class: center, middle, inverse, title-slide # Estadística y Procesos Estocásticos ## <br><br><br> ### <br><br><br><br>Grado en Ingeniería en Tecnologías de la Telecomunicación<br>Curso 2017-18 --- layout: true background-image: url(http://www.dma.ulpgc.es/profesores/personal/stat/uni2.png) background-size: 50% 75% --- background-image: url(http://www.dma.ulpgc.es/profesores/personal/asp/images/TelecoHome.jpg) background-size: cover background-position: 15% 45% class: inverse, bottom, left # 1. Resumen del <br> Proyecto Docente --- ### Campus virtual: material para seguir la asignatura. .pull-left[ [](https://www2.ulpgc.es/aplicaciones/proyectosdocentes/pdf.php?id_proyecto=51867&NUEVA=1) * [Proyecto docente](https://www2.ulpgc.es/aplicaciones/proyectosdocentes/pdf.php?id_proyecto=51867&NUEVA=1) ] -- .pull-right[ [](http://www.dma.ulpgc.es/profesores/personal/stat/TELECO/Estadistica_y_Procesos_Estocasticos_v2014.pdf) * [Guía didáctica](http://www.dma.ulpgc.es/profesores/personal/stat/TELECO/Estadistica_y_Procesos_Estocasticos_v2014.pdf) ] --- ## Profesorado <img src="http://www.dma.ulpgc.es/profesores/personal/stat/logoDMAtransp1.png" alt="logo DMA" height="80" style="float:right"> <br> -- * __Profesora de Problemas grupo T1__: Begoña González Landín correo: .blue[bego.landin@ulpgc.es] -- * __Profesora de Teoría grupo T2 y de prácticas T1 y T2__: Carmen Nieves Hernández Flores correo: .blue[carmennieves.hdezflores@ulpgc.es] -- * __Profesor de Problemas grupo T2__: Juan Rocha Martín correo: .blue[juan.rocha@ulpgc.es] -- * __Profesor de Teoría grupo T1__: Angelo Santana del Pino correo: .blue[angelo.santana@ulpgc.es] --- # Sistema de evaluación ## Convocatoria Ordinaria -- .red[__Fuente 1.1.__] Asistencia a las clases y al Laboratorio (5%) -- .red[__Fuente 1.2.__] Actividades de Laboratorio. (15%) -- .red[__Fuente 1.3.__] Realización de un examen parcial y un final de la asignatura (40% primer parcial y 60% el final).(80%) -- En el caso que la nota del examen sea inferior a 5 la nota global de la asignatura será ésta. En otro caso, la calificación global se obtendrá mediante el siguiente criterio: `$$0.05\cdot\textrm{Fuente1.1} +0.15\cdot\textrm{Fuente1.2}+0.8\cdot\textrm{Fuente1.3}$$` --- # Sistema de Evaluación ## Convocatorias Especiales -- .red[__Fuente 2.1.__] Realización de una prueba escrita orientada a la resolución de problemas. (15%) -- .red[__Fuente 2.2.__] Realización de un examen final de la asignatura. (85%) -- En las convocatorias especiales los alumnos se presentarán a un examen final. Para sumar el resto de las fuentes el alumno debe tener bien el 50% del examen final. Y el criterio utilizado es: `$$0.15\cdot\textrm{Fuente 2.1}+0.85\cdot\textrm{Fuente 2.2}$$` --- ## Prácticas de Laboratorio. Se realizarán en el aula de clase y consistirán en la resolución de problemas cuya solución deberá subirse al campus virtual. -- * __Práctica Nº 1:__ Semana del 19 al 23 de febrero de 2018. -- * __Práctica Nº 2:__ Semana del 12 al 16 de marzo de 2018. -- * __Práctica Nº 3:__ Semana del 9 al 13 de abril de 2018. -- * __Práctica Nº 4:__ Semana del 7 al 11 de mayo de 2018. -- ### Exámenes -- * __1er Parcial__ el jueves 22 de marzo de 2018. -- * __Final__ el 25 de mayo de 2018 --- ## Tutorías <img src="http://www.dma.ulpgc.es/profesores/personal/stat/logoDMAtransp1.png" alt="logo DMA" height="80" style="float:right"> <br> ### .red[Pedir cita por correo electrónico] -- .blue[Profesora de Problemas grupo T1:] Begoña González Landín. Martes de 10.30 a 12.30 -- .blue[Profesora de Teoría grupo T2 y de prácticas T1 y T2:] Carmen Nieves Hernández Flores. De Lunes a Jueves de 12.30 a 14.00 -- .blue[Profesor de Problemas grupo T2:] Juan Rocha Martín. Martes y Jueves de 12.00 a 14.00 -- .blue[Profesor de Teoría grupo T1:] Angelo Santana del Pino. Lunes de 12.00 a 14.00 --- background-image: url(http://www.dma.ulpgc.es/profesores/personal/asp/images/TelecoHome.jpg) background-size: cover background-position: 15% 45% class: inverse, bottom, left # 2. Motivación, contenidos <br> y objetivos --- ## ¿Por qué esta asignatura? -- .center[ <video src="http://www.dma.ulpgc.es/profesores/personal/asp/Datos/lol.mp4" width="480" height="270" controls autoplay preload></video> ] -- Jugar a LoL implica: * Disponer de una red que pueda absorver el tráfico generado por el juego. * Que las conexiones sean rápidas y sin errores (ping<50 ms). * Si además la partida se transmite por Twich hay que tener un buen ancho de banda disponible. --- .pull-left[ <br>  ] -- .pull-right[ ### Tráfico soportado por la red * Telefonía * Navegación web * Aplicaciones de mensajería * Redes sociales * Streaming de video y audio * Juegos multijugador online * Discos virtuales en la nube * Navegación asistida con GPS * Reservas de viajes y alojamiento. * Internet de las cosas * ... ] --- ### Conexiones entre usuarios de facebook  Este gráfico, debido a [un empleado de Facebook](https://paulbutler.org/archives/visualizing-facebook-friends/) muestra las conexiones entre usuarios de esta aplicación. Probablemente Twitter e Instagram y las redes de juegos online muestren perfiles parecidos. Las aplicaciones de mensajería muestran un perfil de conexiones más local. --- ### ¿Cómo conseguimos que todo ésto funcione? Requisitos. * Instalar una red de comunicaciones: centralitas, routers, repetidores, cables, antenas, ... -- * Implementar mecanismos de transmisión de señales sobre esta infraestructura, con filtrado de ruido y corrección de errores. -- * Implementar protocolos de gestión de la red para controlar el tráfico. -- * Recopilar datos de manera constante para mantener la infraestructura adecuadamente dimensionada a las necesidades de cada momento. -- * Analizar datos de uso de sistemas y aplicaciones: estimar tendencias, descubrir nuevos mercados, desarrollo de nuevas ideas `\(\longrightarrow\)` Big data. --- ### El papel del azar. * Gestión y protocolos de red. .center[  ] A cada nodo de la red van llegando paquetes de datos que deben hacer cola para saltar al siguiente nodo de la red, hasta que llegan a destino. -- - Los instantes de llegada, el tiempo que tardan en ser despachados, el tiempo que deben permanecer en espera __son magnitudes aleatorias__ - El tamaño de los paquetes, y el número de nodos que deben atravesar también __son aleatorios__ -- .center[ Problemas de Teletráfico `\(\rightarrow\)` .red[__Teoría de colas__] ] --- ### Ejemplo: Medidas obtenidas en un router de red: instantes de llegada de los paquetes de datos, número de paquetes que encuentran en cola a su llegada, tiempo de servicio, tiempo de espera y tiempo total en el sistema: <div id="htmlwidget-f29db41061cc5e626b58" style="width:100%;height:auto;" class="datatables html-widget"></div> <script type="application/json" data-for="htmlwidget-f29db41061cc5e626b58">{"x":{"filter":"none","fillContainer":false,"data":[["1","2","3","4","5","6","7","8","9","10","11","12","13","14","15","16","17","18","19","20","21","22","23","24","25","26","27","28","29","30","31","32","33","34","35","36","37","38","39","40","41","42","43","44","45","46","47","48","49","50","51","52","53","54","55","56","57","58","59","60","61","62","63","64","65","66","67","68","69","70","71","72","73","74","75","76","77","78","79","80","81","82","83","84","85","86","87","88","89","90","91","92","93","94","95","96","97","98","99","100","101","102","103","104","105","106","107","108","109","110","111","112","113","114","115","116","117","118","119","120","121","122","123","124","125","126","127","128","129","130","131","132","133","134","135","136","137","138","139","140","141","142","143","144","145","146","147","148","149","150","151","152","153","154","155","156","157","158","159","160","161","162","163","164","165","166","167","168","169","170","171","172","173","174","175","176","177","178","179","180","181","182","183","184","185","186","187","188","189","190","191","192","193","194","195","196","197","198","199","200","201","202","203","204","205","206","207","208","209","210","211","212","213","214","215","216","217","218","219","220","221","222","223","224","225","226","227","228","229","230","231","232","233","234","235","236","237","238","239","240"],[0.00134,0.001508,0.004176,0.00814,0.011036,0.015072,0.017892,0.020604,0.022032,0.0243,0.024752,0.027356,0.028692,0.03084,0.031608,0.035844,0.038468,0.042524,0.044044,0.045324,0.047296,0.049248,0.05036,0.052184,0.05382,0.056356,0.059044,0.063028,0.0659,0.070032,0.07276,0.076728,0.079616,0.083732,0.086476,0.090484,0.093332,0.097492,0.101832,0.105752,0.110332,0.11434,0.118832,0.122752,0.125692,0.129772,0.132552,0.136512,0.139408,0.140512,0.143348,0.14406,0.146268,0.147108,0.15028,0.150368,0.153124,0.157056,0.159984,0.164112,0.166844,0.1709,0.1737,0.177716,0.18056,0.183884,0.184504,0.187416,0.188372,0.19206,0.19242,0.195916,0.197352,0.199944,0.202776,0.204056,0.206744,0.209636,0.211068,0.213024,0.2136,0.214992,0.21588,0.216944,0.217088,0.218892,0.220848,0.220932,0.221088,0.2227,0.223948,0.2256,0.227356,0.227908,0.230808,0.234808,0.237664,0.241648,0.244524,0.249108,0.249272,0.250376,0.251904,0.253104,0.253252,0.254348,0.255736,0.257924,0.258008,0.260724,0.262444,0.262592,0.267308,0.271092,0.271176,0.275112,0.275316,0.277952,0.281908,0.294076,0.29522,0.296692,0.2978,0.299104,0.3003,0.302484,0.30458,0.305688,0.321908,0.325828,0.329888,0.333212,0.361956,0.365908,0.371488,0.374912,0.378068,0.381456,0.396568,0.419628,0.422604,0.427136,0.44004,0.45922,0.462332,0.46684,0.485596