Simulation

1
Simulation

• Based on
• Law Kelton, Simulation Modeling Analysis,
  McGraw-Hill

2
Why Simulation?

• Test design when cannot analyze
• 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
• 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).

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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
Examplea 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

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No. of customers in queue.
Q(t)
4
3
2
1
0
t
14
No. of customers in queue.
Q(t)
4
3
2
1
0
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).

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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).

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2nd Event customer arrival

• Arrival event is selected and time is advanced to
  this event
• Server busy gt put customer in queue with arrival
  time
• Set No. in queue to 1.
• Generate next-arrival events (dont mess with
  dep.)
• Add 0 to the area under Q(t).
• Add (t2-t1) to the area under B(t).

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Remarks

• While handling an event time is standing
• 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

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Event Graph

• States are bulbs, connected with arrows
• Bold arrow event may occur after nonzero time.
• Thin arrow event may occur immediately.

depar- ture
arrival
22
Alternative Event Graph
Enter service
depar- ture
arrival

• This design is correct as well!
• One more event gt more complexity
• Simplification rule
• If an event node has incoming arcs that are all
  thin and smooth, it can be eliminated.
• A strongly connected component that has no
  incoming arcs from other nodes must be
  initialized.

23
Take care

• GIGO garbage in garbage out
• Realistic scenarios (arrival process, service
  time)
• Full cover of system behavior
• Statistical confidence
• What to model
• Not enough details gt hurts accuracy
• Too many details gt 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.

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Confidence Interval

• X1, X2, Xn are IID random variables with finite
  mean ? and finite variance ?2gt0.

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

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1
0.9
0.8
0.7
0.6
0.5
Area 1-?
0.4
0.3
0.2
0.1
0
-5
-3
0
3
5
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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 (314159269b32 453860245)
• if (b32lt0)
• b32 -b32
• return(0.00000000046566128730 b32)

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Generating Non-Uniform Distributions

• F(x), the PDF, of any distribution is a function
  to 0,1, gt use the inverse transform
• Generate UU(0,1)
• Return XF-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).
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1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
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
Wet Exercise No. 1

• Assume an NxN switch. Time is slotted. At each
  time slot up to N cells (cell size is CZ) are
  forwarded from the input to the output. For this
  exercise we assume all packets are destined to
  port 0, and the scheduler is a simple round robin
  among inputs. Packets arrive at the N input ports
  with inter packet space that is Exponentially
  distributed with parameter ? (normalized to time
  slot size). Packet length is uniformly
  distributed between 64 and 1500. Packets should
  be segmented in the switch for switching.
  100Mbps, buffer size 400 cells. CellCZ 8
  byte header.
• Hand-ins
• Code transcript
• Utilization vs. load plot with confidence
  interval of 90, with 5 executions, and with 20
  executions
• Average delay (queueing service) vs. load
• For each simulation select the range, and explain
  why.
• At what point the system become unstable? Give a
  mathematical analysis that explain this.

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 amirshay_at_eng.tau.ac.il
• Specify development env. (Unix/Visual C/..)
• One tar/zip file with makefile etc.