2. 892
group of sources, requir~ for
service ( channels (k=1,u;(=1,k);
Nk - number of sources in k-th group
of sources of traffic (k=1,u).
2.2. The Analitical Model of System
Let us assume that u traffics with
parameters a
k
, k=1,u are offered to
an EIG hav~ V channel and characte-
rised by accessibility d. A call from
k-th group of sources requiring for
service ( channel is offered with pro-
bability wk,i' Evidently, u~. Farther.
we will name call. requir~ for ser-
'vice ( channels an (-channel call.
Operation of system can be described
by Marcovian process
,X _{ (1). (2) ( 2 ) (u ) (u )}
, - x 1 .x 1 .x2
: ••• :x 1 •... xu
(J)
!where Xi - number of (-channel calls
'from J-th group of sources served by
system. Let S be a set of states of
process:
'We define two sets of neighbour states
for each state X: X+. X-.
x:{x+ ={x(1):"'X(J~1 ... ;x (~?.,x(u)}}J.i 1 i 1 u
i~-lxl
x={x- ={x(1): ... x(J~1 ... :x(U~ ...• x(u)}1
J.i 1 ( 1 u r...
x(J )~O
(
rherefore it is possible to write the
' follow~ linear equation system:
[~ ~a W (N _:(J;& +
J=1 (=1 J J.t J i Ixl.i
U J] (1)(I) u ~ (J)
+I IT (x (4 " Px=I IT (x "+1)P + +
J=1(=1 J. J.i ( XJ (
J=1(=1 •
u J (J)
+I Ia w (N -x +1)~ P
J J.i J i Ixl-1.i xJ
-. i
J= 1 i= 1
where l=min(u. v-IXI). The conditional
probability of transition from state
with Ixl busy channels to sta~: e with
IXI+i busy channels due to offer~ an
i-channel call is
Ixl+i-1
~ = r-r ~(J)
Ixl. i J=lxl
cl, d
where ~(J)=1- CJ/Cv '
(2 )
Solv~ this system yields
probability of state
stationary
Ixl-1
[
(1) (u) (u) nx
1
: ... :x
1
..... x
u
]=[0] ~(k) x
k=O
[
w a ]xiJ)
u u J J.i J
,rT NJ I nn T
J .( (3)
J (J) (J)
J=1 (N _, x ) J=1i=1 Xi
J i~1 i
With the condition that the sum of all
probability of states is unity one ob-
tains
Ixl-1 u
[0]-1=~ n~(k)rT
8
.&=0 J=1
(j)
x mUJ [W;;~:J]:I
(J)
J=1 i=1 Xi 1
NJ. J
x
J (J)
-2(N
J Xi )
i=1
' ( 4)
3. Therefore the probability that K chan-
nels are busy is
[KL=I [,<1 ~••• ::c :~: ... :c: ..)] (5)
BK
where BK={X. IXI=K}
Intensity of traffic from (-channel
calls is defined by
AI=II[t h-i~:J)h wJ·lK=O BK J=t
)( [x(1 ~ ••• i x(~~ ...x(u)]
1 1 u
(6)
So. the probability of t-channel calls
congestion when IXI~-( is
u J
V Ih L(J)]x t
Cl.
J wJ • (
IIJ:I (=1
Pv . (=
K=V-(+1BK
A t
x [x(1 ~ ••• i x(~~ ...x(u)]
1 1 u
Total call congestion in a route is
defined by
u
Pv=L O(Pv . (
(=1
u
(7 )
(8 )
where 0t= ~ W • Internal blocking
k={. k. t
probability for t-channel call is
V-t
L=d-t+1
where
(9)
x
m(n(L.d)
(t) "
0' (L)=L
k=d-(+1
Total internal block~ is
u
Pb=IOtPb. t
t=1
Carried traffic is defined by
{.=1
Total call . congestion is
P=Pb+PV
893
(10)
(11 )
(12 )
(13 )
Consider a special case of the mo-
del. Let wk. t=w{.; 'T k. {.='T (;Cl.k=CI.. k=1.u;
NJ=O. J=1 .u-1; Nu=N; O(=w(. Then equa-
tions (3)-(7) for this case is
Ixl-1
h····· :c..]=(o]nl'(k)
N
)(
(14)
4. 894
For calculalion [X]vthe follow~ algo-
rithm can be propo.sed. Let
O.K<O
i(X)= 1.K=O (18)
:t~[=:,]§(X-' +-X:' ]. X>O
(=1
where symbol [ ] denotes integer part
of number. Then
[X] =§(XJ I}(J/?§(mJI}(J)
Finally. for ' Poissonian t~affic
(GHQ. N~oo. o..N~A)
where A :A. ( IT,. >..,:A,w • ,=1., u.
