,786,786,8
L(SR SB)0,050,230,280,270,260,260,250,260,260,260,260,260,260,26
Tabelul de probabiliti pentru lanul Markovian neergodic(cu ntrri externe deplasate cu 3 stri):012345678910111213C
100,020,030,020,010,010,010000000150
200,020,080,130,130,120,120,120,120,120,120,120,120,12160
30,30,210,240,320,360,390,40,40,410,410,410,420,420,42-155
400,240,260,230,250,260,270,270,270,270,270,280,280,28157
50,30,060,090,050,050,050,050,050,050,050,050,050,050,05-160
600,30,110,070,050,040,030,030,030,030,020,020,020,02-153
700,030,0100000000000168
80,300,020,020,010,0100000000170
900,040,030,020,020,010,010,01000000-171
100,10,030,020,010,010,0100000000173
11000,10,110,10,10,10,110,110,110,110,110,110,11180
1200,050,010,010,01000000000179
Pi(SB)0,40,170,120,080,060,040,020,01000000
Pi(SR)0,60,830,880,910,940,960,970,980,990,990,98111
C(SB)68,315,39,76,53,33,22-0,2-1,7000000
C(SR)-95-473,198,46,63,4556,85,255,256,786,86,86,8
C(k)-26-3212,914,99,96,674,795,095,255,256,786,86,86,8
L(SR SB)00,140,260,280,260,250,250,260,260,260,260,260,260,26
Concluzie: n procesul elaborrii lucrrii de laborator am fcut cunotin cu studiul lanurilor Markov timp discret i evaluarea caracteristicilor de performan.
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_1127137883.xlsDiagram3
0.40.6
0.630.73
0.360.63
0.360.65
0.390.59
0.410.59
0.440.56
0.440.42
0.450.54
0.460.53
0.460.54
0.460.54
0.450.54
0.450.54
P(Sb)
P(Sr)
Foaie1
012345678910111213C
10.300.130.080.140.120.140.140.150.150.150.150.150.15150
200.090.090.130.110.130.120.130.130.130.130.130.130.13160
30.30.120.230.210.210.20.190.180.180.170.170.170.170.17-155
400.270.170.180.160.150.140.030.130.130.130.130.130.13157
50.30.090.050.040.040.040.040.060.030.030.030.030.030.03-160
600.160.080.080.060.060.060.010.060.060.070.070.070.07-153
70.10.40.020.020.010.010.010.050.010.010.010.010.010.01168
8000.050.030.040.040.050.10.050.050.050.050.050.05170
900.120.040.090.080.10.10.060.10.110.110.110.110.11-171
1000.100.070.040.060.060.010.060.060.060.060.060.06173
11000.010.010.010.010.010.010.010.010.010.010.010.01180
1200.010.120.070.080.080.080.080.080.080.080.080.070.07179
Pi(SB)0.40.630.360.360.390.410.440.440.450.460.460.460.450.45
Pi(SR)0.60.730.630.650.590.590.560.420.540.530.540.540.540.54
C(SB)61.865.774629.7137.0434.0838.7852.1940.2838.5738.5738.5736.7838.28
C(SR)-94.5-0.69-13-0.33-3.61-0.43-2.05-11.721.132.681.151.151.151.15
C(k)-32.765.083329.3833.4333.6536.7340.4741.4141.2539.7239.7237.9339.43
L(Sb Sr)0.40.48253968250.37777777780.28055555560.30.28048780490.29090909090.41136363640.29111111110.28478260870.28478260870.28478260870.28222222220.2822222222
L(SR SB)00.20684931510.19365079370.22923076920.21864406780.23389830510.23035714290.2309523810.24444444440.24905660380.24629629630.24629629630.24629629630.2462962963
Foaie1
00
00
00
00
00
00
00
00
00
00
00
00
00
00
P(Sb)
P(Sr)
Foaie2
000
000
000
000
000
000
000
000
000
000
000
000
000
000
C(Sb)
C(Sr)
C(k)
Foaie3
_1127138669.xlsDiagram6
0.5750
0.31176470590.1373493976
