The Exponential Distribution

1. The Exponential Distribution Probability density function

2. The Exponential Distribution Cumulative Distribution function Mean E(X) = 1/λ Variance V(X) = 1/ λ2

3. The Exponential Distribution Memoryless property Pr{X>s+t|X>s} = Pr{X>t}, for all s, t >=0 • Exponential Density often used in queueing systems for: • Elapsed time between arrivals to a system • Time required to service a transaction

4. The Poisson Distribution Probability Density Function Mean E(X) = λ Variance V(X) = λ Paramter λ is thought of as a rate

5. The Poisson Distribution • Important property: If the Poisson models the number of events that occur in a block of time, the exponential with the same parameter λ models the time elapsing between events.

6. Hypothesis Testing H0 = Null hypothesis= assertion to be tested H1 = Alternative hypothesis = Opposite of the null hypothesis The intent of hypothesis testing is to develop a statistical decision rule using the available data to choose H0 or H1 Note: Failure to reject H0 is not the same as accepting H0 – it just means that the doesn’t support rejecting it

7. Hypothesis Testing Type 1 Error: Accepting the alternative hypothesis H1 when the null hypothesisH0 is actually true Type 2 Error: Accepting the null hypothesis H0 when the alternative hypothesisH1 is actually true

8. Simulation Model Validation - Definition Substantiation that a computerized model within its domain af applicability possesses a satisfactory degree of accuracy consistent with the intended application of the model

9. The Need for Context in Validation • Experimental frame - …a limited set of circumstances under which the real system is to be observed or experimented with Note: A model may be valid in one experimental frame, but invalid in another

10. Validation Terminology • Acceptable range of accuracy - acceptable agreement between the simulation model and the real system under a given experimental frame • Validity measure - a specific quantifier of the amount of agreement between the model and the system • Acceptable validity range - Acceptable range of accuracy as measured by the validity measure

11. Hypothesis Testing in Validation H0 : Model is valid for the acceptable range of accuracy under the experimental frame H1 : Model is invalid for the acceptable range of accuracy under the experimental frame

12. Hypothesis Testing in Validation