Simulation Output Analysis

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

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

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

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.

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.

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.

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!

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).

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

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

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.

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

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.

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

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

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,

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

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.

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!

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.

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

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