INFORMATION AND COMMUNICATION TECHNOLOGIES AND SERVICES, VOL. 9, NO. 1,MARCH 2011 1
©2011 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING ISSN 1804-3119
Sampling o Random Da a S eams
Gus á CEPCIANSKY.1, Ladisla SCHWARTZ 1
1 Depa men o Telecommunica ions and Mul imedia, Uni e si y o Zilina, Uni e zi na 1, 01026 Zilina, Slo akia
[email protected], schwa z@ el.uniza.sk
Abs ac . Mode n elecommunica ion ne wo ks wo k on
he ansmission me hod o common da a s eams in which
da a bu s s consis ing o packe s ha u he consis o
pa icula bi s a e mul iplexed om a ious a ic
sou ces. The la ge amoun o da a is ansmi ed h ough
a ansmission medium (op ical ib e), he mo e equen ly
bu s s occu , and he lowe amoun o da a, he mo e
a ely hey ollow. I i is equi ed o moni o how la ge
amoun o da a is being ansmi ed in a ne wo k b anch in
o de o ind ou , o which measu e o his b anch is
occupied, i is no necessa y o ake each in o ma ion uni
(each packe o e en each pa icula bi ). I will do i
in o ma ion whe he a da a bu s occu s in he
ansmission o does i no occu is aken in ce ain ime
in e als – wi h a ce ain sampling equency. The pape
deals wi h his sampling in e als.
Keywo ds
Sampling, a ic, da a bu s , passi e op ical
ne wo k.
1. In oduc ion
The e a e many ypes o da a bu s s on a ansmission
ou e be ween wo ne wo k nodes om a ious a ic
sou ces. The da a bu s leng h is conside ed as a andom
a iable. Acco ding o Figu e 1, he bu s ain in a
common da a s eam can be desc ibed as a andom p ocess
X( ) consis ing o se ies o andom occu ing ec angula
pulses wi h a andom ampli ude ha only gains 2 alues o
x0 = 0 andx1
= 1 and wi h a andom ime leng h
Q
ha
gains alues o
u
k,k = 1, 2, ...
Fig. 1. Packe ain in he da a s eam
The mean le el o he a e age alue o his andom
p ocess
å
=
¥®
=N
k
k
TT
y
1
1
lim
J
(1)
also exp esses he a e age a ic use o a communica ion
channel, e.g. wi h which pa o he o al ime T
heo e ically in ini ely long he channel was occupied by
he ansmi ed da a bu s s. In o he wo ds, he alue y
also exp esses he p obabili y ha he communica ion
channel will be in any momen o sampling ound as
occupied.
Fu he , le
l
be he coun o seizu es o he
communica ion channel by he ansmi ed da a bu s s and
m
he coun o eleases o he communica ion channel
du ing a ime uni . The a io
m
l
=a(2)
ep esen s he a ic o e ed o he communica ion
channel. The in e se alue o
m
å
¥
=
¥®
==
1
1
lim
1
k
k
NN
J
m
J
(3)
is he a e age bu s leng h.
2. Theo y
2.1 Random P ocess Cha ac e is ics
The occu ence o da a bu s s in he common da a
s eam can be conside ed as he Ma ko andom p ocess
because he ea u es o such a p ocess a e ul illed:
·a da a bu s occu ence is ully andom,
·bu s s las o a andom ime,
· he p obabili y o a da a bu s occu ence o end o a
bu s du ing a e y small ime in e al D will be
p opo ional o he du a ion o his in e al.
u
1
u
2
u
3
u
k
u
N-1
u
N
T
2INFORMATION AND COMMUNICATION TECHNOLOGIES AND SERVICES, VOL. 9, NO. 1,MARCH 2011
©2011 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING ISSN 1804-3119
These ea u es ep esen a hidden de e minis ic
elemen in he andom p ocess. The consequence o his is
ha i is possible o explici ly de i e basic cha ac e is ics –
co ela ion and co- a ia ion unc ions, dispe sion and
spec um o ha andom p ocess [1], [2], [3], [4].
1) Solu ion o Random P ocess
The andom p ocess gains only 2 s a es: S0 –
communica ion channel is ee and S1 – communica ion
channel is occupied.
