Snm Mat-347 Simulacion Ev Discretos

8/12/2019 Snm Mat-347 Simulacion Ev Discretos

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Gua de Referencia

MAT-347SIMULACION

M.Sc. Ing. Sandro Nieto Mndez

2014


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Generacin de VariablesAleatorias


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Variables Aleatorias

Solucin:

Generar variables aleatorias que sigan algn tipode distribucin que mejor represente ese suceso.

Transformar un nmero aleatorio, en una variablealeatoria.

Adems de tener como entrada valoresconstantes (parmetros, condicionesexperimentales, etc.), para reproducir elcomportamiento de un sistema no determinstico

(estocstico), un modelo de simulacin requierevalores de variables aleatorias, cuya calidadincidir en la calidad de los resultados obtenidoscomo salida del modelo.


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Mtodos para generar valores de

variables aleatorias

Mtodo de la Transformada Inversa


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La funcin dedensidad

La funcin dedistribucinacumulativa

)()( xXPxf

)()( xXPxF


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xx

t

t

xxPxF )()(

dtxPxF t)()(

Caso Discreto

Caso Contnuo

Luego hacer:

)()( 1 uFxuxF


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F(x)

1

u

0x

)(1 uFx


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Variables Discretas

Distribucin de Bernoulli

casootroen

xp

xp

xXPxf

0

1

01

)()(

Cual ser la Funcin de Distribucin Acumulada?


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Variables Discretas

Distribucin de Bernoulli

11

101

00

)()(

xxp

x

xXPxF

El algoritmo se reduce a:

Generar U ~Uni(0,1).Escogerx = 0 si F(-1) U < F(0)

0 U < 1- p

Escogerx = 1 si F(0) U < F(1)

1- p U < 1

1-p

1

10 2

u

u

0


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Variables Continuas

La distribucin exponencial

0)( xexf x

Cual ser la Funcin de Distribucin Acumulada?

Cmo se calcular x?


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Variables Discretas- Poisson

ei

iXPifi

!)()(

)(*1)1(

!3*

4!4)4(

!2*

3!3)3(

1*

2!2)2(

1)1(

)0(

34

23

2

iXPi

iXP

eeXP

eeXP

eeXP

eXP

eXP

Desarrollando:

n

i

iXPiF0

1)()(


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Algoritmo

1. Generar U

2. i = 0; p = ; F = p

3. Mientras F


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Un proceso de Poisson de intensidad, corresponde a una serie deeventos tal que los intervalos detiempo entre eventos sucesivos sonvariables aleatorias independientesexponenciales de parmetro .

T=1

e1 e2 e3 e4 e5 e60 tiempo

d.exp d.exp d.exp d.exp d.exp d.exp


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Algoritmo

1. i=0

2. t=-1/ *ln(U)3. Mientras t


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Variables Continuas

Uniforme en el intervalo (a,b)

La distribucin uniforme en elintervalo (a,b) (U(a,b)) tiene comofuncin de densidad la siguiente:


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Variables Continuas

Uniforme en el intervalo (a,b)

Algoritmo

1. Introducir a, b

2. Generar u

3. x=a+(b-a)*u

4. retornar x

uabax

ab

axu

xFu

)(

)(

Haciendo:


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Variables Continuas

Distribucin Normal

12

21

n

n

uZ

n

i

i

xZ


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Variables Continuas

Distribucin Normal

Aprovechando que la suma de 12 variablesaleatorias [0,1], tiene el valor esperado de6 y varianza 1.

12

1

1

6*

:Entonces12

2*

i

i

n

i

i

ux

n

nu

x

Algoritmo:

1. Introducir S (valor esperado)

2. Introducir V (desviacin estandar3. suma =0

4. Para k = 1 hasta 12 hacer

5. suma = suma + u(0,1)

6. retornar s+(v *(suma-6))

x=0

PARA i=1 HASTA 12

Genera u ( nmero aleatorio en (0,1))

x=x+u (para conseguir una varianza 1)

FIN_PARA

x=x-6 (para conseguir una media 0)

Salida x


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Types of simulation


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Types of simulation

There are three major ways to approach discrete

simulation.

Each approach offers a different way to look at a

simulation problem. Each, in its own way, suggestsmechanisms to model real situations.

These are:

event scheduling,

activity scanning,

and process interaction ( object orientation).


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Event Scheduling

An eventis anything that changes the system

statistics (also known as the stateof the

system) other than the passage of time.

The essential idea of event scheduling is to

move along the time scale until an event

occurs and then, depending on the event,

modify the system state and possiblyschedule new events.


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Event Scheduling-Algorithm

ALGORITMO PARA EVENT SCHEDULING

Inicio ES

perform initialization (set beginning_time and ending_time, file initial events

into event_list, initialize component state descriptors);

retrieve (and remove) event notice with earliest time from event_list

the event notice consists of : (event_time, event_type,component_indicator);

TIME:= event_time;

while (TIME < ending_time) do

case event_type of

1: execute event_routine_1;

2: execute event_routine_2;

.....................

n: execute event_routine_n;

endcase;retrieve (and remove) event notice with earliest time from event_list;

TIME:= event_time;

endwhile;

endES.

cada procedimiento event_routine_i modeliza la transicin de estado del sistema

resultante de los cambios de estado de las componentes provocados por la ocurrencia de un

suceso de tipo i.

