From b3033a6a34beac57b03893e8a96a766752ba5cfa Mon Sep 17 00:00:00 2001 From: fede Date: Wed, 24 Jun 2026 16:10:36 -0300 Subject: [PATCH] probando notebooklm --- Modelar/resumen.parcial.2.org | 401 ++++++---------------------------- Modelar/resumen.parcial.2.pdf | Bin 0 -> 229892 bytes 2 files changed, 64 insertions(+), 337 deletions(-) create mode 100644 Modelar/resumen.parcial.2.pdf diff --git a/Modelar/resumen.parcial.2.org b/Modelar/resumen.parcial.2.org index d6e0f71..cb513af 100644 --- a/Modelar/resumen.parcial.2.org +++ b/Modelar/resumen.parcial.2.org @@ -1,373 +1,100 @@ -#+title: Resumen Segundo Parcial -#+author: Fede -#+email: federico.nicolas.polidoro@gmail.com +#+title: Resumen Parcial 2 +#+author: fede & notebooklm -* M/G/1 -- Llegadas Poisson. -- Tiempo de servicio General -- 1 Solo servidor +* Intro +El análisis de los sistemas de colas permite identificar cuellos de botella, optimizar recursos y mejorar la experiencia del usuario mediante la reducción de los tiempos de espera. Los hallazgos principales destacan que: - Surge porque el modelo M/M/1 asume tiempos exponenciales, algo que no es realista en muchos sistemas. +- Eficiencia del Sistema M/M/2 +La transición de un sistema de servidor único (M/M/1) a uno doble con fila única reduce drásticamente el tiempo de permanencia, superando en eficiencia a la configuración de dos colas separadas. -*** Condicion de estabilidad -$\rho = \frac{\lambda}{\mu} < 1$ +- Impacto de la Variabilidad (M/G/1) +En sistemas con un solo servidor, la varianza del tiempo de servicio es un factor crítico; a mayor variabilidad, mayor es la longitud de la cola, incluso si la velocidad media se mantiene constante. -\rho representa el porcentaje de tiempo que el servidor esta ocupado. +- Criterios de Intervención +La decisión de añadir servidores (especialmente si son de distinta velocidad) depende de un valor límite de utilización denominado "Rho Crítico" (\rho_c). +- Gestión de Prioridades +Los sistemas no expulsivos permiten dar atención preferencial a clases de clientes específicas sin interrumpir el servicio en curso. -*** Formula -Permite calcular clientes esperados y tiempo esperado en el sistema +* El Modelo M/M/2: Eficiencia y Estructura -$E(n) = (\frac{\rho}{1 - \rho}) [1 - \frac{\rho}{2} (1 - \mu^2 o^2)]$ +El modelo M/M/2 se define por tener arribos que siguen un proceso de Poisson, tiempos de servicio exponenciales y dos servidores que trabajan en paralelo atendiendo una fila única. -*** Interpretacion \mu^2 o^2 -Equivale a CV^2 (coeficiente de variacion al cuadrado). +** Características y Beneficios +- Detección de Cuellos de Botella: Permite identificar dónde se satura el proceso. +- Sensación de Equidad: Al utilizar una fila única, se evita la percepción de que otras filas avanzan más rápido, lo cual es común en sistemas de filas múltiples. +- Uso de Recursos: Maximiza la ocupación de los servidores, evitando que uno permanezca ocioso mientras el otro tiene clientes en espera. -Donde si: -- CV^2 = 1, comportamiento M/M/1. -- CV^2 = 0, servicio deterministico M/D/1. -- CV^2 > 1, mucha variabilidad, peores colas. +** Parámetros Críticos y Estabilidad -* Sistema M/M/2 -*** Definición -- Modelo de teoría de colas con: +Para que el sistema sea estable, se debe cumplir la condición \lambda < 2\mu (la tasa de llegadas debe ser menor a la capacidad total de servicio). - - llegadas Poisson - - tiempos de servicio exponenciales - - 2 servidores en paralelo - - una única cola +| Característica | M/M/1 | M/M/2 | +|------------------------------+---------------------------+---------------------------| +| Número de servidores | 1 | 2 | +| Intensidad de tráfico (\rho) | $\frac{\lambda}{\mu}$ | $\frac{\lambda}{2\mu}$ | +| Tiempo en el sistema (W) | $\frac{1}{(\mu(1-\rho))}$ | $\frac{1}{\mu(1-\rho^2)}$ | +| Tiempo de espera | Mayor | Menor | +| | | | -*** Notación de Kendall -Formato: =A/B/s= +* Comparativa Estratégica: M/M/1 vs. M/M/2 -- Primera =M= → arribos Markovianos (Poisson) -- Segunda =M= → servicios Markovianos (Exponencial) -- =2= → dos servidores +La intervención en un sistema saturado es necesaria cuando las colas aumentan de forma sostenida o el servidor está cerca de su límite. -** Características -- Llegadas aleatorias independientes -- Servicio exponencial -- Cola única compartida -- Disciplina FIFO -- Capacidad infinita -- Población infinita -- Dos servidores trabajando en paralelo +Diagnóstico de Intervención -*** Servidores iguales -- Cada servidor atiende a tasa =μ= +- Sistema Aceptable: E(n) < 3 clientes. +- Necesita Intervención: E(n) > 5 clientes. -- Servicio total: +Caso de Ejemplo: Bajo una tasa de arribos \lambda = 8 y servicio \mu = 10: - - =μs = 2μ= +1. En M/M/1, el tiempo medio en el sistema es de 30 minutos. +2. Al pasar a M/M/2, el tiempo medio se reduce a 7.1 minutos. -- Intensidad de tráfico: +* Servidores Heterogéneos y Selección de Servidor - - =ρ = λ / (2μ)= +Cuando se dispone de un servidor rápido (\mu_1) y uno lento (\mu_2), la decisión de añadir el segundo servidor depende del Rho Crítico (\rho_c). -*** FIFO -- Una sola fila -- El próximo cliente va al primer servidor libre +- Sin selección de servidor: Los clientes entran al primer servidor disponible. +- Con selección de servidor: Existe una política para decidir a qué servidor enviar al cliente. +- Regla de Decisión: Si \rho \geq \rho_c, es conveniente agregar el segundo servidor. Si \rho < \rho_c, el sistema funciona mejor con uno solo. -*** Diferencia con =2·(M/M/1)= -**** M/M/2 -- 1 cola -- 2 servidores compartidos +* El Modelo M/G/1 y la Fórmula de Pollaczek-Khinchine -**** 2·(M/M/1) -- 2 colas separadas -- cada servidor tiene su propia fila +A diferencia de los modelos anteriores, el M/G/1 permite que el tiempo de atención tenga cualquier distribución, requiriendo únicamente conocer su media (E[S]) y varianza (Var[S]). -M/M/2 es más eficiente. +** La Importancia de la Varianza -** Condición de estabilidad -Para que no colapse: +La fórmula de Pollaczek-Khinchine para el número esperado de clientes en cola (E[Lq]) demuestra que la variabilidad es perjudicial: $E[Lq] = \frac{\rho^2 + \lambda^2 \cdot Var[S]}{2(1-\rho)}$ -\rho = \frac{\lambda}{2\mu} < 1 +- M/D/1 (Servicio constante): Posee \sigma = 0, representando la referencia de mínima congestión. +- M/M/1 (Servicio exponencial): Representa la referencia de máxima congestión por su alta variabilidad inherente. -Entonces: +** Aplicaciones de M/G/1 -- =λ < 2μ= +Este modelo es fundamental en entornos donde los tiempos de tarea varían según el caso, tales como: -** ¿Para qué sirve? -*** Optimización de recursos -- Reducir tiempos de espera -- Evitar servidores innecesarios +- Hospitales: Consultas de duración variable. +- Redes / IT: Transmisión de paquetes de distinto tamaño. +- Manufactura: Piezas que requieren distintos tipos de trabajo. -*** Análisis de rendimiento -- Detectar cuellos de botella -- Medir utilización -- Comparar escenarios +* Sistemas con Prioridades (No Expulsivos) -*** Mejora continua -- Evaluar cambios antes de implementarlos +Este modelo se aplica cuando existen clientes de Clase 1 (prioritarios) y Clase 2. La prioridad es no expulsiva, lo que significa que no se interrumpe al cliente que ya está siendo atendido. -- Decidir si conviene: +** Orden de Cálculo para el Tiempo de Espera (T_q) - - agregar servidores - - aumentar velocidad - - reorganizar la cola +Un cliente de clase 1 debe esperar: -*** Aplicaciones -- Bancos -- Supermercados -- Call centers -- Hospitales -- Sistemas informáticos -- Peajes -- Aeropuertos +1. El residuo del tiempo del cliente que ya está en servicio (independientemente de su clase). +2. El tiempo de servicio de todos los clientes de clase 1 que ya estaban en la fila. +3. Posteriormente, recibe su propia atención. -** Ventajas -- Fórmulas simples -- Fácil de analizar -- Permite planificación -- Muy aplicable -- Mayor tolerancia a fallos +* Casos Prácticos y Modelización Numérica -** Desventajas -- Supuestos poco realistas -- Ignora comportamiento humano -- No encuentra automáticamente la solución óptima -- No contempla eventos externos +Las fuentes detallan diversos escenarios de aplicación real: -** Fórmulas principales -*** Parámetros base -- =λ= → tasa de llegadas -- =μ= → tasa de servicio de UN servidor -- =μs = 2μ= -- =ρ = λ / (2μ)= - -*** Número esperado en el sistema -E(n)=\frac{2\rho}{1-\rho^2} - -- Promedio de clientes esperando + siendo atendidos - -*** Tiempo esperado en el sistema -E(T)=\frac{1}{\mu(1-\rho^2)} - -- Tiempo promedio dentro del sistema - -*** Teorema de Little -N=\lambda W - -- Relación entre: - - - clientes promedio - - tasa de llegada - - tiempo promedio - -** M/M/1 vs M/M/2 -| Característica | M/M/1 | M/M/2 | -|--+---------+----------| -| Servidores | 1 | 2 | -| Colas | 1 | 1 | -| Tráfico | =λ/μ= | =λ/2μ= | -| Espera | Mayor | Menor | -| Costo | Menor | Mayor | -| Estabilidad | =λ < μ= | =λ < 2μ= | - -*** Ejemplo -**** Datos -- =λ = 8= -- =μ = 10= - -**** M/M/1 -- =ρ = 0.8= -- =E(T) = 30 min= - -**** M/M/2 -- =ρ = 0.4= -- =E(T) ≈ 7.1 min= - -** M/M/2 con servidores distintos -*** Caso -- un servidor rápido -- otro lento - -*** Fórmulas -\mu_s = \mu_1 + \mu_2 - -\rho = \frac{\lambda}{\mu_1+\mu_2} - -*** Ventajas -- solución rápida ante aumento de demanda -- menor costo -- buena transición temporal - -*** Problema -- percepción de desigualdad: - - - algunos clientes avanzan más rápido que otros según el servidor - asignado - - -* Sistemas con Prioridades - -Los sistemas con prioridades son modelos de colas en los que los -clientes se clasifican según niveles de importancia o urgencia. El -servidor no necesariamente atiende por orden de llegada, sino que -selecciona primero a los clientes de mayor prioridad. - -Las prioridades suelen organizarse en clases jerárquicas: - -- Clase 1 → mayor prioridad -- Clase 2 → prioridad intermedia -- Clase 3 → menor prioridad - -Cada clase posee su propia cola de espera. El servidor siempre -selecciona clientes de la cola de mayor prioridad disponible. - -** Sistema sin interrupción (Non-Preemptive) - -En este esquema, un cliente que ya está siendo atendido siempre finaliza -su servicio. Aunque llegue un cliente de prioridad superior, el servidor -no interrumpe la atención actual. - -La prioridad únicamente afecta la selección del próximo cliente cuando -el servidor queda libre. - -Tiempo de espera en cola para clase 1: - -W_{q1}=W_0+\frac{Q_1}{\mu_1} - -Tiempo total en el sistema para clase 1: - -W_1=W_0+\frac{Q_1+1}{\mu_1} - -Características: - -- No existe interrupción del servicio en curso -- Menor complejidad operativa -- Mayor estabilidad del procesamiento -- Los clientes prioritarios pueden seguir esperando si el servicio - actual es largo - -** Sistema con interrupción (Preemptive) - -En este esquema, si llega un cliente de prioridad superior, el servidor -interrumpe inmediatamente al cliente actual y comienza a atender al -nuevo cliente prioritario. - -El cliente desplazado vuelve a la cola o reinicia el servicio según la -política utilizada. - -Tiempo de espera en cola para clase 1: - -W_{q1}=W_{01}+\frac{Q_1}{\mu_1} - -Tiempo total en el sistema para clase 1: - -W_1=W_{01}+\frac{Q_1+1}{\mu_1} - -Características: - -- Minimiza la espera de clientes urgentes -- Aumenta la complejidad del sistema -- Puede producir sobrecarga por interrupciones frecuentes -- Puede perjudicar severamente a clases bajas - -** Inanición (Starvation) - -La inanición ocurre cuando clientes de baja prioridad permanecen -indefinidamente en espera debido a la llegada continua de clientes de -mayor prioridad. - -Consecuencias: - -- Espera potencialmente infinita -- Falta de equidad -- Degradación del rendimiento global -- Bloqueo práctico de ciertas tareas - -Mecanismos de mitigación: - -- Aging -- Límites de interrupciones -- Cuotas mínimas de servicio -- Aumento dinámico de prioridad - -* Sistemas Complejos - -Un sistema complejo es un conjunto de componentes interrelacionados cuya -interacción genera comportamientos globales difíciles de predecir -mediante el análisis individual de cada componente. - -El comportamiento del sistema surge de las interacciones entre sus -partes y no únicamente de las propiedades de cada elemento aislado. - -** Características principales - - - -- Gran cantidad de componentes -- Interacciones simultáneas -- Dependencias mutuas -- Retroalimentación (feedback) -- Dinámica no lineal -- Adaptación continua -- Comportamiento emergente -- Sensibilidad a perturbaciones - - - -** Propiedades fundamentales - - - -*** Emergencia - - - -El sistema presenta propiedades globales que no existen en los -componentes individuales. - - - -*** No linealidad - - - -Pequeñas variaciones pueden producir cambios desproporcionadamente -grandes en el comportamiento global. - - - -*** Autoorganización - - - -El sistema puede generar orden interno sin control centralizado. - - - -*** Adaptabilidad - - - -El sistema modifica su comportamiento frente a cambios del entorno. - - - -** Relación con teoría de colas - - - -Los sistemas complejos frecuentemente incorporan: - -- múltiples colas, -- prioridades, -- asignación dinámica de recursos, -- interrupciones, -- competencia entre procesos. - -La interacción entre estos elementos puede generar: - -- congestión, -- cuellos de botella, -- propagación de fallos, -- inestabilidad, -- inanición, -- saturación sistémica. - -La teoría de colas se utiliza para modelar y analizar partes específicas -de estos sistemas, especialmente aquellas relacionadas con espera, -utilización de recursos y rendimiento global. +- Fábrica de Periscopios: Un sistema de dos procesos consecutivos (corte y doblado) que requiere el cálculo de probabilidades de estado para determinar la saturación en cada etapa. +- Sector de Cajas de Supermercado: Operación mediante fila única que conduce a dos cajas registradoras idénticas. Se analizan métricas como la probabilidad de sistema vacío y el tiempo promedio en cola. +- Enrutamiento de Paquetes: Uso del Teorema de Little (L = \lambda \cdot W) para despejar la tasa de arribos y determinar el tiempo de permanencia en buffers de routers con alta capacidad de servicio. +- Sistemas de IA y APIs: Aplicación del modelo M/G/1 para gestionar solicitudes de backend donde la variabilidad del servicio es alta. diff --git a/Modelar/resumen.parcial.2.pdf b/Modelar/resumen.parcial.2.pdf new file mode 100644 index 0000000000000000000000000000000000000000..66881a8a747bc1eec7d85bbd3d7e9198dbafafbf GIT binary patch literal 229892 zcmY!laBR8|4K6cN1p|frq%1BQ8!mmH)V%bP3;L>-?%qdAN zQqXtHNi0cqNlngAN#(M$<0>vGN=?k=s<^c`%sc;%nZUl^;V(QGC!X29`Ef)`a{CREd9JJ~BjcXp+UtF>*<~;w-`!%V&@MqAe7E#M|FSXtN zoZ_K=s@^8LMY^S%vM2FB3AaASetYT6gr=CKF~Y`I&TVl#aQS}u`h$I55~sD7THFfM z(mz29!McDnTnk$FL%bTx91XU=$? zDd!M#aZwcaiTCX%OU`k#Rd&U!d|}(3AelNV=S;$@lYG?+uUPE7)srCg$emx{0{88k 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