Modelos y Simulación

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1 1 Modelos y Simulación Clase 5: Unidad 2: Simulación de eventos aleatorios. Facultad de Ingeniería - Universidad de Buenos Aires Modelos y Simulación / Clase 5

2 2 Temas de hoy Cadenas de Markov - Material de lectura (esta clase): Libros (Mínimo): [15] Probability, Statistics, and Random Processes for Eletrical Engineerging, 3rd_Ed. - Leon-Garcia., Cap 11. [5] Intuitive Probability and Random Processes Using MatLab - Steven M. Kay, Cap 22. Libros (opcional): [16] Performance Modeling and Design of Computer Systems: Queueing Theory in Action, Mor Harchol- Balter,2013, Cap 8-10 [19] Introduction to probability models, Ross Sheldon, 2007, Cap 4. Papers (opcional): The Anatomy of a Large-Scale Hypertextual Web Search Engine, Sergey Brin and Lawrence Page, Computer Networks and IDSN Systems, 30 (1998), Facultad de Ingeniería - Universidad de Buenos Aires Modelos y Simulación / Clase 5

3 Cadenas de Markov

4 Introducción 4 En la práctica los procesos aleatorios que necesitamos modelar no son independientes. Una forma de incluir dependencia entre variables aleatorias es considerar un modelo que limite esta dependencia a los vecinos más cercanos. Proceso de Markov: Eventos pasados X(tk) Evento presente t1 tk tk+1 t

5 Algunos ejemplos 5 (incrementos estacionarios)

6 6 Cadenas de Markov Una cadena de Markov es un proceso de Markov que toma valores X(tk) naturales. Propiedad de Markov: la probabilidad conjunta de un conjunto de variables aleatorias en distintos instantes de tiempo se obtiene como el producto de la probabilidad inicial y las probabilidades de las transiciones subsiguientes:

7 Cadenas de Markov en tiempo discreto 7 Propiedad de Markov: Asumimos probabilidades de transiciones homogéneas (estacionariedad): Matriz de probabilidades de transición: La distribución conjunta de una cadena de Markov queda totalmente definida a partir de la matriz de transiciones y la condición inicial

8 Representación con grafos 8 Nodos: estados Links: probabilidades de transición Ejemplos: Dos estados Tres estados Bernoulli counting process (infinitos estados - contables)

9 Algunos ejemplos 9 Bulbs in reserve Matriz de transiciones Grafo Matriz de transiciones Grafo

10 Probabilidades de transición en n pasos 10 En general, para 2 pasos: En general, para n pasos: Propagación de probabilidades de estados:

11 Análisis de ejemplo 11 Matriz de transiciones Bulbs in reserve La probabilidad de que se acaben las lamparas de repuesto tiende a uno

