1. Simulation Output Analysis

2. Summary Examples Parameter Estimation Sample Mean and Variance
Point and Interval Estimation
Terminating and Non-Terminating Simulation
Mean Square Errors

3. Example: Single Server Queueing System
x(t) t 1 2 3 4 5 6 7 8 9 10 11 12 13 14 S4 S3 S5 S7 S1 S2 S6
Average System Time
Let Sk be the time that customer k spends in the queue, then,
Area under x(t)
IMPORTANT: This is a Random Variable
Estimate of the average system time over the first N customers

4. Example: Single Server Queueing System
x(t) t 1 2 3 4 5 6 7 8 9 10 11 12 13 14
3T3 T0
2T2 T1
Probability that x(t)= i
Let T(i) be the total observed time during which x(t)= i
Probability Estimate
Total observation interval
Average queue length
Utilization

5. Parameter Estimation
Let X1,…,Xn be independent identically distributed random variables with mean θ and variance σ2.
In general, θ and σ2 are unknown deterministic quantities which we would like to estimate.
Random Variable!
Sample Mean:
Τhe sample mean can be used as an estimate of the unknown parameter θ. It has the same mean but less variance than Xi.

6. Estimator Properties Unbiasedness:
An estimator is said to be an unbiased estimator of the parameter θ if it satisfies
Bias:
In general, an estimator is said to be an biased since the following holds
where bn is the bias of the estimator
If X1,…,Xn are iid with mean θ, then the sample mean is an unbiased estimator of θ.

7. Estimator Properties Asymptotic Unbiasedness:
An estimator is said to be an asymptotically unbiased if it satisfies
Strong Consistency:
An estimator is strongly consistent if with probability 1
If X1,…,Xn are iid with mean θ, then the sample mean is also strongly consistent.

8. Consistency of the Sample Mean
The variance of the sample mean is x θ x θ
Increasing n
But, σ is unknown, therefore we use the sample variance
Also a Random Variable!

9. Recursive Form of Sample Mean and Variance
Let Mj and Sj be the sample mean and variance after the j-th sample is observed.
Also, let M0=S0=0.
The recursive form for generating Mj+1 and Sj+1 is
Example: Let Xi be a sequence of iid exponentially distributed random variables with rate λ= 0.5 (sample.m).

10. Interval Estimation and Confidence Intervals
Suppose that the estimator then, the natural question is how confident are we that the true parameter θ is within the interval (θ1-ε, θ1+ε)?
Recall the central limit theorem and let a new random variable
For the sample mean case
Then, the cdf of Zn approaches the standard normal distribution N(0,1) given by

11. Interval Estimation and Confidence Intervals
Let Z be a standard normal random variable, then
Area = 1-a
fZ(x) x
Thus, as n increases, Zn density approaches the standard normal density function, thus

12. Interval Estimation and Confidence Intervals
Substituting for Zn
fZ(x) x
Thus, for n large, this defines the interval where θ lies with probability 1-a and the following quantities are needed
The sample mean
The value of Za/2 which can be obtained from tables given a
The variance of which is unknown and so the sample variance is used.

13. Example
Suppose that X1, …, Xn are iid exponentially distributed random variables with rate λ=2. Estimate their sample mean as well as the 95% confidence interval.
SOLUTION
The sample mean is given by
From the standard normal tables, a =0.05, implies za/22
Finally, the sample variance is given by
Therefore, for n large,
SampleInterval.m

14. How Good is the Approximation
The standard normal N(0,1) approximation is valid as long as n is large enough, but how large is good enough?
Alternatively, the confidence interval can be evaluated based on the t-student distribution with n degrees of freedom
A t-student random variable is obtained by adding n iid Gaussian random variables (Yi) each with mean μ and variance σ2.

15. Terminating and Non-Terminating Simulation
There is a specific event that determines when the simulation will terminate
E.g., processing M packets or
Observing M events, or
simulate t time units,
...
Initial conditions are important!
Non-Terminating Simulation
Interested in long term (steady-state) averages

16. Terminating Simulation
Let X1,…,XM are data collected from a terminating simulation, e.g., the system time in a queue.
X1,…,XM are NOT independent since
Xk=max{0, Xk-1-Yk}+Zk
Yk, Zk are the kth interarrival and service times respectively
Define a performance measure, say
Run N simulations to obtain L1,…,LN.
Assuming independent simulations, then L1,…,LN are independent random variables, thus we can use the sample mean estimate

17. Examples: Terminating Simulation
Suppose that we are interested in the average time it will take to process the first 100 parts (given some initial condition).
Let T100,j j=1,…,M, denote the time that the 100th part is finished during the j-th replication, then the mean time required is given by
Suppose we are interested in the fraction of customers that get delayed more than 1 minute between 9 and 10 am at a certain ATM machine.
Let be the delay of the ith customer during the jth replication and define 1[Dij]=1 if Dij>1, 0 otherwise. Then,

18. Non-Terminating Simulation
Any simulation will terminate at some point m < ∞, thus the initial transient (because we start from a specific initial state) may cause some bias in the simulation output.
Replication with Deletions
The suggestion here is to start the simulation and let it run for a while without collecting any statistics.
The reasoning behind this approach is that the simulation will come closer to its steady state and as a result the collected data will be more representative
warm-up period
Data collection period
time r m

19. Non-Terminating Simulation
Batch Means
Group the collected data into n batches with m samples each.
Form the batch average
Take the average of all batches
For each batch, we can also use the warm-up periods as before.

20. Non-Terminating Simulation
Regenerative Simulation
Regenerative process: It is a process that is characterized by random points in time where the future of the process becomes independent of its past (“regenerates”)
time
Regeneration points
Regeneration points divide the sample path into intervals.
Data from the same interval are grouped together
We form the average over all such intervals.
Example: Busy periods in a single server queue identify regeneration intervals (why?).
In general, it is difficult to find such points!

21. Empirical Distributions and Bootstrapping
Given a set of measurements X1,…,Xn which are realizations of iid random variables according to some unknown FX(x;θ), where θ is a parameter we would like to estimate.
We can approximate FX(x; θ) using the data with a pmf where all measurements have equal probability 1/n.
The approximation becomes better as n grows larger.

22. Example
Suppose we have the measurements x1,…,xn that came from a distribution FX(x) with unknown mean θ and variance σ2. We would like to estimate θ using the sample mean μ. Find the Mean Square Error (MSE) of the estimator based on the empirical data.
The empirical mean is an unbiased estimator of θ. X x1 x2 xn
1/n 1 …
2/n
Empirical distribution
Vector of RVs from the empirical distribution
Based on empirical distribution

23. Example Therefore Xi is a RV from the empirical distribution
Compare this with the sample variance!
Therefore