Transcript Statistical properties of the regenerative processes with

Statistical properties of the
regenerative processes with
networking applications.
Goricheva Ruslana.
•
•
•
•
•
Regenerative processes.
Central Limit Theorem.
Regenerative method of estimation.
Simulation in system G/G/1/m.
Simulation of failures flow (non-standard
moments of regeneration).
Process X(t) is a regenerative process, if it’s segments
{X i : n1  i  n , n  0}
are independent and identically distributed.
0  0  1  ...  n  .... - regeneration points .
f(t) – measurable function.
1
N
1
lim  f ( X i ) 
N  N
i 0
E[ f ( X i )]
i 0
E[ 1 ]
(weak convergence).
r
Regenerative Central Limit Theorem.
Point estimator for : rn 
Theorem:
where:
Yn
n
 r.
1
n 2  n [rn  r ]   N (0,1),
n -number of regenerative cycles,
 n -average length of cycle,
N (0,1) -normal distribution with parameters 0, 1,
 2  E[(Y  r 1 ) 2 ].
1
Confidence interval.
r  [rn 
where:
z s (n)

n
n
, rn 
z s (n)

n
n
].
z   1 ( ),(1   ) -quantile of N(0,1),
2
s 2 (n)   2 .
System G/G/1/m.
• Poisson arrivals,
• Pareto distribution of
service times,
• One server,
• m – buffer size (can
be infinite).
 n  tn  tn 1 ,
0  E 1 
1

 ,

 ,
 1
ES

 1
 1.
E 1   1
0  ES1 
Estimation of average number of
costumers.
i
-number of costumers in a system in the i-th moment,
0  0  1  ....   n -arriving into empty system,
T  {0  t0  t1  ....  t N } -time set,
  { i T}
-a regenerative process over  ,
1
N
E[ i ]
1
lim  i  i 0
 r.
N  N
E[ 1 ]
i 0
Dependence:
number of arrivals-time.
  0, 0305.
  0,9166.
  0,5444.
Confidence intervals.
  0, 0305.
  0,5444.
Main conclusions.
• Our confidence interval becomes more
narrow while we increase number of cycles,
that completely corresponds regenerative
CLT.
• These reduction behaviors depends on
some factors.
• Decreasing of buffer size makes bounds of
interval more close to our estimator.
• With growing of loading the system we get
more wide confidence interval.
System G/M/1/10.
• Pareto distribution of
arrivals,
• Exponential service,
• One server.
 n  tn  tn 1 ,

0  E 1 
 ,
 1
0  ES1 
1
 ,

ES   1
 1
 1.
E 1

Estimation of average number of
costumers.
i
-number of costumers in a system in the i-th moment,
0  0(k )  1( k )  ....  n( k )-arriving, we got k costumers in our
system after arriving, 0  k  m,
T  {0  t0  t1  ....  tN } -time set,
  { i T}
-a regenerative process over 
1( k )
N
E[  i ]
1
lim  i  i 0
 r.
N  N
E[ 1 ]
i 0
|( k )
,
Regeneration points (different
types of regeneration).
G / M /1/10;   0,6140.
G / M /1/10;   0,9983.
Comparison of intervals for different
types of regeneration.
G / M /1/10;   0,6140.
G / M /1/10;   0,9983.
Main conclusions.
Increasing of regenerative cycles causes the
reduction of confidence interval, so get
wider estimator in that type of regeneration,
where we have more regeneration points.
Varying number k, we can achieve better
estimation. This example also illustrates
regeneration property of concerned
process.
Simulation of failure flow.
• Poisson arrivals,
• Pareto distribution of
service times,
• One server,
• m – buffer size.
1, -lost an arrival in the i-th moment,
Ii  
0, -no loses in the i-th moment.
n
n   Ii -number of failures.
i 1
Estimation of probability of failure.
0  0  1  ....   n -arriving into empty system,
T  {0  t0  t1  ....  t N } -time set,
 I  {Ii T }
-a regenerative process over
1
E[ i ]
1
lim  n  i 0
 r.
N  N
E[ 1 ]
,
Failure points.
G / G /1/ 6;   0,6500.
G / G /1/10;   0,9921.
Comparison of intervals for
different size of buffer.
G / G /1/ 6, G / G /1/10;   0,9921.
G / G /1/ 6, G / G /1/10;   0,6500.
Main conclusions.
In this example we also can be sure that our estimation
corresponds regenerative CLT. And we can notice how
parameters of the system affects to the number of failures.

z s (n)

n
n
,
So, rare regeneration (long regeneration
intervals) causes increasing of confidence
intervals, in case of bigger buffer size. But
we get less failures, so s(n) decreases. And
it makes interval for r more narrow.