The p ocess s ays in he ze o s a e S0 i no da a bu s
occu s du ing D , he p obabili y o which is:
(
)
p D-=D+
l
1,
00 .(4)
The p ocess ansi s om he s a e S0 o he s a e S1
(communica ion channel will be seized) i a da a bu s
occu s du ing D . The p obabili y o his is:
(
)
p D=D+
l
,
01 .(5)
The p ocess s ays in he s a e S1 i a da a bu s does
no inish du ing D , he p obabili y o which is:
(
)
p D-=D+
m
1,
11 .(6)
The p ocess ansi s om he s a e S1 o he s a e S0
(communica ion channel will be eleased) i a da a bu s
inishes du ing D . The p obabili y o ha is:
(
)
p D=D+
m
,
10 .(7)
The p obabili y ha mo e han 1 change occu s in he
s a e o he channel du ing D is equal o ze o.
The p obabili ies p00,p01,p10,p11 a e a anged in o
he ma ix o ansi ion p obabili ies:
=
÷
÷
ø
ö
ç
ç
è
æ
D+D+
D+D+
=D+ ),(),(
),(),(
),(
1110
0100
p p
p p
P
=
÷
÷
ø
ö
ç
ç
è
æ
D-D
DD-
=)1)
)1
mm
ll
AJ ×+=D
÷
÷
ø
ö
ç
ç
è
æ
-
-
+
÷
÷
ø
ö
ç
ç
è
æ
=.
10
01
mm
ll
.(8)
The ma ix J is he uni one. The ma ix A is he
ma ix o in ensi ies o ansi ion p obabili ies. In gene al,
he elemen s o he ma ix Acan be ime dependen . Bu i
i is supposed he e godici y o he andom p ocess, e.g.
he Ma ko p ocess is homogenous, he ime dependence
does no exis .
By he solu ion o he basic equa ion o he Ma ko
andom p ocess
A p
p ×= )(
)(
d
d,(9)
i can be calcula ed, wi h which p obabili y p0( ) he
communica ion channel will be ee and wi h which
p obabili y p1( ) he communica ion channel will be
occupied by a da a bu s in a ime . The equa ion is sol ed
by means o Laplace ans o m:
()
ò
¥-
==
0
).()}({ psF p deL s,(10)
})({
)( A p
p ×=
þ
ý
ü
î
í
ìL
d
d
L.(11)
Hence, when he ans o m is ca ied ou
(
)
(
)
[
]
1
0-
-××= AJspsF .(12)
The sough ec o p( ) will be ob ained a e he
in e se Laplace ans o m
(
)
(
)
{
}
(
)
[
]
{
}
1
11 0-
-- -××== AJspLsFL p . (13)
Fi s , le s.J – A be w i en in:
÷
÷
ø
ö
ç
ç
è
æ
+-
-+
=
=
÷
÷
ø
ö
ç
ç
è
æ
-
-
-
÷
÷
ø
ö
ç
ç
è
æ
=-×
mm
ll
mm
ll
s
s
sAJs 10
01
(14)
The in e se ma ix:
[ ]
=
÷
÷
ø
ö
ç
ç
è
æ
+
+
++×
=-× -
lm
lm
ml
s
s
ss
AJs .
)(
1
1
÷
÷
÷
÷
ø
ö
ç
ç
ç
ç
è
æ
++×
+
++×
++×++×
+
=
)()(
)()(
ml
l
ml
mml
l
ml
m
ss
s
ss
ssss
s
.(15)
To ca y ou he in e se Laplace ans o m, he
elemen s o he in e se ma ix shall be decomposed in o
he sum o pa icle ac ions. Then he basic sys em o
equa ions (13) acqui es he o m:
() ()
+
ê
ê
ê
ê
ë
é
÷
÷
÷
÷
ø
ö
ç
ç
ç
ç
è
æ
++
++
=
ml
l
ml
mml
l
ml
m
s
psF 1
.0
ú
ú
ú
ú
û
ù
÷
÷
÷
÷
ø
ö
ç
ç
ç
ç
è
æ
++
-
+
-
+
++
+
ml
m
ml
mml
l
ml
l
ml
s
1.(16)
The esul will be ob ained by he in e se Laplace
ans o m:
() (){ } ()
+
ê
ê
ê
ê
ë
é
÷
÷
÷
÷
ø
ö
ç
ç
ç
ç
è
æ
++
++
==-
ml
l
ml
mml
l
ml
m
0L
1psF p
INFORMATION AND COMMUNICATION TECHNOLOGIES AND SERVICES, VOL. 9, NO. 1,MARCH 2011 3
©2011 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING ISSN 1804-3119
ú
ú
ú
ú
û
ù
÷
÷
÷
÷
ø
ö
ç
ç
ç
ç
è
æ
++
-
+
-
+
++-
ml
m
ml
mml
l
ml
l
ml
.