(19 )
(20)
2.3. Calculation Method for Multistage
L~ System in Group Selection
We will interpret accessibility d
of EIG as an effective accessibility
deff (see Appendix) of trunk group o~
l~ system. Then. the described model
can be used for approximative calcula-
t~on of probability charaoteristics of
Multi-o~annel Link Systems (MLS).
The iteration algorithm presented
below can be proposed ~or practical
usei it is assumed. that MLS structure
and tra~~ics d1stribut~on are known:
1. Let deff=V. where V-total number of
channels in the route.
2. Calculate PV." Pb ,(' p. '=1,u.
See (7)-(10). (13).
3. Calculate Y. See (12)
4. Calculate ~ef!' ~ee . Ap~endix.
5. Calculate P V,('p b,t'P, (=1,u.
See (7) - (1Q ), ( 13) •
6. I~ Ip -P'I~ (& - absolute error),
then end the calculation, otherwise
, , .p=p, PV,(=PV ,(' P b ,t=Pb ,(,t=1,u and ·
go to point 3.
2.4 Example
Monte Carlo simulation was applied
~or chek~ the accuracy of the model.
Simulation and theoretical results were
compared. In F~g. 1 blocking probabili-
t~es for 1-channel. 2-channel and
4-channel calls are demonstrated by
2-stage link system with the following
structure (short notation of system
structure is taken from [2]);
00 1128
16
16 16
128
It is proposed that MLS should have the
number of routes - h=16 with V=128
channels in each route and sequental
hunting mode. Offered traffic is Pois-
sonian with the following parameters:
w1
=O.85, w2
=0.1, w4
=O.05.
1.-------.--------.------~
0.1
0.01r---~--~~----_+------~
0.001
~------~--------~------~
0.70 0.75 0.80 0.85
carried traffic per trunk
Fi9. 1 Internal blockin9 probabilities
versus carried traffic per trt~
(simulation witJl 95-percent con-
5. 3. PERFORMANCE ANALYSIS OF LINK SYSTEMS
WITH POINT-Ta-POINT SELECTION
3.1 The Analitical Model of
Point-to-point System
Let us assume that Poissonian
traffic with parameter A is offered to
an EIG having V channel and characteri-
sed by accessibility a. An t-channel
call t=1,u. that offered to a grading
group hunted, at first, t free channels
from a total number channels V in the
route. If there are not t free chan-
nels the call is lost. If there are t
free channels and the grading group has
access to all choosen free channels the
call is served, otherwise it is lost.
Distribution of service time of t-chan-
nel call is negative exponential with
parameter Tt' t=1,u.
Probability that there are K free
channels accessible to fixed grading
group when J channels are busy is
a-K J-(a-K) J
Ca CV_a ICv (21 )
Probability that from the rest of V-J
free channels, m channels accessible to
fixed grading group, under condition
that there are ~ such channels, will be
found after exactly m attempts:
nJ m m
O-v-J.m=C~CV-J (22 )
The conditional probability of transi-
tion from state with J busy channels to
state with J+m busy channels is due to
offering m-channel call
mtn(a,V-JJ
_ " (K) (K)
~J,m- ~ PJ O-v-J,
K=max(m,a-J)
min(d,V-JJ
~ d-K K-m
d ~ CJ CV - J - m
Cv K=max(m,d-JJ
(23 )
Compar~ the factors of the x a-m term
in the left and the r~t parts of the
identity
J V-J-m V-m
(1+x) (1 +x) = (1 +x)
we have
a-m a (m)
~J,m=CV-m ICV=Fd
895
(24 )
(25 )
The factor F~mJdoes not depend on J .