12 Comportamiento asintótico: ejemplo 12 Dada la siguiente cadena de Markov, encontrar la matriz de transiciones asintótica? Buscamos autovalores y autovectores de la matriz de transición P Si 0 + < 1 <latexit sha1_base64="6jiapbrgn22hwf3kpxqnxshgybm=">aaab9xicbvbns8naej34wetx1aoxxsiiqkle0iohohepfewhnlfmtpt26wytdjdkkf0fxjwo4tx/4s1/47bnqvsfddzem2fmxpgkro3rfjtlyyura+ufjelm1vbobmlvv6gttffwp4livctezqsxrg64eayvkozxkfgzhnxm/oyju5on8t4muxbe2jm84hsnlr58fgkft/2qgbzyoqwyw3gniivey0kzctq6ps+/m9asztjqgvq3ptc1wqiv4vswcdhpneurdrdh2pzkjjkortorx+tykl0sjcqwngsq/p4yyaz1ma5tz4ymr+e9ifif185mdbmmuewzwysdlyoyquxcjhgqllemgjg0bkni9lzc+6iqghtu0ybgzb+8sbpnfc+tehfn5ep1hkcbdueitscdc6jcldsgdhqupmmrvdlpzovz7nzmwpecfoya/sd5/agx8zh4</latexit> <latexit sha1_base64="6jiapbrgn22hwf3kpxqnxshgybm=">aaab9xicbvbns8naej34wetx1aoxxsiiqkle0iohohepfewhnlfmtpt26wytdjdkkf0fxjwo4tx/4s1/47bnqvsfddzem2fmxpgkro3rfjtlyyura+ufjelm1vbobmlvv6gttffwp4livctezqsxrg64eayvkozxkfgzhnxm/oyju5on8t4muxbe2jm84hsnlr58fgkft/2qgbzyoqwyw3gniivey0kzctq6ps+/m9asztjqgvq3ptc1wqiv4vswcdhpneurdrdh2pzkjjkortorx+tykl0sjcqwngsq/p4yyaz1ma5tz4ymr+e9ifif185mdbmmuewzwysdlyoyquxcjhgqllemgjg0bkni9lzc+6iqghtu0ybgzb+8sbpnfc+tehfn5ep1hkcbdueitscdc6jcldsgdhqupmmrvdlpzovz7nzmwpecfoya/sd5/agx8zh4</latexit> <latexit sha1_base64="6jiapbrgn22hwf3kpxqnxshgybm=">aaab9xicbvbns8naej34wetx1aoxxsiiqkle0iohohepfewhnlfmtpt26wytdjdkkf0fxjwo4tx/4s1/47bnqvsfddzem2fmxpgkro3rfjtlyyura+ufjelm1vbobmlvv6gttffwp4livctezqsxrg64eayvkozxkfgzhnxm/oyju5on8t4muxbe2jm84hsnlr58fgkft/2qgbzyoqwyw3gniivey0kzctq6ps+/m9asztjqgvq3ptc1wqiv4vswcdhpneurdrdh2pzkjjkortorx+tykl0sjcqwngsq/p4yyaz1ma5tz4ymr+e9ifif185mdbmmuewzwysdlyoyquxcjhgqllemgjg0bkni9lzc+6iqghtu0ybgzb+8sbpnfc+tehfn5ep1hkcbdueitscdc6jcldsgdhqupmmrvdlpzovz7nzmwpecfoya/sd5/agx8zh4</latexit> <latexit sha1_base64="6jiapbrgn22hwf3kpxqnxshgybm=">aaab9xicbvbns8naej34wetx1aoxxsiiqkle0iohohepfewhnlfmtpt26wytdjdkkf0fxjwo4tx/4s1/47bnqvsfddzem2fmxpgkro3rfjtlyyura+ufjelm1vbobmlvv6gttffwp4livctezqsxrg64eayvkozxkfgzhnxm/oyju5on8t4muxbe2jm84hsnlr58fgkft/2qgbzyoqwyw3gniivey0kzctq6ps+/m9asztjqgvq3ptc1wqiv4vswcdhpneurdrdh2pzkjjkortorx+tykl0sjcqwngsq/p4yyaz1ma5tz4ymr+e9ifif185mdbmmuewzwysdlyoyquxcjhgqllemgjg0bkni9lzc+6iqghtu0ybgzb+8sbpnfc+tehfn5ep1hkcbdueitscdc6jcldsgdhqupmmrvdlpzovz7nzmwpecfoya/sd5/agx8zh4</latexit> (, ) Las probabilidades de estado no dependen de la condición inicial (asintóticamente)

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sha1_base64="73rz/wurql16qutkbvjqzrroe1u=">aaaca3icbvdlssnafj3uv42vqdvddbbbvcmiojtc0y3gpoj9qbpdzdpph84kywyilfbw46+4cagiw3/cnx/jtm1cww9cojxzl/fee6acke2631zpaxllda28bm9sbm3volt7lzvkktamsxgioyfwlloynjxtnhzssbeiow2hw6uj336gurekvtojlpoc92mwmyk1kqlnaelb9lqmgnxyq+p7g+illbjcgksbu3gr7hrwkaccvecbrub8eb2ezilgmncsvbe5qfzzldujni5tl1m0xwsi+7rraiwfvx4+/wemj43sg1eitcuattxfezkwso1eadof1gm1703e/7xupqmlp2dxmmkak9mikonqj3ascowxsynmi0mwkczccskas0y0ic02iad5lxdj67sk3cq6pavul4s4yuaqhietgma5qinr0abnqmajeaav4m16sl6sd+tj1lqyipl98afw5w9d0jva</latexit> Comportamiento asintótico: teoría 13 El comportamiento asintótico está determinado por el límite de la siguiente matriz: 0 = E n n n 0 0 K e =[1, 1,...,1] T Ejercicio 1: demostrar que y siempre son un autovector y un autovalor de P, respectivamente. 1 C A E 1 <latexit 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sha1_base64="ttuf0chy3cy0enxhj9j9wcsuhes=">aaab9hicbvdlsgmxfl1tx7w+qi7dbivgqkxe0iwlohuxfewd2qfkmpk2njmzk0yhtpsdblwo4tapceffmlaz0nydgcm553jvjp8iro3rfjuftfwnza3idmlnd2//ohx41nrxqihr0fjequ0tzqsxrgg4eaydkeyix7cwp7yb+a0ru5rh8tgme+zfpc95yckxvvimxwgzaeknjze4v664vxcotepwtiqqo94rf3wdmkyrk4ykonuhu4nxmqimp4jns91us4tqiemzjqwsrex72fzoktqzsodcwnkndzqrvycyemk9jnybjigz6gvvjv7ndvitxnszl0lqmkslrweqkinrraeucmwoewnlcfxc3orogchcje2pzevay19ejc2lknar+ogyurvn6yjcczzcowc4ghrcqx0aqoejnuev3pyr8+k8ox+lamhjz47hd5zph4ftkeg=</latexit> Donde =[ 1, 2,..., K ] el vector de B probabilidades de estados asintótico.. A [ 1, 2,..., K ] 1 Independientemente de la condición inicial, la cadena de Markov converge a las probabilidades asintóticas: =[ 1, 2,..., K ] Condición de equilibrio o steady state (si existe): KX k=1 k =1