)(
e.(17)
Hence
[ ]
[ ]
ml
ml
mm
ml
ml
)0()0(
)0()0(
1
)(
10
)(
100
pp
pp p
-
+
+
++
+
=
+-
e,(18)
[ ]
[ ]
ml
ml
ll
ml
ml
)0()0(
)0()0(
1
)(
10
)(
101
pp
pp p
+-
+
+
++
+
=
+-
e.(19)
These equa ions desc ibe he Ma ko andom p ocess
wi h 1 channel in ha manne ha hey gi e he
p obabili y wi h which he andom p ocess will s ay in one
o bo h s a es in an eligible u u e ime .
2) Co ela ion and Co- a ia ion Func ion
The alue o he andom p ocess X( ) acqui es in a
ime he andom a iable X ha can acqui e only 2
alues: x0 = 0 i a da a bu s is no p esen o x1 = 1 i a
da a bu s is p esen . Thus, he e a e 4 a ious
combina ions o p oduc s o de e mine he co ela ion
unc ion R(
):
() ( ){ }
() ( ){ }
() ( ){ }
() ( ){ }
() ( ){ }
1111
})()({..
)().()(
1111
0101
1010
0000
1
0
1
0
=+Ç=××=
==+Ç=××+
+=+Ç=××+
+=+Ç=××+
+=+Ç=××=
==+Ç==
=+=
åå
==
u
X XP
x Xx XPxx
x Xx XPxx
x Xx XPxx
x Xx XPxx
x Xx XPxx
X XR
ij
jiji
(20)
I x0 = 0, only he las om 4 a icles has a non-ze o
alue. As he andom p ocess is also s able and e godic, i
does no depend on he alue , so i can be = 0, oo.
Then:
(
)
(
)
(
)
(
)
(
)
() (){ }
110
0
=Ç=
=×=+×=
u
XXP
XXX XR .(21)
The p obabili y P{X(0) = 1 ÇX(
) = 1} can be ge
om he o mula o he condi ional p obabili y:
(
)
(
)
{
}
(){ } () (){ }
10/110
110
==×=
=
=
Ç
=
XXPXP
XXP
.(22)
Supposing s abili y o he andom p ocess wi h
p obabili y P{X(0) = 1}, ha he communica ion channel
will be occupied a he beginning (in he ime = 0). The
condi ional p obabili y is equal o P{X(
) = 1/X(0) = 1}, i
he communica ion channel was occupied a he
beginning, and i i will be also occupied a e a ime
, i
can be calcula ed om he equa ion (19), in which p0(0) =
0 and p1 = 1:
ú
ú
û
ù
ê
ê
ë
é+
+
=
=
ú
ú
ú
û
ù
ê
ê
ê
ë
é
+
÷
÷
ø
ö
ç
ç
è
æ+
=
=
+
+
+
==
=
=
=
=
+-
÷
÷
ø
ö
ç
ç
è
æ+-
+-
J
m
m
l
ml
l
m
m
l
m
l
ml
m
ml
l
).1(
..1
)(
1
.
1
1.
1
.1.
1
.)(
}1)0(/1)({}.1)0({)(
a
aa
a
p
XXPXPR
e
e
e
(23)
ú
ú
û
ù
ê
ê
ë
é+
+
=+-
J
).1(
.
1
1.
1
a
aa
ae.(24)
he eby he equa ions (2) and (3) we e used, oo.
Then he co ela ion unc ion R(
) using (22) and
(24) will be:
ú
ú
û
ù
ê
ê
ë
é
+
+
=+-
J
).1(
.
1
1.
1
.)( a
aa
a
yR e.(25)
The ollowing ela ionship holds be ween a ic
a o e ed o a communica ion channel and hei a ic use
y:
(
)
apy ×-= 1
1.(26)
whe e p1 is he p obabili y ha a channel is occupied in
he s eady s a e:
1
1
1lim
1
)( )1(
11 +
=
ú
ú
û
ù
ê
ê
ë
é
+
+
=¥= +-
¥® a
a
aa
a
pp
a
J
e. (27)
Subs i u ing in o (24) we ob ain:
.