0.39130434780.1779220779
0.3250.2027777778
0.2969696970.2117647059
0.29736842110.2209677419
0.2878048780.2271186441
0.28139534880.2275862069
0.29090909090.2410714286
0.29090909090.2454545455
0.28478260870.2444444444
0.28478260870.2462962963
0.28222222220.2462962963
0.28222222220.2462962963
L(Sb-Sr)
L(Sr-Sb)
Foaie1
012345678910111213C
100.020.040.070.10.120.130.140.140.140.150.150.150.15150
200.020.080.110.120.120.120.120.130.130.130.130.130.13160
30.30.210.230.260.250.230.210.20.190.180.180.170.170.17-155
400.240.260.210.190.170.160.150.140.140.130.130.130.13157
50.30.060.090.050.050.040.040.040.030.030.030.030.030.03-160
600.30.110.080.060.050.050.060.060.060.060.070.070.07-153
700.030.040.020.010.010.010.010.010.010.010.010.010.01168
80.300.020.020.030.040.040.040.050.050.050.050.050.05170
900.040.030.050.070.080.090.10.10.10.110.110.110.11-171
100.10.030.020.030.040.050.060.060.060.060.060.060.060.06173
110000.010.010.010.010.010.010.010.010.010.010.01180
1200.050.080.090.080.080.080.080.080.080.080.080.070.07179
Pi(SB)0.40.170.230.280.330.380.410.430.440.440.460.460.450.45
Pi(SR)0.60.830.770.720.680.620.590.580.560.550.540.540.540.54
C(SB)68.315.3428.7730.0131.0535.7737.2937.0838.7838.7838.5738.5736.7838.28
C(SR)-94.5-47.17-13.26-8.17-5.1-2.01-0.48-2.031.152.71.131.151.151.15
C(k)-26.2-31.8315.5121.8425.9533.7636.8135.0539.9341.4839.739.7237.9339.43
L(Sb Sr)0.5750.31176470590.39130434780.3250.2969696970.29736842110.2878048780.28139534880.29090909090.29090909090.28478260870.28478260870.28222222220.2822222222
L(SR SB)00.13734939760.17792207790.20277777780.21176470590.22096774190.22711864410.22758620690.24107142860.24545454550.24444444440.24629629630.24629629630.2462962963
Foaie1
00
00
00
00
00
00
00
00
00
00
00
00
00
00
Psb
Psr
Foaie2
000
000
000
000
000
000
000
000
000
000
000
000
000
000
C(Sb)
C(Sr)
C(k)
Foaie3
00
00
00
00
00
00
00
00
00
00
00
00
00
00
L(Sb-Sr)
L(Sr-Sb)
_1127139082.xlsDiagram2
62-92.4-30.4
0.254.714.96
19.03-1.1517.88
9.959.2519.2
6.291.868.15
4.86.311.1
5.011.886.89
3.226.559.77
-0.216.86.59
05.255.25
05.255.25
06.786.78
06.786.78
06.86.8
C(Sb)
C(Sr)
C(k)
Foaie1
012345678910111213C
10.300.080.020.030.020.010.010.0100000150
200.090.10.140.120.120.120.120.120.120.120.120.120.12160
30.30.120.280.290.360.360.390.390.40.410.410.410.410.42-155
400.270.170.230.230.260.250.270.270.270.270.270.270.28157
500.090.050.050.050.050.050.050.050.050.050.050.050.05-160
60.30.160.080.070.050.040.040.030.030.030.030.020.020.02-153
700.0100.010000000000168
80.100.050.020.020.010.010.01000000170
900.120.040.040.030.020.010.010.0100000-171
1000.100.040.010.010.010.01000000173
1100.030.110.080.10.090.10.10.110.110.110.110.110.11180
1200.010.030.010.010.010.010000000179
Pi(SB)0.40.240.20.140.10.070.050.040.0200000
Pi(SR)0.60.760.790.860.910.920.950.960.980.990.990.980.981
C(SB)620.2519.039.956.294.85.013.22-0.2100000
C(SR)-92.44.71-1.159.251.866.31.886.556.85.255.256.786.786.8
C(k)-30.44.9617.8819.28.1511.16.899.776.595.255.256.786.786.8
