From 23949b97a2e0cbe445ecca026292e2def450dba3 Mon Sep 17 00:00:00 2001 From: fede Date: Thu, 21 May 2026 12:30:28 -0300 Subject: [PATCH] termine el resumen de esto --- Modelar/resumen.parcial.org | 317 ++++++++++++++++++++++++++++++++++++ Modelar/resumen.parcial.pdf | Bin 0 -> 281698 bytes 2 files changed, 317 insertions(+) create mode 100644 Modelar/resumen.parcial.pdf diff --git a/Modelar/resumen.parcial.org b/Modelar/resumen.parcial.org index 1e27d02..d961b36 100644 --- a/Modelar/resumen.parcial.org +++ b/Modelar/resumen.parcial.org @@ -155,3 +155,320 @@ Diseñados para simulación, como GPSS, Simula o Arena. - Discreta: cambios por eventos. - Continua: cambios continuos mediante ecuaciones diferenciales. - Mixta: combina ambas. + +* Clase 5 - Teoria de colas parte 1 +** ¿Que es la teoria de colas? +Esta consiste en un estudio matematico de los sistemas de espera para optimizar recursos y reducir tiempos de demora. + +** Origen +Fue desarrollado en 1909 al analizar la congestion de llamas telefonicas en copenhague. + +** Elementos +Tiene 3 elementos basicos +- Clientes: Quienes solicitan el servicio. +- Area de espera: Lugar fisico o cirtual donde esperan. +- Servidores: quienes brindan el servicio. + +** Notacion de Kendall +Describe sistemas de colas de forma compacta: + +*** Formato +`A / B /C ` +donde: +- A, distribucion de llegadas. +- B, distribucion de servicio. +- c, cantidad de servidores. + +*** Tipos de principales +- M, markoviana (Poisson/exponencial). +- D, Deterministica. +- G, General. + +*** Ejemplos +- M/M/1, llegadas Poisson, servicio exponencial y 1 servidor. +- M/M/c, varios servidores. +- M/D/1, servicio constante. +- D/D/1, llegadas y servicios contantes. + +*** Disciplinas de servicio +- FIFO, primero en llegar, primero en salir. +- LIFO, ultimo en llegar, primero en salir. +- Prioridad, atencion según importancia. + +** Sistemas M/M/1 +Es el modelo más basicos para modelar. + +*** Caracteristicas +- Llegadas Poisson. +- Servicio exponencial. +- Un solo servidor. +- Cola FIFO. +- Entradas y salidas independientes. + +*** Parametros Importantes +- Tasa de arrivos + $\lambda$ + Consiste en la cantidad promedio de clientes que llegan por unidad de tiempo. + +- Tasa de servicio + $\mu$ + Es la cantidad promedio de clientes que el servidor atiende cada una unidad de tiempo + +- Tiempo medio de servicio + + $T_s = \frac{1}{\mu}$ + +*** Factor de utilizacion +$\rho = \frac{\lambda}{\mu}$ + +indica que tan ocupado esta el servidor. + +- $\rho < 0.5$ sistema subutilizado. +- $0.5 \leq \rho < 0.8$ funcionamiento optimo. +- $\rho$ cercano a 1: colas largas. +- $\rho \geq 1$: congestion. + +Si llegan más clientes que lo que la cola puede atender, la cola crece indefinidamente. + +*** Probabilidad de Estado +Es la probablidad de tener exactamente n clientes en el sistema + +$pi_n = \rho^n (1 - \rho)$ + +teniendo en cuenta que: +- \pi_0 = sistema vacio. +- \pi_1 = un cliente. +- \pi_2 = dos clientes. +*** Numero medio de clientes +Cantidad promedio de clientes en el sistema: + +$N = \frac{\rho}{1 - \rho}$ + +Cuando \rho se acerca a 1, el numero de clientes crece muchísimo. + +*** Equilibrio y Congestión + +Un *Sistema Estable* Ocurre cuando: + +$\lambda < \mu$ + +#+begin_quote +El sistema puede atender a los clientes sin que la cola crezca infinitamente. +#+end_quote + +Mientras