1. © 2003, Carla Ellis Simulation Techniques Overview Simulation environments emulation exec- driven sim trace- driven sim stochastic sim Workload parameters System Config parameters Factor levels Result Data Discrete events

2. © 2003, Carla Ellis The End Game: When to Stop t transient interval steady state final conditions a b

3. © 2003, Carla Ellis 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

4. © 2003, Carla Ellis 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 ! z 1-   Var(x) Var(x) = Var(x) n

5. © 2003, Carla Ellis 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

6. © 2003, Carla Ellis Replications m replications with different seed value each time, of size n+n 0 where n 0 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.

7. © 2003, Carla Ellis Batch means Subsamples – long simulation run of N + n 0 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 x i and x i+1 with bigger n’s until it is small enough t n0n0 n n n C.I width again inversely proportional to mn

8. © 2003, Carla Ellis 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 n 1, n 2, …, n m sizes s.t. C.I. is narrow enough t Regeneration points Regeneration cycle

9. © 2003, Carla Ellis Structure of Discrete Event Simulation eventQ scheduler Event handlers State var results

10. © 2003, Carla Ellis Role of Random Values in Discrete Event Simulation eventQ scheduler Event handlers State var results e random parameters random values in initialization of state

11. © 2003, Carla Ellis 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)

12. © 2003, Carla Ellis Random Number Generators x n = f ( x n-1, x n-2 ) where x 0 is seed Pseudo-random since, given the same seed, the sequence is repeatable and deterministic Cycle length – length of repeating sequence Example: x n = a x n-1 + b mod m seed cycle period

13. © 2003, Carla Ellis 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

14. © 2003, Carla Ellis 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. u i is used for one variable and u i+1 for next (e.g. interarrival time and service time for next event – they would be correlated)

15. © 2003, Carla Ellis Use non-overlapping streams Using Random Number Generators seed 1 cycle period seed 2 cycle period

16. © 2003, Carla Ellis 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 Using Random Number Generators

17. © 2003, Carla Ellis 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)