L(Sb Sr)0.40.12916666670.3550.27857142860.290.27142857140.320.30.1500000
L(SR SB)0.050.23421052630.27594936710.26976744190.25934065930.25543478260.25368421050.256250.26020408160.25757575760.25757575760.25918367350.25918367350.257
Foaie1
00
00
00
00
00
00
00
00
00
00
00
00
00
00
P(Sb)
P(Sr)
Foaie2
000
000
000
000
000
000
000
000
000
000
000
000
000
000
C(Sb)
C(Sr)
C(k)
Foaie3
_1127139611.xlsDiagram3
0.40.6
0.170.83
0.120.88
0.080.91
0.060.94
0.040.96
0.020.97
0.010.98
00.99
00.99
00.98
01
01
01
Pi(Sb)
Pi(Sr)
Foaie1
012345678910111213C
10.300.080.020.030.020.010.010.0100000150
200.090.10.140.120.120.120.120.120.120.120.120.120.12160
30.30.120.280.290.360.360.390.390.40.410.410.410.410.42-155
400.270.170.230.230.260.250.270.270.270.270.270.270.28157
500.090.050.050.050.050.050.050.050.050.050.050.050.05-160
60.30.160.080.070.050.040.040.030.030.030.030.020.020.02-153
700.0100.010000000000168
80.100.050.020.020.010.010.01000000170
900.120.040.040.030.020.010.010.0100000-171
1000.100.040.010.010.010.01000000173
1100.030.110.080.10.090.10.10.110.110.110.110.110.11180
1200.010.030.010.010.010.010000000179
Pi(SB)0.40.240.20.140.10.070.050.040.0200000
Pi(SR)0.60.760.790.860.910.920.950.960.980.990.990.980.981
C(SB)620.2519.039.956.294.85.013.22-0.2100000
C(SR)-92.44.71-1.159.251.866.31.886.556.85.255.256.786.786.8
C(k)-30.44.9617.8819.28.1511.16.899.776.595.255.256.786.786.8
L(Sb Sr)0.40.12916666670.3550.27857142860.290.27142857140.320.30.1500000
L(SR SB)0.050.23421052630.27594936710.26976744190.25934065930.25543478260.25368421050.256250.26020408160.25757575760.25757575760.25918367350.25918367350.257
Foaie1
00
00
00
00
00
00
00
00
00
00
00
00
00
00
P(Sb)
P(Sr)
Foaie2
000
000
000
000
000
000
000
000
000
000
000
000
000
000
C(Sb)
C(Sr)
C(k)
Foaie3
_1329143336.unknown
_1127139605.xlsDiagram1
68.3-94.5-26.2
15.34-47.17-31.83
9.73.1912.89
6.58.414.9
3.36.69.9
3.223.456.67
-0.2154.79
-1.716.85.09
05.255.25
05.255.25
06.786.78
06.86.8
06.86.8
06.86.8
C(Sb)
C(Sr)
C(k)
Foaie1
012345678910111213C
100.020.030.020.010.010.010000000150
200.020.080.130.130.120.120.120.120.120.120.120.120.12160
30.30.210.240.320.360.390.40.40.410.410.410.420.420.42-155
400.240.260.230.250.260.270.270.270.270.270.280.280.28157
50.30.060.090.050.050.050.050.050.050.050.050.050.050.05-160
600.30.110.070.050.040.030.030.030.030.020.020.020.02-153
700.030.0100000000000168
80.300.020.020.010.0100000000170
900.040.030.020.020.010.010.01000000-171
100.10.030.020.010.010.0100000000173
11000.10.110.10.10.10.110.110.110.110.110.110.11180
1200.050.010.010.01000000000179
Pi(SB)0.40.170.120.080.060.040.020.01000000
Pi(SR)0.60.830.880.910.940.960.970.980.990.990.98111
C(SB)68.315.349.76.53.33.22-0.21-1.71000000
C(SR)-94.5-47.173.198.46.63.4556.85.255.256.786.86.86.8
C(k)-26.2-31.8312.8914.99.96.674.795.095.255.256.786.86.86.8
L(Sb Sr)0.5750.31176470590.31666666670.3250.26666666670.30.150000000
L(SR SB)00.13734939760.25795454550.2780219780.26382978720.25416666670.25360824740.26020408160.25757575760.25757575760.25918367350.2570.2570.257