1
1
1
1
1.
1
.).1()(
)1(
2
)1(
1
ú
ú
û
ù
ê
ê
ë
é+
÷
ø
ö
ç
è
æ
+
=
=
ú
ú
û
ù
ê
ê
ë
é+
+
-=
+-
+-
J
J
a
a
aa
a
aa
a
apR
e
e
(28)
Co- a ia ion unc ion is:
(
)
(
)
2
mRK -=
.(29)
whe e m is he mean alue (mean le el) o he andom
p ocess. Then
4INFORMATION AND COMMUNICATION TECHNOLOGIES AND SERVICES, VOL. 9, NO. 1,MARCH 2011
©2011 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING ISSN 1804-3119
[ ]
2
2
2
1
2
1
.
1
1).1(. ÷
ø
ö
ç
è
æ
+
=
ú
û
ù
ê
ë
é÷
ø
ö
ç
è
æ
+
-=-== a
a
a
a
a
apyym .(30)
So he co- a ia ion unc ion will be:
.
)1(
1
1
1
1
)(
)1(
2
2
)1(
2
J
J
+-
+-
+
=
=
÷
ø
ö
ç
è
æ
+
-
ú
ú
û
ù
ê
ê
ë
é
+
÷
ø
ö
ç
è
æ
+
=
a
a
a
a
a
a
aa
a
K
e
e
(31)
I a will be exp essed by means o y om (26) and
(27), we ob ain:
y
y
a-
=10
<
y
,
and hen
J
J
.
).1(
1
1
1
2).1.(
1
1
1
)( y
y
y
yy
y
y
y
y
K-
-
÷
÷
ø
ö
ç
ç
è
æ+
-
--=
÷
÷
ø
ö
ç
ç
è
æ+
-
-
=ee .(32)
Dispe sion is:
(
)
(
)
yyK -== 10
2
s
.(33)
3) Spec um
The spec um o a andom p ocess can be calcula ed
by means o he Wiene -Khin chin ans o m which is he
Fou ie ans o m o a co- a ia ion unc ion:
ò
¥
¥-
-
=
w w
de j
).()( KS.(34)
Le in he equa ion (32)
J
a
).1(
1
y-
=.(35)
and
s
2 is gi en by (33). Then
== ò
¥
¥-
-
-
sw w
a
dee j
..)( 2
S
=
ú
ú
û
ù
ê
ê
ë
é+= ò ò
¥ ¥ +---
0 0
)()(2
s wa wa
dede jj
22
22 211
wa
a
s
wawa
s
+
=
÷
÷
ø
ö
ç
ç
è
æ
+
+
-
=jj . (36)
0 0
R(
)
a
2
S(
w
)
0
w
a) b) c)
X
( )
Fig. 2. Shape, co- a ia ion unc ion and spec um o he andom
p ocess
The da a bu s s ain, he shape o he co- a ia ion
unc ion and he spec um belonging o i a e d awn on
Figu e 2 o he compa ison. The spec um on Figu e 2 c)
has a shape o a p obabili y dis ibu ion which can
ad an ageously be used o de e mina ion o he sampling
equency.
2.2 Sampling F equency
I he spec um S(
w
) is o be conside ed o a
p obabili y dis ibu ion, i mus be ul illed:
ò
¥
¥-
=1)(.
ww
dSc,(37)
hence
2
2
2
22
22
1
1
2
1
2
1
ps
sw
wa
a
s
=
+
=
+
=
òò ¥
¥-
¥
¥- x
x
cd
d
. (38)
So he dis ibu ion will ha e he shape:
22
1
.)(
wa
p
a
w
+
= ,(39)
which is he Cauchy dis ibu ion and i exp esses he
p obabili y densi y o occu ence o ampli udes o
pa icula spec al componen s. Then he elemen a y
ec angle (
w
). .D
w
in Figu e 3 gi es he p obabili y wi h
which he componen s in he in e al om
w
o
w
+ D
w
will occu in he spec um. All spec al componen s in he
ange om -¥ o +¥ occu wi h he p obabili y P = 1
because he in eg al om (
w
) in his ange equals o 1.