tanto un *Sistema Congestionado*: + +$\rho \geq 1$ + +#+begin_quote +la cola tiende a infinito. +#+end_quote + +** Ecuacion de estado estable +Relaciona probab lidades entre estados consecutivos. + +$\rho * \pi_{n-1} = \pi_n$ + +** Teorema de Little +relaciona la cantidad promedio de cleintes, junto con la tasa de llegada y el tiempo promedio del sistema $N = \lambda W$. + +Donde: +- N: clientes promedio en el sistema. +- \lambda : tasa de llegadas. +- W: tiempo promedio de permanencia. + +*** Variantes +- $Q = \lambda Wq$. clientes en cola. +- $N_s = \lambda Ts$. clientes siendo atendidos. + +* Clase 6 - Teoria de colas parte 2 +** M/M/1/N +Son modelos de colas en los cuales: +- Llegadas aleatorias tipo Poisson. +- Tiempos de servicio exponenciales. +- Un unico servidor. +- Capacidad maxima finita de N clientes. + +A diferencia del modelo M/M/1 un M/M/1/N los clientes pueden ser rechazados cuando el sistema esta lleno. + +** Caracteristicas Principales +Componentes del modelo: +- M: llegadas de poisson. +- M: Servicio exponencial. +- 1: un servidor. +- N: capacidad maxima del sistema. + +Funcionamiento: +- Si el sistema tiene menos de N clientes, el nuevo cliente entra. +- Si el sistema está lleno (n = N), el cliente es rechazado. + +Propiedades: +- llegadas de Poisson. +- Servicio Exponencial. +- Un unico servidor. +- Disciplina FIFO. +- Independencia entre entradas y salidas. +- Capacidad Limitada. + +*** Aplicaciones +El modelo M/M/1/N se utiliza en sistemas con capacidad fisica limitada. + +Ejemplos: +- Sala de espera medicas. +- Routers. +- Centrales telefonicas. +- Estacionemientos. +- Ascensores. +- Plataformas de streaming. + +En todos los casos existe un limite de capacidad y posibilidad de bloqueo. + +** Probabilidad de Bloqueo +La probablidad de bloqueo (P_B) representa la probabilidad de que u cliente llegue y encuentre el sistma lleno. + +$P_B = \pi_N = \frac{\rho^N (1 - \rho)}{1 - \rho^{N + 1}}$ + +Donde: +- $\rho = \frac{\lambda}{\mu}$. +- \lambda Tasa de llegada. +- \mu tasa de servicio. +- N, capacidad maxima. + +Un P_B bajo significa que no hay muchos clientes rechazados mientras que uno alto es que se rechazan muchos clientes. + +** Tasa de rechazo +Indica cuantos cliente son rechazados por unidad de tiempo. + +$\tau = \lambda * P_B$ + +** Rendimiento del Sistema +Es la tasa efectiva de clientes atendidos. + +Rendimiento de entrada: +$\gamma = \lambda (1 - P_B)$ + +Rendimiento de salida: +$\gamma = \mu (1 - \pi_0)$ +** Deduccion de la formula de Probablidad +En un Modelo M/M/1/N solo existen los estados: +#+begin_center +$n = 0,1,2,...,N$ +#+end_center + +Por esto la distribucion de probabilidad debe corregirse para que la suma total sea 1. +** Comparación con M/M/1 +| Caracteristicas | M/M/1 | M/M/1/n | +|------------------------+-------------------+------------------| +| Capacidad | Infinita | Finita | +| Rechazo | No | Si | +| Probablidad de bloqueo | 0 | x>0 | +| Estados | Infinitos | 0 a N | +| Rendimiento | \gamma = \lambda | \gamma < \lambda | +| Estabilidad | Requoiere \rho <1 | Siempre estable | + +** Ventajas del Modelo M/M/1/N +- Representa sistemas reales con límites físicos. +- Permite calcular pérdidas por rechazo. +- Ayuda al dimensionamiento de capacidad. +- Evita crecimiento infinito de colas. + +** Desventajas +- Puede generar pérdida de clientes. +- Afecta