Foaie1
00
00
00
00
00
00
00
00
00
00
00
00
00
00
Pi(Sb)
Pi(Sr)
Foaie2
000
000
000
000
000
000
000
000
000
000
000
000
000
000
C(Sb)
C(Sr)
C(k)
Foaie3
_1127139058.xlsDiagram1
0.40.6
0.240.76
0.20.79
0.140.86
0.10.91
0.070.92
0.050.95
0.040.96
0.020.98
00.99
00.99
00.98
00.98
01
P(Sb)
P(Sr)
Foaie1
012345678910111213C
10.300.080.020.030.020.010.010.0100000150
200.090.10.140.120.120.120.120.120.120.120.120.120.12160
30.30.120.280.290.360.360.390.390.40.410.410.410.410.42-155
400.270.170.230.230.260.250.270.270.270.270.270.270.28157
500.090.050.050.050.050.050.050.050.050.050.050.050.05-160
60.30.160.080.070.050.040.040.030.030.030.030.020.020.02-153
700.0100.010000000000168
80.100.050.020.020.010.010.01000000170
900.120.040.040.030.020.010.010.0100000-171
1000.100.040.010.010.010.01000000173
1100.030.110.080.10.090.10.10.110.110.110.110.110.11180
1200.010.030.010.010.010.010000000179
Pi(SB)0.40.240.20.140.10.070.050.040.0200000
Pi(SR)0.60.760.790.860.910.920.950.960.980.990.990.980.981
C(SB)620.2519.039.956.294.85.013.22-0.2100000
C(SR)-92.44.71-1.159.251.866.31.886.556.85.255.256.786.786.8
C(k)-30.44.9617.8819.28.1511.16.899.776.595.255.256.786.786.8
L(Sb Sr)0.40.12916666670.3550.27857142860.290.27142857140.320.30.1500000
L(SR SB)0.050.23421052630.27594936710.26976744190.25934065930.25543478260.25368421050.256250.26020408160.25757575760.25757575760.25918367350.25918367350.257
Foaie1
00
00
00
00
00
00
00
00
00
00
00
00
00
00
P(Sb)
P(Sr)
Foaie2
000
000
000
000
000
000
000
000
000
000
000
000
000
000
C(Sb)
C(Sr)
C(k)
Foaie3
_1127138667.xlsDiagram4
0.40.6
0.170.83
0.230.77
0.280.72
0.330.68
0.380.62
0.410.59
0.430.58
0.440.56
0.440.55
0.460.54
0.460.54
0.450.54
0.450.54
Psb
Psr
Foaie1
012345678910111213C
100.020.040.070.10.120.130.140.140.140.150.150.150.15150
200.020.080.110.120.120.120.120.130.130.130.130.130.13160
30.30.210.230.260.250.230.210.20.190.180.180.170.170.17-155
400.240.260.210.190.170.160.150.140.140.130.130.130.13157
50.30.060.090.050.050.040.040.040.030.030.030.030.030.03-160
600.30.110.080.060.050.050.060.060.060.060.070.070.07-153
700.030.040.020.010.010.010.010.010.010.010.010.010.01168
80.300.020.020.030.040.040.040.050.050.050.050.050.05170
900.040.030.050.070.080.090.10.10.10.110.110.110.11-171
100.10.030.020.030.040.050.060.060.060.060.060.060.060.06173
110000.010.010.010.010.010.010.010.010.010.010.01180
1200.050.080.090.080.080.080.080.080.080.080.080.070.07179
Pi(SB)0.40.170.230.280.330.380.410.430.440.440.460.460.450.45
Pi(SR)0.60.830.770.720.680.620.590.580.560.550.540.540.540.54
C(SB)68.315.3428.7730.0131.0535.7737.2937.0838.7838.7838.5738.5736.7838.28
C(SR)-94.5-47.17-13.26-8.17-5.1-2.01-0.48-2.031.152.71.131.151.151.15
C(k)-26.2-31.8315.5121.8425.9533.7636.8135.0539.9341.4839.739.7237.9339.43
L(Sb Sr)0.5750.31176470590.39130434780.3250.2969696970.29736842110.2878048780.28139534880.29090909090.29090909090.28478260870.28478260870.28222222220.2822222222