14 Usando la condición de equilibrio 14 Usamos : Nota importante: Esto sólo vale si (condición de convergencia)!!

15 Qué pasa si no se cumple la condición de convergencia? 15 Por ejemplo = =1 <latexit sha1_base64="qo/ces76wszeu1bim8jhvzimfgs=">aaab+nicbvdlsgnbejz1genro0cvg0hwfhzf0isq9oixgnladgm9k9lkyoydmv4lrpkulx4u8eqxepnvncr70mscpouqbqanglqkjy7zba2srq1vbja2yts7u3v7duwgpznmmd5kiuxujwdnpyh5ewvk3kkvhyiqvb2mbqz++4erlzl4hscp9ymyxciudnbipbviguyhqk+of3a03e3zvafmzecxivuqkinq6nlfxj9hwcrjzbk07rpoin4ocgwtffl2ms1tycmy8k6hmurc+/ns9ak9muqfhokyfsodqb83coi0hkebmywah3rrm4r/ed0mw0s/f3gaiy/z/kewkxqtos2b9oxidoxyegbkmfspg4ichiatsgnbxfzymmmd1vyn5t6dv+vxrrwlcksoyslxyqwpk1vsie3cycn5jq/kzxqyxqx362m+umivo4fkd6zph1plkro=</latexit> <latexit sha1_base64="qo/ces76wszeu1bim8jhvzimfgs=">aaab+nicbvdlsgnbejz1genro0cvg0hwfhzf0isq9oixgnladgm9k9lkyoydmv4lrpkulx4u8eqxepnvncr70mscpouqbqanglqkjy7zba2srq1vbja2yts7u3v7duwgpznmmd5kiuxujwdnpyh5ewvk3kkvhyiqvb2mbqz++4erlzl4hscp9ymyxciudnbipbviguyhqk+of3a03e3zvafmzecxivuqkinq6nlfxj9hwcrjzbk07rpoin4ocgwtffl2ms1tycmy8k6hmurc+/ns9ak9muqfhokyfsodqb83coi0hkebmywah3rrm4r/ed0mw0s/f3gaiy/z/kewkxqtos2b9oxidoxyegbkmfspg4ichiatsgnbxfzymmmd1vyn5t6dv+vxrrwlcksoyslxyqwpk1vsie3cycn5jq/kzxqyxqx362m+umivo4fkd6zph1plkro=</latexit> <latexit sha1_base64="qo/ces76wszeu1bim8jhvzimfgs=">aaab+nicbvdlsgnbejz1genro0cvg0hwfhzf0isq9oixgnladgm9k9lkyoydmv4lrpkulx4u8eqxepnvncr70mscpouqbqanglqkjy7zba2srq1vbja2yts7u3v7duwgpznmmd5kiuxujwdnpyh5ewvk3kkvhyiqvb2mbqz++4erlzl4hscp9ymyxciudnbipbviguyhqk+of3a03e3zvafmzecxivuqkinq6nlfxj9hwcrjzbk07rpoin4ocgwtffl2ms1tycmy8k6hmurc+/ns9ak9muqfhokyfsodqb83coi0hkebmywah3rrm4r/ed0mw0s/f3gaiy/z/kewkxqtos2b9oxidoxyegbkmfspg4ichiatsgnbxfzymmmd1vyn5t6dv+vxrrwlcksoyslxyqwpk1vsie3cycn5jq/kzxqyxqx362m+umivo4fkd6zph1plkro=</latexit> <latexit sha1_base64="qo/ces76wszeu1bim8jhvzimfgs=">aaab+nicbvdlsgnbejz1genro0cvg0hwfhzf0isq9oixgnladgm9k9lkyoydmv4lrpkulx4u8eqxepnvncr70mscpouqbqanglqkjy7zba2srq1vbja2yts7u3v7duwgpznmmd5kiuxujwdnpyh5ewvk3kkvhyiqvb2mbqz++4erlzl4hscp9ymyxciudnbipbviguyhqk+of3a03e3zvafmzecxivuqkinq6nlfxj9hwcrjzbk07rpoin4ocgwtffl2ms1tycmy8k6hmurc+/ns9ak9muqfhokyfsodqb83coi0hkebmywah3rrm4r/ed0mw0s/f3gaiy/z/kewkxqtos2b9oxidoxyegbkmfspg4ichiatsgnbxfzymmmd1vyn5t6dv+vxrrwlcksoyslxyqwpk1vsie3cycn5jq/kzxqyxqx362m+umivo4fkd6zph1plkro=</latexit> Necesitamos más teoría para poder determinar convergencia!!

