Ministerul Educaiei al Republicii MoldovaUniversitatea Tehnic a Moldovei

Lucrare de laborator Nr1.La Procese stochastice

Tema: Lanul Markov timp discret

A efectuat: st. grupei TI-093 Overcenco Marin

A verificat: Palii Diana

Chiinu 2010Scopul lucrarii: Studierea metodelor de redare, descriere, analiza aproprietatilor de comportare ale lanturilor Markov timp discret si evaluarea caracteristicelor numerice de performanta.Consideraii Teoretice: Lanul aleator de timp Markov este un ir de variabile aleatoare, care satisface condiia lui Markov i anume: probabillitatea sistemului c sistemul discret la momentul (k+1) (deseori numit i epoc sau perioad), s se gseasc n starea discret (ik+1), condiia de faptul c sistemul s-a gsit, respectiv, la momentele 1,2,....,k-1,k n strile i1,i2,....,ik, nu depinde de ultima stare,adic

Probabilitatea c sistemul va fi n starea i la momentul k , o vom nota:

Probabilitatea c sistemul va trece n starea j la momentul (k+1), tiind c n momentul precedent k el se afla n starea i, adic probabilitatea condiionat

poart numele de probabilitate de trecere. Un lan Markov este complet determinat dac cunoatem: mulimea strilor discrete S={si, i=1,n}, vectorul-linie al probabilitilor de stare iniial (0) i matricea stochastic a probabilitilor de trecere:

Relaia prin care determinm probabilitile de stare la momentul (k+1) cu ajutorul proba-bilitilor de trecere i a vectorului de stare corespunztor momentului k, este descris de ecuaia Kolmogorov(9):

Dac la fiecare stare j se va ataa o funcie const cj(k) de aflare a lanului DLM n aceast stare, atunci costul mediu c(k) de funcionare a lanului este:

n continuare va fi prezentat lanul Markov, care este un graf cu 12 noduri. Pentru Lanul Marcov dat trebuie de calculate 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 stri , astfel nct .

n acest caz ;

; C=Csb+Csr; Unde k este numrtorul perioadelor , probabilitatea c sistemul la momentul de timp k se afl ntr-o submulime de stri.

Lant Markov ergodic

121

11

2

10

3

9

67854

Lant Markov neergodic

121

11

2

10

3

9

67854

Tabelul valorilor grafului ergotic:

Xx1x2x3x4x5x6x7x8x9x10x11x12

x10.20.2000.200.200000.2

x20.300000.6000.1000

x30000.4000.30000.30

x4000.100.6000000.30

x50000.40.10.100.40000

x600000.200.100.7000

x70000.400.300.30000

x80000000.100.100.80

x90000.70000.10.10.100

x100.400.3000000.20.100

x1100.10.400.20000.3000

x1200.10.40.3000000.200

Tabelul valorilor gragului neergotic:

Xx1x2x3x4x5x6x7x8x9x10x11x12

x10.20.2000.200.200000.2

x2000100000000

x3000100000000

x4000.700.30000000

x50000.50.50000000

x600000.200.100.7000

x70000.400.300.30000

x80000000.100.100.80

x90000.70000.10.10.100

x100.400.3000000.20.100

x1100.10.400.20000.3000

x1200.10.40.3000000.200

Tabela fara mutari de adrese[ergotic]

Tabela fara mutari de adrese[ergotic](cu conditile initiale schimbate)

Tabela fara mutari de adrese[neergotic]:

Tabel cu mutati de adrese [neergolic]:

Concluzie: In urma efectuarii lucrarii de laborator Nr.1 am studiat metode de redare,descriere,analiza a proprietatilor de comportare ale lanturilor markov timp discret si evaluare a caracteristicilor numerice de performanta

1Si/k012345678910CostulS100.080.040.020.020.020.010.010.010.010.0110S200.040.020.020.020.020.020.020.020.020.0225S30.10.240.050.090.10.080.10.090.090.090.0970S40.20.080.30.160.210.210.20.210.20.210.21-15S500.20.110.230.170.190.190.180.190.190.19100S600.030.050.050.050.040.040.040.040.040.04150S70.10.030.090.040.040.050.040.050.040.050.05-10S800.030.10.080.110.090.10.10.10.10.110S900.160.070.10.120.10.110.10.110.110.11120S100.20.020.020.010.010.010.010.010.010.010.01-20S110.40.090.120.190.140.190.160.170.170.170.1780S12000.020.010000000-5Psr0.30.460.480.50.50.50.510.50.50.510.51Psb0.70.440.510.50.490.50.470.480.480.490.49Csr436.61029.424.525.126.525.825.825.825.8Csb2732.325.6353333.620.832.33232.132.1CS3169.935.664.457.558.747.357.257.857.957.91Si/k01234567891011CostulS100.030.030.020.020.010.010.010.010.010.010.0110S20.10.020.040.020.010.030.020.020.020.020.020.0225S300.080.160.080.080.110.080.10.090.090.090.0970S400.180.090.310.170.210.210.20.210.20.210.21-15S50.20.020.20.120.230.180.180.190.180.190.190.19100S600.110.030.060.040.050.040.040.040.040.040.04150S70.10.040.050.060.050.040.050.040.050.040.040.04-10S80.40.110.020.120.070.120.10.10.10.10.10.110S900.050.20.10.10.110.10.110.110.110.110.11120S1000.040.010.20.010.010.010.010.010.010.010.01-20S1100.320.170.090.210.130.190.160.170.170.170.1780S120.200.010.0100000000-5Psr(cs)0.30.30.490.530.490.530.490.510.50.50.510.51Psb(cs)0.70.690.520.660.50.470.50.370.50.480.480.48Csr(cs)2310.833.5513.526.223.322.923.521.722.822.622.6Csb(cs)242.942.01253731.833.632.532.333.333.333.3Cs(cs)2553.775.638.563.255.156.5565456.155.955.9

1Si/k0123456789CostulS100.080.020.0100000010S200.040.020.0100000025S30.10.360.220.380.240.370.25350.260.2670S40.20.140.470.320.50.350.5370.490.49-15S500.140.130.220.210.260.24270.250.25100S600.0300.01000000150S70.100.040.010.0100000-10S800.030.020.0200000010S900.160.040.010.020.010000120S100.20.0200000000-20S110.400.020.010.010000080S12000.020000000-5Psr0.30.680.840.930.50.980.990.9911Psb0.70.320.160.70.040.10000Csr438.127.4546.330.344.43445.9535.8535.9Csb27.824.46.13.73.11.20000Cs31.862.533.65033.445.63445.9535.935.9

1Si/k01234567891011CostulS10.10.020.0100000000010S200.030.040.010000000025S300.040.320.240.370.260.350.270.340.270.330.3370S400.210.160.470.340.490.370.480.390.480.40.4-15S50.20.120.20.170.230.220.260.240.270.250.270.27100S600.060.020.010.010000000150S70.20.060.020.020.010000000-10S80.40.060.020.020.0100.10000010S900.040.150.050.020.01000000120S1000.020.01000000000-20S1100.320.050.020.020.0100000080S120.10.020000000000-5Psr(cs)0.20.40.720.890.940.970.970.991111Psb(cs)0.80.60.280.120.070.020.0100000Csr(cs)20124129.5543.832.8544.9535.744.9528.644.144.1Csb(cs)2.539.124.99.15.52100000Cs22.551.165.938.6549.334.854635.744.9528.644.144.1