L(SR SB)00.13734939760.17792207790.20277777780.21176470590.22096774190.22711864410.22758620690.24107142860.24545454550.24444444440.24629629630.24629629630.2462962963
Foaie1
00
00
00
00
00
00
00
00
00
00
00
00
00
00
Psb
Psr
Foaie2
000
000
000
000
000
000
000
000
000
000
000
000
000
000
C(Sb)
C(Sr)
C(k)
Foaie3
00
00
00
00
00
00
00
00
00
00
00
00
00
00
L(Sb-Sr)
L(Sr-Sb)
_1127138668.xlsDiagram5
68.3-94.5-26.2
15.34-47.17-31.83
28.77-13.2615.51
30.01-8.1721.84
31.05-5.125.95
35.77-2.0133.76
37.29-0.4836.81
37.08-2.0335.05
38.781.1539.93
38.782.741.48
38.571.1339.7
38.571.1539.72
36.781.1537.93
38.281.1539.43
C(Sb)
C(Sr)
C(k)
Foaie1
012345678910111213C
100.020.040.070.10.120.130.140.140.140.150.150.150.15150
200.020.080.110.120.120.120.120.130.130.130.130.130.13160
30.30.210.230.260.250.230.210.20.190.180.180.170.170.17-155
400.240.260.210.190.170.160.150.140.140.130.130.130.13157
50.30.060.090.050.050.040.040.040.030.030.030.030.030.03-160
600.30.110.080.060.050.050.060.060.060.060.070.070.07-153
700.030.040.020.010.010.010.010.010.010.010.010.010.01168
80.300.020.020.030.040.040.040.050.050.050.050.050.05170
900.040.030.050.070.080.090.10.10.10.110.110.110.11-171
100.10.030.020.030.040.050.060.060.060.060.060.060.060.06173
110000.010.010.010.010.010.010.010.010.010.010.01180
1200.050.080.090.080.080.080.080.080.080.080.080.070.07179
Pi(SB)0.40.170.230.280.330.380.410.430.440.440.460.460.450.45
Pi(SR)0.60.830.770.720.680.620.590.580.560.550.540.540.540.54
C(SB)68.315.3428.7730.0131.0535.7737.2937.0838.7838.7838.5738.5736.7838.28
C(SR)-94.5-47.17-13.26-8.17-5.1-2.01-0.48-2.031.152.71.131.151.151.15
C(k)-26.2-31.8315.5121.8425.9533.7636.8135.0539.9341.4839.739.7237.9339.43
L(Sb Sr)0.5750.31176470590.39130434780.3250.2969696970.29736842110.2878048780.28139534880.29090909090.29090909090.28478260870.28478260870.28222222220.2822222222
L(SR SB)00.13734939760.17792207790.20277777780.21176470590.22096774190.22711864410.22758620690.24107142860.24545454550.24444444440.24629629630.24629629630.2462962963
Foaie1
00
00
00
00
00
00
00
00
00
00
00
00
00
00
Psb
Psr
Foaie2
000
000
000
000
000
000
000
000
000
000
000
000
000
000
C(Sb)
C(Sr)
C(k)
Foaie3
00
00
00
00
00
00
00
00
00
00
00
00
00
00
L(Sb-Sr)
L(Sr-Sb)
_1127137891.xlsDiagram2
61.8-94.5-32.7
65.77-0.6965.08
46-1333
29.71-0.3329.38
37.04-3.6133.43
34.08-0.4333.65
38.78-2.0536.73
52.19-11.7240.47
40.281.1341.41
38.572.6841.25
38.571.1539.72
38.571.1539.72
36.781.1537.93
38.281.1539.43
C(Sb)
C(Sr)
C(k)
Foaie1
012345678910111213C
10.300.130.080.140.120.140.140.150.150.150.150.150.15150
200.090.090.130.110.130.120.130.130.130.130.130.130.13160
30.30.120.230.210.210.20.190.180.180.170.170.170.170.17-155
400.270.170.180.160.150.140.030.130.130.130.130.130.13157
50.30.090.050.040.040.040.040.060.030.030.030.030.030.03-160
600.160.080.080.060.060.060.010.060.060.070.070.070.07-153
70.10.40.020.020.010.010.010.050.010.010.010.010.010.01168
8000.050.030.040.040.050.10.050.050.050.050.050.05170
900.120.040.090.080.10.10.060.10.110.110.110.110.11-171
1000.100.070.040.060.060.010.060.060.060.060.060.06173
