Goricheva Ruslana.

Statistical properties of the regenerative processes with networking applications.

• 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

1{ : , 0}i n nX i n are independent and identically distributed.

0 10 ... ....n - regeneration points .

f(t) – measurable function.

1

0

0 1

[ ( )]1

lim ( )[ ]

iNi

iN

i

E f Xf X r

N E

(weak convergence).

Regenerative Central Limit Theorem.Point estimator for : .n

n

n

Yr r

Theorem:Theorem:12 [ ] (0,1),n nn r r N

-normal distribution with parameters 0, 1,

where:

1

2 21

(0,1)

[( ) ].

n

n

N

E Y r

-number of regenerative cycles,

-average length of cycle,

Confidence interval.

( ) ( )[ , ].

n n

n n

z s n z s nr r r

n n

where:1

2 2

( ), (1 )2

( ) .

z

s n

-quantile of N(0,1),

System G/G/1/m.• Poisson arrivals,• Pareto distribution of

service times,• One server,• m – buffer size (can

be infinite).1

1

1

1

1

,

10 ,

0 ,1

1.1

n n nt t

E

ES

ES

E

Estimation of average number of costumers.

i

100 ....n

{ }i T

-number of costumers in a system in the i-th moment,

-arriving into empty system,

0 1{0 .... }NT t t t

,

-time set,

-a regenerative process over

1

0

0 1

[ ]1

lim .[ ]

iNi

iN

i

Er

N E

Dependence: number of arrivals-time.

0,0305.

0,5444.

0,9166.

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.

1

1

1

1

1

,

0 ,1

10 ,

11.

n n nt t

E

ES

ES

E

Estimation of average number of costumers.

i( ) ( ) ( )0 10 ....k k k

n

-number of costumers in a system in the i-th moment,

-arriving, we got k costumers in our system after arriving,

{ }i T 0 1{0 .... }NT t t t

|( )

,k

-time set,

-a regenerative process over

( )1

0

0 1

[ ]1

lim .[ ]

k

iNi

iN

i

Er

N E

0 ,k m

Regeneration points (different types of regeneration).

/ /1/10; 0,6140.G M

/ /1/10; 0,9983.G M

Comparison of intervals for different types of regeneration.

/ /1/10; 0,6140.G M / /1/10; 0,9983.G M

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,

0,iI

-lost an arrival in the i-th moment,

-no loses in the i-th moment.

1

n

n ii

I

-number of failures.

Estimation of probability of failure.

100 ....n

{ }iI I T

-arriving into empty system,

0 1{0 .... }NT t t t

,

-time set,

-a regenerative process over

1

0

1

[ ]1

lim .[ ]

ii

nN

Er

N E

Failure points.

/ /1/10; 0,9921.G G

/ /1/ 6; 0,6500.G G

Comparison of intervals for different size of buffer.

/ /1/ 6, / /1/10; 0,6500.G G G G

/ /1/ 6, / /1/10; 0,9921.G G G G

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.

( ),

n

z s 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.