© 2003, carla ellis simulation techniques overview simulation environments emulation exec- driven...

© 2003, Carla Ellis

Simulation Techniques Overview

Simulation environments

emulation

exec-driven

sim

trace-driven

sim

stochasticsim

Workloadparameters

SystemConfig

parameters

Factorlevels

Result DataDiscreteevents


The End Game: When to Stop

t

transientinterval

steadystate

finalconditions

a b


Issues• If you stop at a – before the cleanup phase:

– Must be careful in calculating metrics when some events are outstanding

• Example: scheduling simulation. Some jobs completed and some still in queue.

• Average service time must count just completed jobs.

• Average queue length must be over time not queuing events.

• If you stop at b – completion:– Not at steady state again

• Example: jobs are finishing with no new arrivals


• Assuming you choose to stop at a – how to determine a?

• Simulation should be run long enough for the confidence interval for the mean response to narrow to desired width

• Variance of the sample mean of n independent observations uses variance of observations

Stopping Criteria

x z1-Var(x)

Var(x) = Var(x)

n


Correlated Observations

• Often the observations are not independent– Memory access time depends on cache

state built by previous memory request– Waiting time depends on length of previous

job• Solutions

– Independent replications of simulation experiment

– Batch means– Regeneration


Replications

• m replications with different seed value each time, of size n+n0 where n0 is initial transient phase during which data is discarded.

• Confidence interval is inversely proportional to mn– Increase either m or n to get narrower C.I.– Page 431 shows how to calculate overall

mean for all replications, Var(x), and C.I.


Batch means• Subsamples – long

simulation run of N + n0 observations

• Divide N observations into m samples of n observations each

• Batch size n must be large enough so the batch means have little correlation

• Compute covariance of successive batch means xi and xi+1 with bigger n’s until it is small enough

t

n0

n n n

• C.I width again inversely proportional to mn


Regeneration

• Independent phases where the execution returns to an initial state– Flushed cache– Empty job queue

• Regeneration cycles may be of unequal length (complicates math – page 434)

• m cycles of n1, n2, …, nm sizes s.t. C.I. is narrow enough

t

Regenerationpoints

Regenerationcycle


Structure of Discrete Event Simulation

eventQ scheduler

Event handlers

State var

results


Role of Random Values in Discrete Event Simulation

eventQ scheduler

Event handlers

State var

results

e

randomparameters

random values ininitializationof state


Random Values

• Want random values with a specified distribution

• Step 1: produce uniformly distributed numbers between 0 and 1(random number generation)

• Step 2: apply transformation to produce values from desired distribution (random variate generation)


Random Number Generators

xn = f ( xn-1, xn-2) where x0 is seed• Pseudo-random since, given the same seed,

the sequence is repeatable and deterministic• Cycle length – length of repeating sequence• Example: xn = a xn-1 + b mod m

seed

cycle

period


Desirable Properties

• Period should be large• Should be efficiently computable• Successive values should be independent and

uniformly distributed

• Types discussed in Jain:– Linear congruential (LCG)– Tausworthe – long, based on exclusive-or– Extended Fibonacci

• Use “off the shelf” generator that has been tested


Using Random Number Generators

Seed Selection – issue is critical if multistream simulation (need random numbers for more than one variable)

• Do not use zero and avoid even numbers as seeds

• Do not use one stream for two (or more) purposes s.t. ui is used for one variable and ui+1 for next (e.g. interarrival time and service time for next event – they would be correlated)


• Use non-overlapping streams


seed1

cycle

period

seed2

cycle

period


• Random number stream does not have to be reinitialized for replications of simulation, can pick up where last one left off

• Do not use random seeds (e.g. time of day)– Can not be reproduced– Not possible to guarantee multiple streams

do not overlap



Potential Pitfalls

• Testing for randomness – a single test is not sufficient – chap 27, next lecture.

• Implementation matters – overflow and truncation can change the path of the sequence

• Bits of successive words are not guaranteed random (e.g. generating random memory addresses and then using page number field does not necessarily give you random pages)

© 2003, carla ellis simulation techniques overview simulation environments emulation exec- driven...

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