11000.010.010.010.010.010.010.010.010.010.010.010.01180
1200.010.120.070.080.080.080.080.080.080.080.080.070.07179
Pi(SB)0.40.630.360.360.390.410.440.440.450.460.460.460.450.45
Pi(SR)0.60.730.630.650.590.590.560.420.540.530.540.540.540.54
C(SB)61.865.774629.7137.0434.0838.7852.1940.2838.5738.5738.5736.7838.28
C(SR)-94.5-0.69-13-0.33-3.61-0.43-2.05-11.721.132.681.151.151.151.15
C(k)-32.765.083329.3833.4333.6536.7340.4741.4141.2539.7239.7237.9339.43
L(Sb Sr)0.40.48253968250.37777777780.28055555560.30.28048780490.29090909090.41136363640.29111111110.28478260870.28478260870.28478260870.28222222220.2822222222
L(SR SB)00.20684931510.19365079370.22923076920.21864406780.23389830510.23035714290.2309523810.24444444440.24905660380.24629629630.24629629630.24629629630.2462962963
Foaie1
00
00
00
00
00
00
00
00
00
00
00
00
00
00
P(Sb)
P(Sr)
Foaie2
000
000
000
000
000
000
000
000
000
000
000
000
000
000
C(Sb)
C(Sr)
C(k)
Foaie3
_1126608119.unknown
_1126608518.unknown
_1126608762.unknown
_1127137866.xlsDiagram1
0.40
0.48253968250.2068493151
0.37777777780.1936507937
0.28055555560.2292307692
0.30.2186440678
0.28048780490.2338983051
0.29090909090.2303571429
0.41136363640.230952381
0.29111111110.2444444444
0.28478260870.2490566038
0.28478260870.2462962963
0.28478260870.2462962963
0.28222222220.2462962963
0.28222222220.2462962963
L(Sb-Sr)
L(Sr-Sb)
Foaie1
012345678910111213C
10.300.130.080.140.120.140.140.150.150.150.150.150.15150
200.090.090.130.110.130.120.130.130.130.130.130.130.13160
30.30.120.230.210.210.20.190.180.180.170.170.170.170.17-155
400.270.170.180.160.150.140.030.130.130.130.130.130.13157
50.30.090.050.040.040.040.040.060.030.030.030.030.030.03-160
600.160.080.080.060.060.060.010.060.060.070.070.070.07-153
70.10.40.020.020.010.010.010.050.010.010.010.010.010.01168
8000.050.030.040.040.050.10.050.050.050.050.050.05170
900.120.040.090.080.10.10.060.10.110.110.110.110.11-171
1000.100.070.040.060.060.010.060.060.060.060.060.06173
11000.010.010.010.010.010.010.010.010.010.010.010.01180
1200.010.120.070.080.080.080.080.080.080.080.080.070.07179
Pi(SB)0.40.630.360.360.390.410.440.440.450.460.460.460.450.45
Pi(SR)0.60.730.630.650.590.590.560.420.540.530.540.540.540.54
C(SB)61.865.774629.7137.0434.0838.7852.1940.2838.5738.5738.5736.7838.28
C(SR)-94.5-0.69-13-0.33-3.61-0.43-2.05-11.721.132.681.151.151.151.15
C(k)-32.765.083329.3833.4333.6536.7340.4741.4141.2539.7239.7237.9339.43
L(Sb Sr)0.40.48253968250.37777777780.28055555560.30.28048780490.29090909090.41136363640.29111111110.28478260870.28478260870.28478260870.28222222220.2822222222
L(SR SB)00.20684931510.19365079370.22923076920.21864406780.23389830510.23035714290.2309523810.24444444440.24905660380.24629629630.24629629630.24629629630.2462962963
Foaie1
00
00
00
00
00
00
00
00
00
00
00
00
00
00
P(Sb)
P(Sr)
Foaie2
000
000
000
000
000
000
000
000
000
000
000
000
000
000
C(Sb)
C(Sr)
C(k)
Foaie3
00
00
00
00
00
00
00
00
00
00
00
00
00
00
L(Sb-Sr)
L(Sr-Sb)
_1126608204.unknown
_1126449512.unknown
_1126449939.unknown
_1126448963.unknown
