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Lund University / Presentation 2012

Discrete Event Simulation

ETSN01 Advanced Telecommunication


Why simulation?

• We need to study and understand complicated systems to be able to design better ones

• Predict system behaviour under different conditions and parameters• Approaches:

– Analysis– Experiment– Simulation


Why simulation? (cont’d)

Analysis Simulation ExperimentMathematical model (solving equations etc.)

Software implementation of system model

Tinker with actual system

Requires detailed understanding of system properties

For complex systems, usually requires approximations and simplifying assumptions

Usually straightforward

Possible to implement complex details of system without approximation

Still requires modelling of environment

In principle, does not require understanding how the system works

Captures behaviour of environment

If model is good, allows deep insight for broad range of scenarios, including rare events

Allows insight to broad range of scenarios

Study of rare events tricky but possible

Requires deployment of every scenario tested

May be difficult to reproduce

Rare events can be impossible to stimulate


Types of Simulation

• Time-driven simulation– Evaluate changes in system every time step– Can we wasteful if system frequently idle

• Discrete event-driven simulation– Maintain a list of pending “events” at all times– Process each event in turn to update system state and list of future events– Works well for time-independent systems with not too frequent events

• Markovian simulation (aka Roulette simulation)– Only possible for Markovian systems (memoryless and time-independent, transitions among

states only depend on current state)– Only maintain current state, simulate state transitions (no timing information needed)

• Monte-Carlo simulation– Use independent samples from some analytic solution

• Rare event simulation– Artificially boost occurrence of rare events


Simulation approaches

• Code in some HLL: C, C++, Java,….

• Use a simulation package: ns, ssf, Opnet, …– Ns (http://www.nsnam.org): “Ns is a discrete event simulator targeted at networking

research. Ns provides substantial support for simulation of TCP, routing, and multicast protocols over wired and wireless (local and satellite) networks”

– Ssfnet (http://www.ssfnet.org): “A public-domain standard for discrete-event simulation of large, complex systems in Java and C++”

– Opnet (http://www.opnet.com): Commercial network simulation product

– Omnet++ (http://www.omnetpp.org/): Network simulation tool, open source, gaining traction.

http://www.nsnam.org/

http://www.opnet.com/

http://www.opnet.com/

http://www.omnetpp.org/


Discrete Event Simulation: Event List

• Key is the event list– Contains times of scheduled events

– All events must be scheduled

– Assuming t1<t2<t3<t4<t5<t6<t7

t1,t1info

t2,t2info

t3,t3info

t4,t4info

t5,t5info

t6,t6info

t7,t7info

Next event


Discrete Event Simulation: Event List

t1,t1info

t2,t2info

t3,t3info

– Now schedule new event at time tn where t3<tn<t4

t4,t4info

t5,t5info

t6,t6info

t7,t7info

t4t4info

t5,t5info

t6,t6info

t7,t7info

tn,tninfo


Example: Single Server Queue

Events:– Packet arrives and joins queue– Packet begins service– Packet completes service


–Packet arrives and joins queue–Increment queue; if queue was empty, begin service of packet

–Packet begins service–Schedule completion of packet service

–Packet completes service–Decrement queue

Example: Single Server Queue


Implementation Details

• Must implement the event list processing– need to maintain event list, system state, simulation clock

– execute events from list

• Must consider inputs– schedule all arrivals at start

– or use “bootstrapping”: schedule next arrival when prior arrival happens

• Must consider outputs– decide which statistics to collect (e.g. count arrivals, departures)

• Consider length of simulation / termination condition


Single-Server Queue Simulation

Start

Init

MoreEvents?

Arrvl? Dept?

Update varsSchedule next arrival

Update varsSchedule next departure

Yes

Yes Yes

No

End

No

Compute performanceDisplay stats


Statistical Analysis

• Simulation is an experiment of a software model• Results are samples of underlying random variables• We want to obtain insight into the system behaviour

– and predict how it will work in other scenarios• This means that we MUST analyse the simulation results

statistically

Estimated MeanConfidence Interval,with probability x%,true mean lies withinthis range


Understanding statistical analysis

OR??


Independent Samples

• Assume a random variable Z with unknown probability distribution• How to estimate key distribution metrics?

– mean μ– variance σ2

• Obtain n independent samples (data points) {z1,z2,...,zn}


Estimation of mean

• Estimate mean by calculating sample average:

• Sample average is itself a random variable!

• Is a good estimator of the mean μ?

• Since each zi is distributed according to Z,

• therefore


Estimation of mean (cont’d)

• How many samples should we take?• Let’s estimate the variance

• So: the more samples, the smaller the variance of around the mean


Estimation of variance

• Calculate sample variance:

• Expected value of the sample variance:


Estimation of variance (cont'd)

• This gives a biased estimate of the population variance, σ2

• So we instead use unbiased sample variance, denoted s2. Note division by n – 1 instead of n: this is called Bessel's correction.


Confidence Intervals

• Assume is distributed normally (Gaussian)– reasonable approximation if sample size is large

• Then, depends only on α!

– is 0.9 for α≈1.645

– is 0.95 for α≈1.96

– is 0.99 for α≈2.576

• For example: is confidence interval for μ with confidence level of 95%

• In practice, σ is not known – use instead


Transients in the Simulation

• Everything above assumes that the samples are:– independent

– and identically distributed

• Distribution is different during transient phase– must ensure samples are obtained after system has reached steady state

• To avoid transient data:– run long enough so that transients have negligible impact on the estimates of confidence

interval

– or, if only short runs available, manually discard samples from transient phase


Effect of Correlation

• Simulation data is never perfectly independent

• Can estimate correlation between successive samples

– autocovariance of lag k is defined as:

• Autocovariance should be low (close to zero) for any k

• Otherwise, need to do better


Method of Batch Means

• Group raw sample data into batches–so that successive batches are approximately

independent• For example, if you have n data points, group them into m

batches of size p=n/m• Calculate the m batch means• Now treat the batches as if they were independent


Batch Means

• Probably most widely-used method for estimating confidence intervals

• Gives less bias at the expense of more variability in the estimator• The difficult part is choosing the batch size

– batch too large: confidence intervals larger than is justified by the actual data

– batch too small: excessive correlation and misleadingly small confidence intervals


For Further Reading

• B.D. Ripley, Stochastic Simulation, John Wiley and Sons, 1987 • B.D. Ripley, Uses and abuses of statistical simulation, Mathematical Programming Vol 42, pp

53-68, 1988 • K. Pawlikowski, “Steady-state simulation of queueing processes: A survey of problems and solutions”, ACM

Computing Surveys, Vol 22, No 2, pp123-170, June 1990• A.M. Law and W.D. Kelton, Simulation Modelling and Analysis, McGraw-Hill, 1991• R.F. Garzia and M.R. Garzia (eds), Network Modelling, Simulation and Analysis, Marcel Dekke

1990 • E.D. Lazowska, Quantitative System Performance: Computer System Analysis using Queueing

Network Models, Prentice-Hall, 1984 • Telecom Australia, A Course in Teletraffic Engineering, Telecom Australia, 1978 • The Network Simulator - ns-3, http://www.nsnam.org

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