Tha means i we wan o ca ch all changes in he cou se
o he andom p ocess wi h he o al ce ain y, i mus be
sampled wi h he in ini e high equency. The e o e, le
he sampling equency is limi ed by he equency
w
and
hen he p obabili y P wi h which each change will be
caugh in he andom p ocess a his sampling equency
will be calcula ed:
INFORMATION AND COMMUNICATION TECHNOLOGIES AND SERVICES, VOL. 9, NO. 1,MARCH 2011 5
©2011 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING ISSN 1804-3119
(
w
)
0
Dw w
+
w
-
w
Fig. 3. Dis ibu ion o ampli udes o spec al componen s in he
andom p ocess
a
w
p
wa
w
p
aw
w
Pa c g
2
22 =
+
=ò
-
d.(40)
The sampling equency o he sampling pe iod can
be calcula ed om his equa ion o he gi en p obabili y
P:
2
g
)1(2
1
2
g.
2
1P
y
P
p
Jp
p
a
p
-
== ,(41)
2
co g.)1(2 P
yT
p
Jp
-= .(42)
3. Expe imen al
3.1 Analysis o Resul s
The equa ions (41) and (42) gi e he sampling
equency o sampling in e al, espec i ely in he case
da a bu s s occu o ally andomly and hei leng h is
o ally andom, oo. I is also he case when he limi
si ua ion can occu when da a bu s s ollow consecu i ely
each o he and he gap be ween hem is in ini ely small (y
® 1), o he da a bu s s leng h is in ini ely sho (
u
® 0),
o bo h cases occu simul aneously.
3.2 P ac ical Applica ion
An example whe e his heo y could be applied is he
gigabi passi e op ical ne wo k (GPON) [5], [6], [7], [8],
[9]. The GPON is an access ne wo k whe e mo e
subsc ibe s (up o 128) sha e he same op ical ib e. The
ups eam communica ion be ween an op ical ne wo k
e mina ion (ONT) on he subsc ibe side and he
b oadband node – op ical line e mina ion (OLT) on he
ne wo k side goes h ough op ical pulses bu s s called
ansmission con aine s (T-CONTs) ca ying da a s eams.
I is necessa y o hinde o e lapping T-CONTs o
simul aneously communica ing ONTs. The e o e a ime
window is ma ched o each T-CONT om OLT. This
window de ines he T-CONT leng h and he ime ins an s
in which an ONT may send T-CONTs wi h da a. To
de e mine hese p ope ly, The OLT ei he ecei es and
decodes dynamic bandwid h assignmen (DBA) epo s
coming om ONT, o moni o s he a ic load on each
b anch in he op ical ne wo k. The i s me hod is called
he s a us DBA and he second he non-s a us DBA.
The s a us DBA equi es exchange o a ious con ol
in o ma ion be ween an OLT and an ONT in heade s o
da a ba ches. The non-s a us DBA does no equi e any
in o ma ion exchange bu i is necessa y o know how
o en he a ic s a us shall be sampled on a ne wo k
b anch. He e he o mula (41) o (42) can be help ul.
The GPON bi a e is 1 244,16 Mbi /s in he
ups eam. The sho es da a uni is one 125 ms long ame
which is pe o med by 19 440 by es. The longes da a uni
in he ups eam is 216 = 65 536 by es which las s 421,4 ms.
Le ´s ake he mean alue o hese 2 igu es o he a e age
bu s leng h:
s2,273
2
4,421125
mJ
=
+
=.(43)
Conside ing P = 0,95, we ob ain o sampling
in e al om (42):
)1.(89365
2
95,0
co g.2,273).1(2 yyT -=-=
p
p
[ms], (44)
(
)
yT -×» 166 [ms]. (45)
The longes sampling in e al will be when he e is
no a ic load (y = 0):
66
max, =
Tms. (46)
In o de o a oid collision o 2 consecu i e T-
CONTs, he gua d in e al consis ing o 32 bi s is
in oduced in he ups eam. They pe o m he gua d ime
32/1 244 160 000 = = 25,72 ns. Du ing his ime, ONT
emi s no ene gy in o he op ical ib e. Tha means he e
a e 4-by e gaps in he 65 536-by es bu s s. Tha ep esen s
he a ic load o
99994,0
53665
453665 =
-
=y.(47)
and so he sho es sampling in e al will be:
(
)
smsT
m
4004,09994,0166
min, ==-×= . (48)
I is necessa y o ake in o conside a ion ha he
maximum and minimum sampling in e als T ,max and
T ,min a e calcula ed on condi ion ha he da a bu s s occu
andomly on he op ical ibe and hei leng h is also
andom in he ange om 125 o 421,4 ms.