16 Clases de estados 16 El estado j es accesible desde el estado i si pij(n)>0 para algún n: Los estados j e i están conectados si son accesibles el uno con el otro: i! j <latexit sha1_base64="e3/v88xxs706o5/eppfqhoqlc0i=">aaab+hicbvbns8naej34wethox69lbbbu0le0gpri8ck9gpaudbbtbt2kw27e6ww/hivhhtx6k/x5r9x2+agrq8ghu/nmdmvtkuw6hnfzsrq2vrgzmgrul2zu1dy9w8armwa8tptuulwsa2xiuf1fch5k9wcxqhkzxb4pfwbd1wbozi7hku8igk/ezfgfk3uduucdltod5bqrr7jfdctexvvbrjm/jyuiuet6351eoplmu+qswpm2/dsdmzuo2cst4qdzpcusiht87alcy25ccazwyfkxco9eiltk0eyu39pjglszcgobwdmcwawvan4n9fomlomxijjm+qjmy+kmklqkwkkpcc0zyhhllcmhb2vsahvlkhnqmhd8bdfxians4rvvfzb83l1ko+jaedwdkfgwwvu4qzquacggtzdk7w5t86l8+58zftxnhzmep7a+fwbgb2s+q==</latexit> i $ j <latexit sha1_base64="z0wa5lf9zmrm4a3bwt5xo6pr2wm=">aaab/hicbvbns8naen34wetxtecvi0xwvbir9fj04rgc/ya2lm12067d7ibdirjc/stepcji1r/izx/jts1bwx8mpn6bywzemahuwpo+nzxvtfwnzdjwextnd2/fpthsgzvqyppucau7itfmcmmawegwtqiziupb2uh4euq3h5g2xmk7ybiwxgqoecqpasv13qrhpcei0hw4aqk1est3fbfq1bwz8dlxc1jfbrp996s3udsnmqqqidfd30sgyikgtgwblhupyqmhyzjkxuslizkj8tnxe3xilqgollylac/u3xm5iy3j4tb2xgrgztgbiv953rsiyydnmkmbstpffkucg8ltjpcaa0zbzjyqqrm9fdmr0ysczatsq/axx14mrboa79x82/nq/aqio4so0de6rt66qhv0gxqoisjk0dn6rw/ok/pivdsf89yvp5ipod9wpn8atgquza==</latexit> Clase: conjunto de estados conectados entre sí. Cadena de Markov irreducible: cuando todos los estados forman una única clase. Ejemplos: Tres clases: {0}, {1,2} y {3} Una clase: {0,1,2,3} Cadena irreducible