Step 1:

Remove the event notice for

the imminent event from FEL

Step 2:

Advance CLOCK to imminent

event time

Step 3:

Execute imminent event,

update system state

Step 4:

Generate future events (ifnecessary) and place their

event notices

on FEL ranked by event time.

Step 5:

Update cumulative statistics

and counters


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Event

Scheduling

Flowchart


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Event Scheduling-Example

Ar r ival Event

1. Check the status of the ATM(s) (idle or busy)

(a) If there is an idle machine

(i) Start serving the customer, update idle status

(ii) Generate a departure event for this customer (b) If all busy, place customer in queue

2. Generate new arrival event for next customer.

Departure Event

1. Check queue (empty or not)

(a) If empty, set ATM to idle (b) If not empty then do

(i) Choose a waiting customer, start serving the customer

(ii) Generate a departure event for this customer


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Activity Scanning

Uses a fixed time increment and a rule-based

approach to decide whether any activities can

begin at each point in simulated time.

At each clock advance, the conditions for

each activity is checked and if the conditions

are true, then the corresponding activity

begins.(Condition, action).


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Activity Scanning

In this approach, events are considered to beactivities of duration zero time units, and all activitiesare classified as B or C activities, where B activitiesare bound to occur, but C activities are conditional

upon certain conditions being true. Using this classification, the B-type activities and

events can be scheduled ahead of time (just as inthe event scheduling approach). This allows creationof a FEL containing B-type activities, where

variable times can be used for these activities.Scanning to check if any C-type activities can beginhappens at the end of each time advance, after allB-type events have been completed


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Activity Scanning

ALGORITMO PARA ACTIVITY SCANNING

Inicio AS;

perform intialization (set beginning_time and ending_time, set component clock T[i]

for each ative_type component i, initialize component state descriptors);

TIME:= event_time;

while (TIME < ending_time) do

SCAN: for j:= highest_priority tolowest_priority do

set i to component with priority j;

if (T[i]= TIME and cond_routine_i evaluates true)

then execute act_routine_i;

exit to SCAN;

endif;

endfor;

TIME:= minimum of all T[i]such that T[i]> TIME;

endwhile;endAS.

Cada procedimiento cond_routine_i evalua las condiciones especficas de activacin para

la componente i, devuelve como resultado los valores cierto (true) o falso (false).

Cada procedimiento act_routine_i modeliza la transicin de estado del sistema resultante

del cambio de estado de la componente i.

Se supone que cada componente tiene una prioridad nica.


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Activity Scanning

Phase A:

Remove the imminent event from the FEL and advance the clockto its event time.

Remove any other events from the FEL that have the same

event time.Phase B:

Execute all B-type events that were removed from the FEL.

Phase C:

Scan the conditions that trigger each C-type activity and activate

any whose conditions are met. Re-scan until no additional C-type activities can begin or events

occur


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Activity Scanning-Example

In this model, there are three activities:

1. An arrival occurs

2. A service is completed

3. A service begins

The actions associated with these activities are

1. (Arrival) Put customer in queue, generate next arrival2. (Completion) Declare server idle

3. (Begin) Make server busy, remove customer from queue, generate

completion.

There is one new activity: the beginning of service. It is not triggered

by any other activity. Instead, it occurs when the following twoconditions are satisfied:

1. There are customers in the queue

2. A server is idle


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Process Interaction

The simulation model is defined in terms of entities or objectsand their life cycle as they flow through the system, demandingresources and queueing to wait for resources.

Since in this view, a process is the life cycle of one entity,constraints on different processes cause them to interact.

Those simulation languages that are written with this view inmind, allow the analyst to describe the process flow in terms ofhigh level block or network constructs, while the interactionamong processes is handled automatically.

The process interaction approaches use a variable time advance;that is, when all events and system state changes have occurredat one instant of simulated time, the simulation clock is advancedto the time of the next imminent even on FEL

ALGORITMO PARA PROCESS INTERACTION


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Process

Interaction

ALGORITMO PI ;

perform initialization (set beginning_time and ending_time, file initial events into

future events list (FEL) and current events list (CEL) as appropiate,

initialize component state descriptors);

TIME:= beginning_time;

while (TIME ending_time) do (CEL scan and time advance)

retrieve first CEL entry (i.e. event not ice);

(CEL scan)

while (last entry in CEL has not been processed) do

while (cond_seg evaluates true and current component has not

terminated and rescan not indicated) do

execute act_seg;

determine next step for this component;

endwhile;

if (current component changed steps)

then remove old CEL entry;

if (current co mponent did not terminate)

then file new CEL entry;

endif;

endif;

if (rescan ind icated) then position to first CEL entry;

else position to next CEL entry in sequence;

endif;

endwhile;


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Process Interaction - Example

1. A SOURCE that generates arriving customers

2. A QUEUE to hold waiting customers

3. A FACILITY to serve customers

If a program provides these components as basic buildingblocks, then the modeler need only provide the parameters: theinterarrival times, the queue discipline, and the service times.

At this point, the computer must do all of the following work:

SOURCE must generate customers

When a customer arrives, it must either be placed in QUEUE orsent to FACILITY

When a customer arrives at FACILITY, a processing time mustbe generated

When FACILITY becomes idle, the QUEUE must be checked formore customers.

Snm Mat-347 Simulacion Ev Discretos

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