,0.485736,0.486504,0.4899,0.495856,0.507976,0.512644,0.514396,0.530968,0.53576,0.550968,0.55212,0.560224,0.563544,0.568056,0.611888,0.614964,0.620368,0.621656,0.623308,0.624384,0.632492,0.634736,0.638312,0.668368,0.680604,0.684548,0.684712,0.685688,0.689104,0.690084,0.69204,0.694004,0.694172,0.694312,0.695952,0.697908,0.698056,0.699864,0.7015,0.702068,0.70398,0.70408,0.704912,0.705816,0.70876,0.708984,0.711772,0.712892,0.715984,0.716568,0.718628,0.722572,0.725488,0.72614,0.729404,0.732344,0.736728,0.740844,0.744908,0.747704,0.751896,0.754564,0.758484,0.76142,0.765508,0.768276,0.769512,0.770352,0.772116,0.772192,0.772544,0.775136,0.775668,0.77912,0.779728,0.781996,0.783264,0.785964,0.788856,0.792976,0.793168,0.795712,0.79968,0.802568,0.806552,0.809428,0.813628,0.817652,0.817928,0.818232,0.823064,0.828068,0.832296],[0,1,2,0,0,0,0,0,0,0,1,1,1,2,2,2,3,1,0,1,1,1,2,2,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,0,0,1,2,1,0,1,1,1,0,0,1,1,2,0,1,0,1,1,1,0,1,2,0,1,0,0,0,0,1,0,0,0,0,1,2,2,0,1,2,2,3,1,0,0,0,0,0,0,0,0,1,1,1,2,3,3,4,2,3,0,0,1,0,0,1,0,0,0,0,0,1,2,1,0,0,0,1,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,0,0,0,0,0,1,1,1,1,0,0,0,1,0,1,2,2,2,3,3,4,5,6,6,7,6,6,5,6,6,3,2,3,4,0,0,0,0,0,0,1,2,3,0,0,0,1,1,2,3,2,3,3,4,3,3,3,4,5,6,6,5,5,3,2,2,1,2,2,0,0,0],[0.004036,0.000398,1.1e-05,0.002812,0.000625,0.000145,0.00133,0.000327,0.001352,0.001227,0.003033,0.002575,0.002676,0.004925,0.002825,5.1e-05,0.001415,2.4e-05,0.002961,0.000838,0.003221,0.001175,0.000619,0.001517,0.000393,0.001633,6e-06,0.000571,0.001201,0.001504,0.000701,1.5e-05,0.002598,1.3e-05,0.001314,4.7e-05,0.006977,0.001493,2.8e-05,0.005527,0.000208,0.003371,0.004064,6.2e-05,0.001985,0.00355,0.001574,0.000544,0.000159,0.000143,0.003155,0.000348,0.002689,0.000186,0.002354,0.002322,0.00446,3.7e-05,0.001442,0.005938,0.004691,0.00021,0.001314,0.003259,0.00235,0.001607,0.002625,0.002301,0.001162,0.004343,0.000136,0.000312,0.004187,0.000281,1.1e-05,0.001474,0.002268,0.002235,0.000523,0.000315,0.000425,0.000562,0.001598,0.001518,0.00078,0.000213,0.000256,0.003863,0.000262,0.000329,0.00042,0.001213,0.000443,0.0012,0.001604,0.00083,0.000272,0.000674,0.001503,0.001248,0.001461,0.001504,0.003706,2.5e-05,0.000427,0.001506,0.000783,0.000448,0.000143,0.001039,0.00108,0.000751,0.000591,0.001554,0.000864,8.5e-05,0.00075,0.001491,0.002113,0.003262,0.000207,0.000833,0.000215,0.000714,0.001591,0.004481,0.001171,0.001666,0.002347,0.001495,0.000959,0.000216,0.003989,0.00258,0.001392,0.003007,0.001761,0.000595,0.000312,0.001208,0.001309,0.003422,0.000204,0.001198,0.010045,0.000289,6.5e-05,0.001112,0.005892,0.003468,0.005626,0.005414,0.000799,0.00181,0.000836,0.001931,0.000966,0.006631,2.1e-05,0.001037,0.000144,0.001226,0.001729,0.001888,0.001996,0.000767,0.001317,0.00144,0.001035,0.000501,0.002174,0.000272,0.000832,0.000625,0.003701,0.001783,0.000497,0.001332,0.000156,0.000867,0.000455,0.002135,0.002509,0.003162,0.003104,0.002141,0.001752,0.00108,0.003184,0.000171,0.001695,0.001875,0.001327,0.000821,0.000218,0.001996,0.00192,0.006051,0.000304,3.8e-05,0.000541,5e-05,0.000786,0.002308,0.000674,0.000544,0.000865,0.009932,0.000187,0.001487,0.000261,0.000725,0.000296