17 <latexit sha1_base64="n6fm69hencrndubfv+stebsudwk=">aaab63icbva9swnbej2lxzf+rs1tfongfe5e0djoyxnbxebyhl3nxrjkd+/ynrncyf+wsvde1j9k579xl7lcex8mpn6bywzeleph0fe/vdla+sbmvnm7sro7t39qptxq2yqzjldyihptiajlumjeqogsd1ldqyokf4zgt7n/+msnfyl+wenkq0whwsscucylnojjfrxm1/05ycojclkdas1+9as3sfimueymqbxdwe8xnfkdgkk+q/qyy1pkxntiu45qqrgnp/nbz+tmkqmsj8avrjjxf09mqbj2oilxqsio7lkxi/953qzj63aqdjoh12yxkm4kwytkj5obmjyhndhcmrhuvsjg1fcglp6kcyfyfnmvtc/qgv8p7i9rjzsijjkcwcmcqwbx0ia7aeilgizggv7hzvpei/fufsxas14xcwx/4h3+aafijjm=</latexit> <latexit sha1_base64="n6fm69hencrndubfv+stebsudwk=">aaab63icbva9swnbej2lxzf+rs1tfongfe5e0djoyxnbxebyhl3nxrjkd+/ynrncyf+wsvde1j9k579xl7lcex8mpn6bywzeleph0fe/vdla+sbmvnm7sro7t39qptxq2yqzjldyihptiajlumjeqogsd1ldqyokf4zgt7n/+msnfyl+wenkq0whwsscucylnojjfrxm1/05ycojclkdas1+9as3sfimueymqbxdwe8xnfkdgkk+q/qyy1pkxntiu45qqrgnp/nbz+tmkqmsj8avrjjxf09mqbj2oilxqsio7lkxi/953qzj63aqdjoh12yxkm4kwytkj5obmjyhndhcmrhuvsjg1fcglp6kcyfyfnmvtc/qgv8p7i9rjzsijjkcwcmcqwbx0ia7aeilgizggv7hzvpei/fufsxas14xcwx/4h3+aafijjm=</latexit> <latexit sha1_base64="n6fm69hencrndubfv+stebsudwk=">aaab63icbva9swnbej2lxzf+rs1tfongfe5e0djoyxnbxebyhl3nxrjkd+/ynrncyf+wsvde1j9k579xl7lcex8mpn6bywzeleph0fe/vdla+sbmvnm7sro7t39qptxq2yqzjldyihptiajlumjeqogsd1ldqyokf4zgt7n/+msnfyl+wenkq0whwsscucylnojjfrxm1/05ycojclkdas1+9as3sfimueymqbxdwe8xnfkdgkk+q/qyy1pkxntiu45qqrgnp/nbz+tmkqmsj8avrjjxf09mqbj2oilxqsio7lkxi/953qzj63aqdjoh12yxkm4kwytkj5obmjyhndhcmrhuvsjg1fcglp6kcyfyfnmvtc/qgv8p7i9rjzsijjkcwcmcqwbx0ia7aeilgizggv7hzvpei/fufsxas14xcwx/4h3+aafijjm=</latexit> <latexit sha1_base64="n6fm69hencrndubfv+stebsudwk=">aaab63icbva9swnbej2lxzf+rs1tfongfe5e0djoyxnbxebyhl3nxrjkd+/ynrncyf+wsvde1j9k579xl7lcex8mpn6bywzeleph0fe/vdla+sbmvnm7sro7t39qptxq2yqzjldyihptiajlumjeqogsd1ldqyokf4zgt7n/+msnfyl+wenkq0whwsscucylnojjfrxm1/05ycojclkdas1+9as3sfimueymqbxdwe8xnfkdgkk+q/qyy1pkxntiu45qqrgnp/nbz+tmkqmsj8avrjjxf09mqbj2oilxqsio7lkxi/953qzj63aqdjoh12yxkm4kwytkj5obmjyhndhcmrhuvsjg1fcglp6kcyfyfnmvtc/qgv8p7i9rjzsijjkcwcmcqwbx0ia7aeilgizggv7hzvpei/fufsxas14xcwx/4h3+aafijjm=</latexit> <latexit