Fluctuation Mechanism and Control on System Instantaneous Availability

38 ◾ System Instantaneous Availability

1. Critical repair time τ is a positive constant.

In this case,

∫∫

∫

∫

∫

∫

{}

{}

{}

()

{}

{}

{}

{}

() () ()

() ()

()

() ()() () () ()

()

() ()

() ()

{}

() () {}

() () ()

()() () ()

>= ==−

≤< +==

=≤<+ <τ =

=−−=τ− −

+≤ ==

===+=+≤

===+≤

=−==

=− =

ττ

,0|001

,0|00

,|00

,0|00

0| 00,

0| 0

0| 00

*.

11

1111

1111

00

11 1

11111

0

11

1

0

1

0

11

0

PX tZ tZ Ft

PX tXYZtZ

PX tXYY Z

Ft Ft sdGs FtGFtsdG s

PX YtZt Z

PZtZ XYudPX Yu

PZtZudPX Yu

PZtu ZdQu

At udQu Qt At

t

t

t

t

en, the instantaneous availability of the new model satisfies

∫

() () () () ()() ()

()

=− +τ−− +

τ

At Ft FtGFtsdG sQtAt

1*

1 1

0

. (3.17)

2. Critical repair time τ is a nonnegative random variable.

Suppose the distribution function of τ is H(t), like the conclusion of

part (1), there is

∫∫

{}

{}

{}

()

() () ()

() ()

() ...