1. Based on Law & Kelton, Simulation Modeling & Analysis, McGraw-Hill

2. Why Simulation? Test design when cannot analyze Verify analysis
System too complex
Can analyze only for certain cases (Poisson arrivals, very large N, etc.)
Verify analysis
Fast production of results

3. Simulation types Static vs. dynamic Deterministic vs. stochastic
Continuous vs. discrete
Simulation model  system model

4. Discrete Event System Discrete system Example: a queueing system
The system state can change only in a countable number of points in time!
Event = an instantaneous change in state.
Example: a queueing system
System state: Number of customer
At each time number of customer can only change by an integer

5. Continuous Simulation
Simulating the flight of a rocket in the air
System state: rocket position and weight
State changes continuously in time (according to a partial differential equation)

6. Components of a discrete-event simulation model
System state: the collection of state variables necessary to describe the system at a particular time.
Simulation clock: a variable giving the current value of simulated time
Event list: a list containing the next time when each type of event will occur.
Statistical counters: variables used for storing statistical information about system performance
How many moments?

7. Components of a discrete-event simulation model (2)
Initialization routine: a subprogram to initialize the simulation model at time 0.
Timing routine: a subprogram that determines the next event from the event list and then advances the simulation clock to the time when the event is to occur.
Event routine: A subprogram that updates the system state when a particular type of event occurs (one routine per event).

8. Components of a discrete-event simulation model (3)
Library routine: A set of subprograms used to generate random observations from probability distributions that were determined as part of the simulation model.
Report generator: A subprogram that computes estimates (from the statistical counter) of the desired performance measures when simulation ends.

9. Components of a discrete-event simulation model (4)
Main program: a subprogram that invokes the timing routine to determine the next event and then transfers control to the corresponding event routine to update the system state. May also check for termination and invoke the report generator.

10. Flow control
Init
Determine
next event
Event 1
Event n
Done? no
report

11. Example a single server queuing system
System to be simulated
Ai Interarrival times are I.I.D.
Si Service time are I.I.D.
FIFO service
Work preserving
Initial state
Empty and idle
First arrival after Ai time units from time 0
Termination after n customers left queue.

12. Performance measures d(n) = expected average delay in queue.
Customer point of view
q(n) = expected time-average number of customers in queue.
System point of view
Note: average over time which is continuous!!
u(n) = proportion of time server is busy

13. No. of customers in queue.
Q(t) 4 3 2 1 t

14. No. of customers in queue.
Q(t) 4 3 2 1 t

15. Performance measures (2)
Averages are not always enough.
Register max/min values.
Register the entire “pdf”
histograms

16. Initialization Set simulation time to 0. Set initial state:
Server idle
Queue empty
Last event time
Init event list
Generate 1st arrival
Zero stat counters
Total delay and number delayed
Area under Q(t) and B(t).

17. 1st Event: customer arrival
After init finished the arrival event is selected and time is advanced to this event
Change server from idle to busy
Add 0 to total delay, increment No of delayed.
Generate two events:
This customer departure
Next arrival (generate as you go)
Add 0 to the area under Q(t).
Add 0 to the area under B(t).

18. 2nd Event: customer arrival
Arrival event is selected and time is advanced to this event
Server busy => put customer in queue with arrival time
Set “No. in queue” to 1.
Generate next-arrival events (don’t mess with dep.)
Add 0 to the area under Q(t).
Add (t2-t1) to the area under B(t).

19. Remarks While handling an event time is sanding
But, order of operation is still important:
First update area under Q(t) then update “No. in queue”
Two events at the same time
Order may change simulation result!

20. Determining Events
In complex systems events sequence may not be trivial.
Use of event graph to aim us is designing the events

21. Event Graph States are bulbs, connected with arrows
Bold arrow: event may occur.
Thin arrow: event must occur.
depar-
ture
arrival

22. Alternative Event Graph
Enter
service
depar-
ture
arrival
This design is correct as well!
One more event => more complexity
Simplification rule:
If an event node has incoming arcs that are all thin and smooth, it can be eliminated.
A strongly connected connected component that have no incoming arcs from other nodes must be initialized.

23. Take care GIGO = garbage in garbage out Statistical confidence
Realistic scenarios (arrival process, service time)
Full cover of system behavior
Statistical confidence
What to model
Not enough details => hurts accuracy
Too many details => slows simulation
Attempt to verify correctness
Simulate cases you can analyze

24. Random Generator
Make sure you can regenerate your random sequence, or debugging is hell.
For long simulations, use 32 bit pseudo random generator. 16 bit is too short!

25. Statistical Confidence
Better a few short runs than one long one.
However, make sure run time is long enough to make end conditions negligible.
Given a set of IID random variables we can calculate the confidence interval.

26. Confidence Interval
X1, X2, …Xn are IID random variable with finite mean  and finite variance 2>0.
Fn(z)=P(Zn z) is Dist. Func. of Zn
By central limit theorem Fn(z)(z) (the std. Normal r.v.)
For sufficiently large n, approx. 100(1-) percent confidence interval for  is given by

27. 1
0.9
0.8
0.7
0.6
0.5
Area:
1-
0.4
0.3
0.2
0.1 -5 -3 3 5

28. Confidence Interval for Small n
For small number of observations, we need to correct the coefficient of the interval width
The small n the larger tn-1,1-/2 is.
tn-1,1-/2 is taken from a table.

29. Confidence Interval
In general all plots has to have a confidence interval on every mark
Exception: high confidence (mark size)

30. A good pseudo random generator
float myrand() {
b32 = ( *b );
if (b32<0)
b32 = -b32;
return( * b32); }

31. Generating Non-Uniform Distributions
F(x), the PDF, of any distribution is a function to [0,1], => use the inverse transform
Generate U~U(0,1)
Return X=F-1(U)
P(Xx) = P(F-1(U)  x)=P(U F(x))=F(x)
Since U is U(0,1) and
0 F(x) 1,
P(U y) = y

32. Generating the Exponential distribution
Note we can use - ln(u).

33. 1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1 20 40 60 80
100
120
140
160
180
200

34. Simulating complex systems
Systems often have many identical components
A queueing system with multiple queues
A switch with multiple ports running a distributed algorithm
Must keep state for each component
Array of structures in C,
multiple instances of the basic object (C++)
Keep the connectivity (when applicable)

35. Exercise No. 1 Write an event simulator for a single queue Hand ins:
M/M/1
M/D/1
Hand ins:
Code transcript
Utilization vs. load plot with confidence interval of 90%, and the theoretical expectation
With 5 executions, and with 20 executions
Average delay (queueing +service) vs. load

36. Remarks
Code has to work anywhere (not just at home), especially in TAU.
Make it portable.
Oral exams through the semester.
Send code to
Specify development env. (Unix/Visual C/..)
One tar/zip file with makefile etc.

37. Exercise No.2
Give an exact expression (neglect history) to the bound of the throughput of an NxN switch given that the scheduler can select any of the two packets at the HOL. What happens when N.
Plot the bound as a function of the depth of the queue the scheduler can look at.