sha1_base64="n6fm69hencrndubfv+stebsudwk=">aaab63icbva9swnbej2lxzf+rs1tfongfe5e0djoyxnbxebyhl3nxrjkd+/ynrncyf+wsvde1j9k579xl7lcex8mpn6bywzeleph0fe/vdla+sbmvnm7sro7t39qptxq2yqzjldyihptiajlumjeqogsd1ldqyokf4zgt7n/+msnfyl+wenkq0whwsscucylnojjfrxm1/05ycojclkdas1+9as3sfimueymqbxdwe8xnfkdgkk+q/qyy1pkxntiu45qqrgnp/nbz+tmkqmsj8avrjjxf09mqbj2oilxqsio7lkxi/953qzj63aqdjoh12yxkm4kwytkj5obmjyhndhcmrhuvsjg1fcglp6kcyfyfnmvtc/qgv8p7i9rjzsijjkcwcmcqwbx0ia7aeilgizggv7hzvpei/fufsxas14xcwx/4h3+aafijjm=</latexit> <latexit sha1_base64="n6fm69hencrndubfv+stebsudwk=">aaab63icbva9swnbej2lxzf+rs1tfongfe5e0djoyxnbxebyhl3nxrjkd+/ynrncyf+wsvde1j9k579xl7lcex8mpn6bywzeleph0fe/vdla+sbmvnm7sro7t39qptxq2yqzjldyihptiajlumjeqogsd1ldqyokf4zgt7n/+msnfyl+wenkq0whwsscucylnojjfrxm1/05ycojclkdas1+9as3sfimueymqbxdwe8xnfkdgkk+q/qyy1pkxntiu45qqrgnp/nbz+tmkqmsj8avrjjxf09mqbj2oilxqsio7lkxi/953qzj63aqdjoh12yxkm4kwytkj5obmjyhndhcmrhuvsjg1fcglp6kcyfyfnmvtc/qgv8p7i9rjzsijjkcwcmcqwbx0ia7aeilgizggv7hzvpei/fufsxas14xcwx/4h3+aafijjm=</latexit> <latexit sha1_base64="n6fm69hencrndubfv+stebsudwk=">aaab63icbva9swnbej2lxzf+rs1tfongfe5e0djoyxnbxebyhl3nxrjkd+/ynrncyf+wsvde1j9k579xl7lcex8mpn6bywzeleph0fe/vdla+sbmvnm7sro7t39qptxq2yqzjldyihptiajlumjeqogsd1ldqyokf4zgt7n/+msnfyl+wenkq0whwsscucylnojjfrxm1/05ycojclkdas1+9as3sfimueymqbxdwe8xnfkdgkk+q/qyy1pkxntiu45qqrgnp/nbz+tmkqmsj8avrjjxf09mqbj2oilxqsio7lkxi/953qzj63aqdjoh12yxkm4kwytkj5obmjyhndhcmrhuvsjg1fcglp6kcyfyfnmvtc/qgv8p7i9rjzsijjkcwcmcqwbx0ia7aeilgizggv7hzvpei/fufsxas14xcwx/4h3+aafijjm=</latexit> <latexit sha1_base64="n6fm69hencrndubfv+stebsudwk=">aaab63icbva9swnbej2lxzf+rs1tfongfe5e0djoyxnbxebyhl3nxrjkd+/ynrncyf+wsvde1j9k579xl7lcex8mpn6bywzeleph0fe/vdla+sbmvnm7sro7t39qptxq2yqzjldyihptiajlumjeqogsd1ldqyokf4zgt7n/+msnfyl+wenkq0whwsscucylnojjfrxm1/05ycojclkdas1+9as3sfimueymqbxdwe8xnfkdgkk+q/qyy1pkxntiu45qqrgnp/nbz+tmkqmsj8avrjjxf09mqbj2oilxqsio7lkxi/953qzj63aqdjoh12yxkm4kwytkj5obmjyhndhcmrhuvsjg1fcglp6kcyfyfnmvtc/qgv8p7i9rjzsijjkcwcmcqwbx0ia7aeilgizggv7hzvpei/fufsxas14xcwx/4h3+aafijjm=</latexit> Estados recurrentes y transitorios 17 Estado recurrente: si el proceso retorna a ese estado con probabilidad igual a 1. () Un estado recurrente se repite infinitas veces. Estado transitorio: si el proceso retorna a ese estado con probabilidad menor que 1. () Un estado transitorio se repite un número finito de veces. Ejemplo:

18 <latexit sha1_base64="czm0h1hliw/6mjyefvglci/vkre=">aaab9hicbvbns8naej3ur1q/qh69bivgqsqi6euoiucxqr+gdwwz3brln5u4oyme0n/hxymixv0x3vw3btsctpxbwoo9gwbm+bhggh3n2yqsrw9sbhw3szu7e/sh5cojlo4srvmtrijshz9ojrhkteqowcdwjis+yg1/fdfz2xomni9ka9oyesezsh5wstbi3n230efety/lann+uejuntnsvelmpai56v3yv28q0srkeqkgwnddj0yviwo5fwxa6iwaxysoyzb1dzukznrl5kdp7tojdowguqyk2np190rgqq3t0dedicgrxvzm4n9en8hg2su4jbnkki4wbymwmbjncdgdrhhfkrpcqolmvpuoiciutu4le4k7/piqav1uxafqpl5ward5heu4gvm4bxeuoaypuicmuhicz3ifn2tivvjv1seitwdlm8fwb9bnd2sdkdg=</latexit> <latexit 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sha1_base64="hpk9vbqfov2gccybw3cp9ayio5k=">aaab8hicbvbnswmxej34wetx1aoxybhqpeykoccpevfywx5iu5rsmm1dk+ysziwy9fd48aciv3+on/+nabshbx0w8hhvhpl5ysk4sz73jvzw19y3ngtbxe2d3b390sfh08sppqxbyxhrdkgme1yxhuvwshaigzghyk1wddv1w09mgx6rbztowcdjqpgiu2kd9fjpjrzhr72zxqnsvb0z8dlxc1kghpve6avbj2kqmbjuegm6vpfyicpacirypnhndusihzeb6ziqigqmygyht/cpu/o4irurzffm/t2rewnmwiauuxi7nivevpzp66q2ugoyrplumkxni6juybvj6fe4zzwjvowdivrzdyumq6ijts6jogvbx3x5mttpq75x9e8vyrwbpi4chmmjvmchs6jbhdsharqkpmmrvcgnxta7+pi3rqb85gj+ah3+aijaj5a=</latexit> <latexit sha1_base64="hpk9vbqfov2gccybw3cp9ayio5k=">aaab8hicbvbnswmxej34wetx1aoxybhqpeykoccpevfywx5iu5rsmm1dk+ysziwy9fd48aciv3+on/+nabshbx0w8hhvhpl5ysk4sz73jvzw19y3ngtbxe2d3b390sfh08sppqxbyxhrdkgme1yxhuvwshaigzghyk1wddv1w09mgx6rbztowcdjqpgiu2kd9fjpjrzhr72zxqnsvb0z8dlxc1kghpve6avbj2kqmbjuegm6vpfyicpacirypnhndusihzeb6ziqigqmygyht/cpu/o4irurzffm/t2rewnmwiauuxi7nivevpzp66q2ugoyrplumkxni6juybvj6fe4zzwjvowdivrzdyumq6ijts6jogvbx3x5mttpq75x9e8vyrwbpi4chmmjvmchs6jbhdsharqkpmmrvcgnxta7+pi3rqb85gj+ah3+aijaj5a=</latexit> <latexit sha1_base64="hpk9vbqfov2gccybw3cp9ayio5k=">aaab8hicbvbnswmxej34wetx1aoxybhqpeykoccpevfywx5iu5rsmm1dk+ysziwy9fd48aciv3+on/+nabshbx0w8hhvhpl5ysk4sz73jvzw19y3ngtbxe2d3b390sfh08sppqxbyxhrdkgme1yxhuvwshaigzghyk1wddv1w09mgx6rbztowcdjqpgiu2kd9fjpjrzhr72zxqnsvb0z8dlxc1kghpve6avbj2kqmbjuegm6vpfyicpacirypnhndusihzeb6ziqigqmygyht/cpu/o4irurzffm/t2rewnmwiauuxi7nivevpzp66q2ugoyrplumkxni6juybvj6fe4zzwjvowdivrzdyumq6ijts6jogvbx3x5mttpq75x9e8vyrwbpi4chmmjvmchs6jbhdsharqkpmmrvcgnxta7+pi3rqb85gj+ah3+aijaj5a=</latexit> <latexit sha1_base64="hpk9vbqfov2gccybw3cp9ayio5k=">aaab8hicbvbnswmxej34wetx1aoxybhqpeykoccpevfywx5iu5rsmm1dk+ysziwy9fd48aciv3+on/+nabshbx0w8hhvhpl5ysk4sz73jvzw19y3ngtbxe2d3b390sfh08sppqxbyxhrdkgme1yxhuvwshaigzghyk1wddv1w09mgx6rbztowcdjqpgiu2kd9fjpjrzhr72zxqnsvb0z8dlxc1kghpve6avbj2kqmbjuegm6vpfyicpacirypnhndusihzeb6ziqigqmygyht/cpu/o4irurzffm/t2rewnmwiauuxi7nivevpzp66q2ugoyrplumkxni6juybvj6fe4zzwjvowdivrzdyumq6ijts6jogvbx3x5mttpq75x9e8vyrwbpi4chmmjvmchs6jbhdsharqkpmmrvcgnxta7+pi3rqb85gj+ah3+aijaj5a=</latexit> Cadenas de Markov ergódicas 18 Período: El estado i tiene período d si puede repetirse solamente en tiempos múltiplos de d, es decir, cuando pii(n) = 0 si n no es múltiplo de d. Una cadena de Markov es aperiódica cuando todos sus estados tienen período d=1. Estado recurrente positivo: Estado recurrente nulo: E[T i ] < 1 ( <latexit sha1_base64="cqzxz9tuzr16f7y/9lqu2gnmxia=">aaab9hicbvbns8naej3ur1q/qh69bivgqsqi6mfduqspffofbsib7azdutne3ukhhp4olx4u8eqp8ea/cdvmok0pbh7vztazz48f1+g431zhbx1jc6u4xdrz3ds/kb8etxsukmqanbkr6vhem8elayjhwtqxyit0bwv747uz354wpxkkg5jgzavjupkau4jg8u67jt73bnpcbpj2yxwn6sxhrxi3jxxiue+xv3qdicyhk0gf0brrojf6gvhiqwdtui/rlcz0tiasa6gkidnenj96ap8zzwahktil0z6rvycyemqdhr7pdamo9li3e//zugkg117gzzwgk3sxkeiejze9s8aecmuoitqqqhu3t9p0rbshahiqmrdc5zdxseui6jpv9/gyurvn4yjcczzcobhwbtv4gdo0gcitpmmrvfkt68v6tz4wrqurnzmgp7a+fwbi+zhx</latexit> i > 0) E[T i ]=1 ( <latexit sha1_base64="czm0h1hliw/6mjyefvglci/vkre=">aaab9hicbvbns8naej3ur1q/qh69bivgqsqi6euoiucxqr+gdwwz3brln5u4oyme0n/hxymixv0x3vw3btsctpxbwoo9gwbm+bhggh3n2yqsrw9sbhw3szu7e/sh5cojlo4srvmtrijshz9ojrhkteqowcdwjis+yg1/fdfz2xomni9ka9oyesezsh5wstbi3n230efety/lann+uejuntnsvelmpai56v3yv28q0srkeqkgwnddj0yviwo5fwxa6iwaxysoyzb1dzukznrl5kdp7tojdowguqyk2np190rgqq3t0dedicgrxvzm4n9en8hg2su4jbnkki4wbymwmbjncdgdrhhfkrpcqolmvpuoiciutu4le4k7/piqav1uxafqpl5ward5heu4gvm4bxeuoaypuicmuhicz3ifn2tivvjv1seitwdlm8fwb9bnd2sdkdg=</latexit> i = 0) Cadena de Markov ergódica: cuando es irreducible, aperiódica y recurrente positiva. Todos los estados de la cadena son visitados con frecuencia suficiente para que su probabilidad de visita sea distinta de cero. Una cadena ergódica converge a una distribución asintótica con probabilidades distintas de cero