,0.001476,0.002055,0.000837,0.004344,0.002224,0.000509,0.00141,0.003282,0.009107,0.003484,0.000604,0.003118,0.000774,0.00075,0.000492,0.002779,0.0006,0.005924,0.00187,0.000601,0.002194,0.000829,0.00089,0.000237,0.003059,0.000833,0.004134],[0,0.003868,0.001599,-0,0,0,0,0,0,-0,0.000775,0.001204,0.002444,0.002972,0.007129,0.005717,0.003145,0.000504,0,0.001681,0.000547,0.001816,0.001879,0.000674,0.000555,0,0,0,-0,0,-0,0,-0,0,-0,-0,0,0.002817,0,0,0.000947,-0,0,0.000144,-0,0,0.00077,0,0,-0,0,0.002443,0.000583,0.002432,0,0.002266,0.001832,0.00236,-0,-0,0.003206,0.003841,0.001251,-0,0.000415,0,0.000987,0.0007,0.002044,-0,0.003983,0.000623,-0,0.001595,0,0,-0,0,0.000803,0,0,0,-0,0.000534,0.001908,0.000884,0,0.000172,0.003878,0.002529,0.001609,0.000378,-0,-0,-0,-0,0,0,-0,-0,0.001084,0.001441,0.001417,0.003923,0.003799,0.003131,0.003249,0.001844,0.002208,0,-0,0.000932,-0,0,0.00147,0,0,0,-0,0,0.002118,0.000853,0.000577,0,0,0,0.002385,0.002448,0,-0,-0,0,0,3.7e-05,0,-0,-0,-0,-0,-0,-0,0,-0,-0,-0,0.005537,0,-0,0.000344,0.00284,0.000352,-0,0.000746,0,-0,0,0,-0,0,0,-0,0,-0,0,0.0006,0.000943,0.000635,0,-0,-0,0,-0,-0,0.000668,0.000317,0.000602,0.001405,-0,-0,-0,0.000727,-0,0.000179,0.00254,0.003894,0.005362,0.006935,0.006775,0.007755,0.010107,0.009374,0.008125,0.009776,0.008314,0.008015,0.005141,0.006553,0.006413,0.00852,0.005908,0.005295,0.002571,-0,0,0,0,-0,0,0.007264,0.003531,0.002082,0,0,-0,0.000636,0.000927,0.001688,0.00568,0.005312,0.005289,0.003248,0.005922,0.012761,0.014976,0.01288,0.013106,0.00976,0.010318,0.008266,0.007078,0.00479,0.00673,0.005724,0.002125,0.000294,0.000848,0.001434,0,-0,-0],[0.004036,0.004267,0.001609,0.002812,0.000625,0.000145,0.00133,0.000327,0.001352,0.001227,0.003808,0.00378,0.00512,0.007897,0.009953,0.005769,0.00456,0.000527,0.002961,0.002519,0.003768,0.002991,0.002498,0.002191,0.000948,0.001633,6e-06,0.000571,0.001201,0.001504,0.000701,1.5e-05,0.002598,1.3e-05,0.001314,4.7e-05,0.006977,0.004311,2.8e-05,0.005527,0.001155,0.003371,0.004064,0.000206,0.001985,0.00355,0.002344,0.000544,0.000159,0.000143,0.003155,0.002791,0.003272,0.002618,0.002354,0.004588,0.006292,0.002397,0.001442,0.005938,0.007897,0.004051,0.002564,0.003259,0.002764,0.001607,0.003612,0.003,0.003206,0.004343,0.004119,0.000935,0.004187,0.001877,1.1e-05,0.001474,0.002268,0.002235,0.001326,0.000315,0.000425,0.000562,0.001598,0.002052,0.002688,0.001097,0.000256,0.004034,0.004141,0.002857,0.00203,0.001591,0.000443,0.0012,0.001604,0.00083,0.000272,0.000674,0.001503,0.001248,0.002545,0.002945,0.005123,0.003947,0.004227,0.004637,0.004032,0.002292,0.002351,0.001039,0.00108,0.001683,0.000591,0.001554,0.002334,8.5e-05,0.00075,0.001491,0.002113,0.003262,0.002325,0.001685,0.000793,0.000714,0.001591,0.004481,0.003556,0.004114,0.002347,0.001495,0.000959,0.000216,0.003989,0.002618,0.001392,0.003007,0.001761,0.000595,0.000312,0.001208,0.001309,0.003422,0.000204,0.001198,0.010045,0.005826,6.5e-05,0.001112,0.006236,0.006308,0.005978,0.005414,0.001545,0.00181,0.000836,0.001931,0.000966,0.006631,2.1e-05,0.001037,0.000144,0.001226,0.001729,0.001888,0.002595,0.001711,0.001951,0.00144,0.001035,0.000501,0.002174,0.000272,0.000832,0.001293,0.004018,0.