la reputación del servicio. +- Requiere cálculos más complejos. +- Definir capacidad adecuada puede ser difícil. + +* Clase 7 - Teoria de colas parte 3 +Introduccion a los sistemas dependientes de estado, Estos son modelos de colas donde las tasas de llegadaa y servicio no son constantes, sino que cambian segun la cantidad de clientes presentes en el sistema. + + +En estos sistemas \lambda_n representa la tasaq de arrivos cuando hay n clientes. y \mu_n representa la tasa de servicio cuando hay n clientes. + +** Concepto de Dependencia +En estos modelos, las condiciones del sistema responde dinamicamente al estado actual. + +por ejemplo: +En una autopista, el transito es fluido y rapido. Con muchos vehiculos aparece congestion y dismunuye la velocidad. Cuando la autopista esta saturada, algunos conductores buscan rutal alternas. + +De Forma similar en un sistemas de colas las tasas de llegada y servicio pueden variar según la cantidad de clientes presentes. + +** Analogia con sistemas de nacimiento y muerte +Los sistemas dependientes de estado tambien se conocen como sistemas de nacimiento y muerte debido a su similitud con modelos poblacionales. + +*** Nacimientos +En la poblacion es Nacimientos y en las colas es \lambda_n. + +*** Muertes +representan las salidas del sistema: +- En poblacion, Muertes. +- En colas, servicios completados \mu_n. + +** Deduccion de probabilidades de estado +*** Principio de Balance +El analisis se basa en el principio de balance de flujo, en estado estable, la tasa de entradas a yun estado debe ser igual a la tasa de salidas de ese estado. + +*** Desarrollo de formula +Estado 0, el equilibrio entre entradas y salidas. + +$\mu_1 \pi_1 = \lambda_0 \pi_0$ + +Despejado + +$\mu_1 = \frac{\lambda_0}{\mu_1}\pi_0$ + + + +* Clase 8 - M/G/1 y M/D/1 +** M/G/1 +- Llegadas Poisson. +- Tiempo de servicio General +- 1 Solo servidor + + Surge porque el modelo M/M/1 asume tiempos exponenciales, algo que no es realista en muchos sistemas. + +*** Condicion de estabilidad +$\rho = \frac{\lambda}{\mu} < 1$ + +\rho representa el porcentaje de tiempo que el servidor esta ocupado. + + +*** Formula +Permite calcular clientes esperados y tiempo esperado en el sistema + +$E(n) = (\frac{\rho}{1 - \rho}) [1 - \frac{\rho}{2} (1 - \mu^2 o^2)]$ + +*** Interpretacion \mu^2 o^2 +Equivale a CV^2 (coeficiente de variacion al cuadrado). + +Donde si: +- CV^2 = 1, comportamiento M/M/1. +- CV^2 = 0, servicio deterministico M/D/1. +- CV^2 > 1, mucha variabilidad, peores colas. + +** M/D/1 +- Llegadas Poisson. +- Tiempo de servicio deterministico. +- 1 servidor. + +es el sistema más eficiente para un solo servidor. + + +$E(n) = (\frac{\rho}{1 - \rho})(1 - \frac{\rho}{2})$ + +** Sistemas sin interrupcion +Es un sistema que termina un gtrabajo antes de empezar uno con prioridad más alta. + +$W_1 = W_0 + \frac{Q_1 + 1}{\mu_1}$ + +$Wq_1 = W_0 + \frac{Q_1}{\mu_1}$ + +** Sistemas con interrupcion +Si llega alguien de prioridad alta se interrumpe inmediatamente el trabajo actual y el nuevo cliente toma el servidor. + +$W_1 = W_{01} + \frac{Q_1 + 1}{\mu_1}$ + + +** Inanicion +Clientes/procesos de baja prioridad nunca reciben atención porque siempre llegan otros más prioritarios. + +El sistema sigue funcionando pero abandona gente en la cola eternamente. 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