19 Algoritmo Google Page Rank (1)

20 <latexit sha1_base64="73rz/wurql16qutkbvjqzrroe1u=">aaaca3icbvdlssnafj3uv42vqdvddbbbvcmiojtc0y3gpoj9qbpdzdpph84kywyilfbw46+4cagiw3/cnx/jtm1cww9cojxzl/fee6acke2631zpaxllda28bm9sbm3volt7lzvkktamsxgioyfwlloynjxtnhzssbeiow2hw6uj336gurekvtojlpoc92mwmyk1kqlnaelb9lqmgnxyq+p7g+illbjcgksbu3gr7hrwkaccvecbrub8eb2ezilgmncsvbe5qfzzldujni5tl1m0xwsi+7rraiwfvx4+/wemj43sg1eitcuattxfezkwso1eadof1gm1703e/7xupqmlp2dxmmkak9mikonqj3ascowxsynmi0mwkczccskas0y0ic02iad5lxdj67sk3cq6pavul4s4yuaqhietgma5qinr0abnqmajeaav4m16sl6sd+tj1lqyipl98afw5w9d0jva</latexit> <latexit sha1_base64="73rz/wurql16qutkbvjqzrroe1u=">aaaca3icbvdlssnafj3uv42vqdvddbbbvcmiojtc0y3gpoj9qbpdzdpph84kywyilfbw46+4cagiw3/cnx/jtm1cww9cojxzl/fee6acke2631zpaxllda28bm9sbm3volt7lzvkktamsxgioyfwlloynjxtnhzssbeiow2hw6uj336gurekvtojlpoc92mwmyk1kqlnaelb9lqmgnxyq+p7g+illbjcgksbu3gr7hrwkaccvecbrub8eb2ezilgmncsvbe5qfzzldujni5tl1m0xwsi+7rraiwfvx4+/wemj43sg1eitcuattxfezkwso1eadof1gm1703e/7xupqmlp2dxmmkak9mikonqj3ascowxsynmi0mwkczccskas0y0ic02iad5lxdj67sk3cq6pavul4s4yuaqhietgma5qinr0abnqmajeaav4m16sl6sd+tj1lqyipl98afw5w9d0jva</latexit> <latexit sha1_base64="73rz/wurql16qutkbvjqzrroe1u=">aaaca3icbvdlssnafj3uv42vqdvddbbbvcmiojtc0y3gpoj9qbpdzdpph84kywyilfbw46+4cagiw3/cnx/jtm1cww9cojxzl/fee6acke2631zpaxllda28bm9sbm3volt7lzvkktamsxgioyfwlloynjxtnhzssbeiow2hw6uj336gurekvtojlpoc92mwmyk1kqlnaelb9lqmgnxyq+p7g+illbjcgksbu3gr7hrwkaccvecbrub8eb2ezilgmncsvbe5qfzzldujni5tl1m0xwsi+7rraiwfvx4+/wemj43sg1eitcuattxfezkwso1eadof1gm1703e/7xupqmlp2dxmmkak9mikonqj3ascowxsynmi0mwkczccskas0y0ic02iad5lxdj67sk3cq6pavul4s4yuaqhietgma5qinr0abnqmajeaav4m16sl6sd+tj1lqyipl98afw5w9d0jva</latexit> <latexit sha1_base64="73rz/wurql16qutkbvjqzrroe1u=">aaaca3icbvdlssnafj3uv42vqdvddbbbvcmiojtc0y3gpoj9qbpdzdpph84kywyilfbw46+4cagiw3/cnx/jtm1cww9cojxzl/fee6acke2631zpaxllda28bm9sbm3volt7lzvkktamsxgioyfwlloynjxtnhzssbeiow2hw6uj336gurekvtojlpoc92mwmyk1kqlnaelb9lqmgnxyq+p7g+illbjcgksbu3gr7hrwkaccvecbrub8eb2ezilgmncsvbe5qfzzldujni5tl1m0xwsi+7rraiwfvx4+/wemj43sg1eitcuattxfezkwso1eadof1gm1703e/7xupqmlp2dxmmkak9mikonqj3ascowxsynmi0mwkczccskas0y0ic02iad5lxdj67sk3cq6pavul4s4yuaqhietgma5qinr0abnqmajeaav4m16sl6sd+tj1lqyipl98afw5w9d0jva</latexit> Algoritmo Google Page Rank (2) 20 O resolviendo para el estado estacionario: con KX k=1 k =1 Nota importante: Este algoritmo simplificado de Google Page Rank tiene el problema que no garantiza que la cadena de Markov sea irreducible y aperiódica. Ver qué modificación es necesaria para garantizar una cadena ergódica (ver paper original).

21 Material de Lectura - Sobre la clase de hoy: Libros (Mínimo): [15] Probability, Statistics, and Random Processes for Eletrical Engineerging, 3rd_Ed. - Leon-Garcia., Cap 11. [5] Intuitive Probability and Random Processes Using MatLab - Steven M. Kay, Cap 22. Libros (opcional): [16] Performance Modeling and Design of Computer Systems: Queueing Theory in Action, Mor Harchol-Balter,2013, Cap 8-10 [19] Introduction to probability models, Ross Sheldon, 2007, Cap 4. Papers (opcional): The Anatomy of a Large-Scale Hypertextual Web Search Engine, Sergey Brin and Lawrence Page, Computer Networks and IDSN Systems, 30 (1998), Leer para la próxima clase: Libros (Mínimo): [2] Introduction to the Modeling and Analysis of Complex Systems - Sayama, Cap 1, 3 y 4. Libros (opcional): [1] Introduction to Modeling and Simulation with Matlab and Python - Gordon and Guilfoos, Cap. 9