E en he minimum sampling in e al o 4 ms is
5000- imes longe han i each pa icula bi should be
sampled which would gi e he sampling in e al o 1/1
244 160 000 = 0,8 ns.

6INFORMATION AND COMMUNICATION TECHNOLOGIES AND SERVICES, VOL. 9, NO. 1,MARCH 2011
©2011 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING ISSN 1804-3119
4. Conclusion
Applica ion o his heo y on da a s eams con ol
may sa e many wo king ope a ions o digi al
mic ocon olle s and hus enhance hei pe o mance.
Acknowledgmen s
The au ho s g a e ully acknowledge suppo om he
VEGA p ojec No. 1/0375/08 “The Analy ic Model Home
Access Ne wo k New Gene a ion”.
Re e ences
[1] LEVIN, B., L.:Teó ia náhodných p oceso a jej aplikácie
ádio echnike. SNTL P aha, 1965.
[2] SVEŠNIKOV, A. A.: Sbí ka úloh z eo ie p a děpodobnos i,
ma ema ické s a is iky a eo ie náhodných unkcí. SNTL P aha, 1971.
[3] VENTCEĽOVÁ, J., S.: Teó ia p a depodobnos i. Al a-SNTL
B a isla a, 1973.
[4] PRCHAL, J.: Teo ie p a děpodobnos i e sdělo ací echnice. NADAS
P aha, 1975.
[5] A b oadband op ical access sys em wi h inc eased se ice capabili y
using dynamic bandwid h assignmen . ITU-T Recommenda ion
G.983.4, 2001.
[6] Gigabi -capable Passi e Op ical Ne wo ks (G-PON): Physical Media
Dependen (PMD) laye speci ica ion. ITU-T Recommenda ion
G.984.2, 2003.
[7] Gigabi -capable Passi e Op ical Ne wo ks (G-PON): T ansmission
con e gence laye speci ica ion. ITU-T Recommenda ion G.984.3,
2004.
[8] Gigabi -capable Passi e Op ical Ne wo ks (G-PON): ONT
managemen and con ol in e ace speci ica ion. ITU-T
Recommenda ion G. 984.4, 2004.
[9] Gigabi -capable Passi e Op ical Ne wo ks (G-PON): Enhancemen
band. ITU-T Recommenda ion G. 984.5, 2007.
Abou Au ho s ...
Gus a CEPCIANSKY was bo n in 1947.
1971 – Mas e o sciences (MSc) in
elecommunica ions on Uni e si y o
Žilina. 1971 – 1976 – employed a Slo ak
Telecom as a echnical de elopmen
specialis . 1976 – 2002 – employed a Slo ak Telecom as
he head o he ope a ional esea ch cen e. 2002 – 2009 –
employed a T-Com as a echnical assis ance cen e
specialis and a he same ime as a lec u e on he
Telecommunica ion Depa men o he Elec ical Facul y,
Uni e si y o Žilina. Nex o ma ion: 1984 – he 1s doc o
deg ee (CSc). 1995 – he s udy s ay a Technical
Uni e si y Supélec in Pa is. 1998 – he 2nd doc o deg ee
(PhD). 2006 – he associa ed p o esso o Elec ical
Facul y, Uni e si y o Žilina.
Ladisla SCHWARTZ was bo n on 1950
in Žilina. In 1974 g adua ed om he
Uni e si y o Žilina, Depa men o
Telecommunica ions wi h he Mas e ’s
deg ee (MSc). F om 1974 o 1991 he wo ked
in he Resea ch Ins i u e o Compu e
Technology in Žilina as he head o he Depa men o
Da a Communica ions. In 1986 he was awa ded he i le
PhD and in 1991 he became a ull ime eache and
esea ch wo ke o he Uni e si y o Žilina. In 1999 he
was appoin ed an associa e p o esso . His main ocuses a e
unc ions, eliabili y, secu i y elecommunica ion and
compu e ne wo ks and da a communica ions.