002385,0.001902,0.001332,0.000156,0.000867,0.001183,0.002135,0.002688,0.005702,0.006998,0.007503,0.008687,0.007855,0.010939,0.010278,0.011069,0.01,0.011102,0.009135,0.008233,0.007137,0.008473,0.012464,0.008824,0.005947,0.005835,0.002622,0.000786,0.002308,0.000674,0.000544,0.000865,0.009932,0.007451,0.005018,0.002343,0.000725,0.000296,0.001476,0.002691,0.001764,0.006032,0.007904,0.005821,0.0067,0.00653,0.015029,0.016244,0.01558,0.015998,0.01388,0.01051,0.01081,0.011046,0.007678,0.010714,0.0086,0.006325,0.004318,0.001124,0.001738,0.001671,0.003059,0.000833,0.004134],[0.005376,0.005775,0.005785,0.010952,0.011661,0.015217,0.019222,0.020931,0.023384,0.025527,0.02856,0.031136,0.033812,0.038737,0.041561,0.041613,0.043028,0.043051,0.047005,0.047843,0.051064,0.052239,0.052858,0.054375,0.054768,0.057989,0.05905,0.063599,0.067101,0.071536,0.073461,0.076743,0.082214,0.083745,0.08779,0.090531,0.100309,0.101803,0.10186,0.111279,0.111487,0.117711,0.122896,0.122958,0.127677,0.133322,0.134896,0.137056,0.139567,0.140655,0.146503,0.146851,0.14954,0.149726,0.152634,0.154956,0.159416,0.159453,0.161426,0.17005,0.174741,0.174951,0.176264,0.180975,0.183324,0.185491,0.188116,0.190416,0.191578,0.196403,0.196539,0.196851,0.201539,0.201821,0.202787,0.20553,0.209012,0.211871,0.212394,0.213339,0.214025,0.215554,0.217478,0.218996,0.219776,0.219989,0.221104,0.224966,0.225229,0.225557,0.225978,0.227191,0.227799,0.229108,0.232412,0.235638,0.237936,0.242322,0.246027,0.250356,0.251817,0.253321,0.257027,0.257051,0.257479,0.258985,0.259768,0.260216,0.260359,0.261763,0.263524,0.264275,0.267899,0.272646,0.27351,0.275197,0.276066,0.279443,0.284021,0.297338,0.297545,0.298377,0.298593,0.299818,0.301891,0.306965,0.308136,0.309802,0.324255,0.327323,0.330847,0.333428,0.365945,0.368526,0.37288,0.377919,0.379829,0.382051,0.39688,0.420836,0.423913,0.430558,0.440244,0.460418,0.472377,0.472666,0.485661,0.486848,0.49274,0.496208,0.501834,0.51339,0.514189,0.516206,0.531804,0.537691,0.551934,0.558751,0.560245,0.564581,0.5682,0.613114,0.616693,0.622256,0.624251,0.625019,0.626335,0.633932,0.635771,0.638813,0.670542,0.680876,0.68538,0.686005,0.689706,0.691489,0.691986,0.693372,0.69416,0.695039,0.695495,0.698087,0.700596,0.703758,0.706862,0.709003,0.710755,0.711835,0.715019,0.71519,0.716885,0.71876,0.720086,0.720907,0.721125,0.723121,0.725041,0.731092,0.731396,0.731435,0.731975,0.732026,0.73313,0.739036,0.741518,0.745452,0.748569,0.761828,0.762015,0.763502,0.763763,0.766233,0.768572,0.770988,0.773043,0.77388,0.778224,0.780448,0.780957,0.782368,0.78565,0.794757,0.79824,0.798844,0.801962,0.802736,0.803486,0.803978,0.806758,0.807358,0.813282,0.815152,0.815753,0.817946,0.818776,0.819666,0.819903,0.826123,0.828901,0.83643]],"container":"<table class=\"display\">\n <thead>\n <tr>\n <th> <\/th>\n <th>Arrival<\/th>\n <th>NrQueue<\/th>\n <th>Service<\/th>\n <th>Waiting<\/th>\n <th>System<\/th>\n <th>Departure<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"pageLength":5,"columnDefs":[{"className":"dt-right","targets":[1,2,3,4,5,6]},{"orderable":false,"targets":0}],"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[5,10,25,50,100]},"selection":{"mode":"multiple","selected":null,"target":"row"}},"evals":[],"jsHooks":[]}</script> (El