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```Output Analysis and
Experimentation for
Systems Simulation
Performance Measure
• Simulation output analysis concerns with using
simulation to estimate the quantities of interest in a
simulation model
• Quantities are referred as Performance Measure
• Examples of performance measures include
– The average system time of the first n
customers
– The long-run average system time per customer
– The first time the system breaks down
– The long-run average fraction of time the
system is down.
Types of simulation
• There are two types of simulation:
1) Transient Simulation (Short-term simulation)
simulation)
Transient simulation is easier to study than the
simulate the system until infinity.
Transient Simulation
• We are interested in studying the
performance measure of the system for a
finite time horizon.
• This time could be deterministic such as:
– The performance of a system for one day,
one month etc.
• Or Stochastic (the time a certain event
occurs) such as:
– The performance of the system until the first
time it breaks down.
Transient Simulation
• Suppose we want to estimate
m= E{f(X(t)): 0 < t < T} Where
– f is a deterministic function,
– {X(t)} is a stochastic process (a sequence of
random variables),
– T is a stopping time either deterministic or
stochastic.
• Example. X(t) =N(t): the number of
customers in an M/M/1 queuing system at
time t.
Estimating Performance Measure in
the Transient Simulation
Suppose we want to estimate the average number of
customers in system; N, before it breaks down
• We simulate the system n times until it breaks
down.
• Each run, we get an estimate of the number of
customers as Ni (Note that Ni is a random
variable).
• We estimate the average number of customers in
n
1
the system by the estimator Z 
Ni

n
n i 1
Properties of estimators
1) Unbiased: E{Zn} = m
lim Z n  μ
2) Strongly consistent: n 
almost surely (i.e. with probability = 1)
3) The sequence {Zn} is asymptotically
normal:
 Zn  m 
n
  N (0,1)
   n 
The convergence here in distribution.
Confidence Interval for the mean
• The 100 (1-a) % confidence interval Ia is
an interval that satisfies: P{m  Ia} = 1 – a
• Use the asymptotic normal property to
build the confidence interval as
a
In
Z a / 2
Z a / 2 

 Z n 
, Zn 

n
n 

• Where Za/2 can be found in the standard
normal distribution table that represent
the z value where the area under the
normal distribution is 1- a/2 .
  is the standard deviation of the mean. It
can be estimated by s where:
n
1
2
s2 
(N

Z
)

i
n
n  1 i 1
• Some values for Za/2: :are as follows
• For 90% C.I, use Z.05 =1.645
• For 95% C.I, use Z.025 =1.96
• We are interested in studying the
performance measure of the system for a
long run period of time.
• We assume that the system will eventually
settle down (becomes stable).
• Two methods:
– Multiple Replications with Initial Deletion.
– Batch Means Method.
Multiple Replications with Initial Deletion
• Run the simulation for a long simulation run
• Delete initial observations (Initial Bias
Contamination)
• Evaluate the average of the remaining data as a
one sample of the mean.
• Repeat for several times to get estimate for the
mean an confidence interval.
Each replication we delete some observations, so we
do loose some expenses of simulation
Batch Means Method
• Do only one very long simulation run, say on the
interval [0,T].
• Delete few contaminated observations at the
beginning [0,T0]
• Divide the remaining [T0, T] into k subintervals
[T0, T1), [T0, T2), …, [Tk-1, T], and
• Find a sample of the performance measure Zj in
each subinterval j
• Construct the 100(1-a) % confidence interval for
the data Z1 , Z2 , …, Zk
```
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