archivo original tiene 250.000 filas) --- El análisis de estos datos produce las siguientes figuras  --- La tabla siguiente muestra un resumen de las variables anteriores: <table> <thead> <tr> <th style="text-align:right;"> Mean time between arrivals </th> <th style="text-align:right;"> Mean service time </th> <th style="text-align:right;"> Mean waiting time </th> <th style="text-align:right;"> Mean Nr in queue </th> <th style="text-align:right;"> Max Nr in queue </th> </tr> </thead> <tbody> <tr> <td style="text-align:right;"> 0.0026891 </td> <td style="text-align:right;"> 0.0016028 </td> <td style="text-align:right;"> 0.0099626 </td> <td style="text-align:right;"> 6.225505 </td> <td style="text-align:right;"> 106 </td> </tr> </tbody> </table> Nótese: * El elevado tiempo de espera en cola frente al tiempo de servicio: resulta aconsejable aumentar la velocidad de servicio o instalar más servidores. * Por término medio, cada paquete antes de ser enviado debe esperar por algo más de 6 paquetes, lo que ralentiza notablemente el sistema. * El tamaño máximo de cola es de 106 paquetes; si el sistema no dispone de un buffer de capacidad adecuada, se perderían muchos paquetes de datos. <br> .red[__¡Es importante saber analizar este sistema para poder optimizarlo!__] --- ### El papel del azar: un día cualquiera en la ciudad .pull-left[   ] .pull-right[ * ¿Cuántas personas hay en cada momento haciendo uso del móvil en distintos puntos de una ciudad? * ¿Cómo se mueven? ¿Caminan, van en coche? * ¿Cuánto duran sus conversaciones? * ¿Qué tipo de tráfico están generando, voz, video, datos? * ¿Cuántas antenas necesitamos? ¿De qué potencia? ¿Cuántos canales? ¿Donde las colocamos? * ¿Cuántas lineas fijas? ¿Cuál es la demanda de velocidad del usuario? ¿Fibra óptica, cable coaxial? ] --- ### El papel del azar. * Tratamiento de señales: señal sin ruido <br> <audio src="imagenes/sheeran-Photograph.mp3" controls></audio>  --- ### El papel del azar. * Tratamiento de señales: la misma señal pero con ruido constante superpuesto <br> <audio src="imagenes/sheeranRuido.mp3" controls></audio>  --- ### El papel del azar. * Tratamiento de señales: Ruido extraído de la señal anterior <br> <audio src="imagenes/ruido.mp3" controls></audio>  --- ### El papel del azar. * Tratamiento de señales: el ruido puede adoptar muchas formas. <br> <audio src="imagenes/sheeranPulse.mp3" controls></audio>  (La imagen muestra solo uno de los "clicks" que suenan en el audio) --- ### Algunos de los objetivos reseñados en el proyecto docente. __1.__ Conocer los fundamentos de la teoría de la probabilidad. -- __3.__ Conocer y ser capaz de identificar en la práctica las distribuciones de probabilidad de las variables aleatorias adecuadas para modelar problemas de aplicación en Telecomunicaciones. -- __5.__ Conocer las cadenas de Markov y reflexionar sobre su aplicación en el ámbito de las telecomunicaciones. -- __6.__ Distinguir y modelar distintos sistemas de colas. -- __8.__ Comprender los fundamentos del Análisis Espectral y ser capaz de aplicarlos a problemas en el campo de las telecomunicaciones, en particular a aquellos relacionados con el tratamiento y filtrado de señales. -- __9.__ Asumir la necesidad y utilidad de la Estadística y los Procesos Estocásticos como herramienta en su futuro ejercicio profesional. ---  <br> __En resumen:__ el objetivo principal de la asignatura es dotar al futuro graduado en ingeniería de Telecomunicación de los conocimientos y herramientas básicos necesarios para entender y modelar adecuadamente el efecto del azar en los ámbitos de su competencia. --- background-image: url(http://www.dma.ulpgc.es/profesores/personal/asp/images/TelecoHome.jpg) background-size: cover background-position: 15% 45% class: inverse, bottom, left # 3. Temario --- ## Temario __Capítulo 1. Espacios de Probabilidad__ Fenómenos aleatorios. Diferentes conceptos de probabilidad. Espacios muestrales. Axiomática y propiedades de la medida de probabilidad. Espacios finitos de probabilidad: introducción a la combinatoria. Probabilidad condicional. Independencia de sucesos. Teoremas notables de la probabilidad. <br> -- __Capítulo 2. Variables aleatorias__ Concepto de variable aleatoria y su distribución de probabilidad: función de distribución. Distribuciones discretas. Distribuciones absolutamente continuas: función de densidad de probabilidad. Momentos de una distribución de probabilidad. Función característica: cálculo de momentos. Simulación de variables aleatorias. --- ## Temario __Capítulo 3. Vectores aleatorios__ Distribución de probabilidad conjunta de dos variables aleatorias. Distribuciones marginales. Distribuciones condicionales. Independencia estocástica. Correlación lineal. Distribuciones multivariantes. Sucesiones de variables aleatorias independientes. <br> -- __Capítulo 4. Procesos de Markov__ Recorridos aleatorios. Condición de Markov. Cadenas homogéneas de Markov. Matriz de probabilidades de transición. Distribuciones estacionarias. Comportamiento límite de cadenas de Markov. --- ## Temario __Capítulo 5. Procesos de nacimiento y muerte__ El proceso homogéneo de Poisson. Proceso de nacimiento puro. Proceso de Yule. Procesos de nacimiento y muerte. Sistemas de colas markovianos. Condiciones de equilibrio. <br> -- __Capítulo 6. Procesos estacionarios __ Estacionariedad. Funciones de autocovarianza y autocorrelación. Procesos especiales: puramente aleatorios, autorregresivos (AR), medias móviles (MA), ARMA y armónicos. Análisis espectral. Transformaciones lineales y problemas de filtraje. --- background-image: url(http://www.dma.ulpgc.es/profesores/personal/asp/images/TelecoHome.jpg) background-size: cover background-position: 15% 45% class: inverse, bottom, left # 4. Por último ... --- .center[] ### Consejos prácticos. * Traten de llevar la asignatura al día. * No sean tímidos, ¡pregunten lo que no entiendan! * Usen las tutorías del profesorado. * Consulten material: libros, contenido online... * Resuelvan todos los problemas planteados en la guía de la asignatura. --- background-image: url(http://www.dma.ulpgc.es/profesores/personal/asp/images/TelecoHome.jpg) background-size: cover background-position: 15% 45% class: inverse, bottom, left # ¡¡Ánimo y <br> buena suerte!!