Therefore in this case, system (1)
transforms to system of l~ear equati-
ons like for the case of full-access
trunk group ([4]-[6]) and probability
of state will be
The t-channel call congestion is
Pvf ~ [J]v
J=V-t+1
Internal blocking probability of
i-channel calls
PO.t=(Pt-PV,t)/(1-PV,t)=1 - F~t)
(28)
(29 )
3.2. Calculation Method for Multistage
Link System in Point-to-point
Selection
Analogously to Section 2.3 we will
use the ideas of EIG and effective ac-
cessibility for approximative calcula-
tion probability characteristics of MLS
in point-to-point selection.
For practical using of discribed
model can be proposed the following
iterationing algorithm~ it is assumed,
that MLS structure and traffics distri-
butions are known:
1. Let PV,t=O' t=1,u.
6. 896
2. Calculate A' z=A z(1-P
V
z),'=1,u,
C,lo C,lo ,10
where A
c
,,- traffic per multiple of
last but one stage.
,
3 . Us~ the traffic Ac.~ and results
of section 2 calculate probability n
as point-to-point s~le time slot
congestion.
4. Calculate a 11' See Appendix.
(,)8 _
5. Calculate Fa ,pb",PV .' • '=1.u.
See (28). (29).
6. If IPV.1-P~.11~ c (c - absolute er-
ror). then end the calculation.
" ,
otherwise PV,,=Pv,t' Pb,,=Pb ,(' '=1.u
and go to point 2.
3.3. Example
Monte Carlo.' simulation was applied
for chek~ the accuracy of the mode~.
Simulation and theoretical results were
compared. In Fig. 2 and Fig. 3 blocking
probabilities and call congestions for
1-channel. 2-channel and 4-channel
calls are demonstrated by 3-stage link
system with the follow~ structyre:
00 I 64
48
48 48
64
64 64
48
I t is proposed that MLS should have the
number of routes - h=64 with V=48 chan-
nels in each route and sequental hunt-
~ mode. Offered traffic is Poissonian
with the follow~ parameters: w1
=0.8S.
w2
=0.1. w4=0.OS. Simulation results
show that it is possible to neglect the
-losses in the first stage of this link
system.
1
y
0.001~------~------~------~
0.65 0.70 0.75 0.80
carried traffic per trt~
Fig. 2 Internal blocking probabilities
versus carried traffic per trunk
(simulation witJl 95-percent con-
fidence interval)
1
p
0.1
o.01 ~"------..,,......c:::;.-------I----------l
0.001
0.65 0.70 0.75
carried traffic per trt~
y
0.80
Fig. 3 Call congestions versus carried
traffic per trunk <:silnulation
with 95-percent confidence in-
terval)
APPENDIX. EFFECTIVE ACCESSIBILITY
In [3] it was shown that for a
link system with arbitrary structure
the effective accessibility de!! is
00 (In 11) (-1
del!=~m, ,! (30)
(=1
where m( - semi-invariant of (-th or-
der, 11 - carried traffic per trunk in
7. the route. If d
eff
is approximated by
two f .irst terms of (30) (the approxima-
tion of second order). then
(2 )
defl ~ m1 + 0.5 m2ln(y) (31 )
Where m
1
=E(d) - the mean of accesiibi-
lity. ~=var(d) - variance of accessi-
bility. Thus
(2 )
defl ~ E(d) + 0.5 var(d)ln(y) (32 )
Values E(d) and var(d) were investiga-
ted in (1]. It was shown that
ny
E (d) ~ ( 1-rr )V + g +
g-1
g
(33 )
-1 -2
var(d)~rr(1-rr)(1+g -2g ) (34)
where
rr - the point-to-point time conges-
tion;
V· - the number of channels in the ro-
ute;
Y1- the carried traffic per multiple
in stage number one;
n
1
- the number of inlets per multiple
in stage number one;
Y1= Y1 /n1;
~ - a number of traffic groups.
Number of traffic groups is
g=N/N (35)g
where N -total number of link system
inlets. N - a maximal number of linkg
system inlets when link system is still
non blocking.
Consider the effective accesibili-
ty factor fd
'd =defllV
From (33) • (34 ) we have
rr 1 g - 1
fd~(1-rr)+8+
V g
-1 -2
+ rr(1 - rr)(1+g -2g )
(36)
1-
n -1
Y1 1
~ Y
1
+
n1 y
1
1
(37 )
As follows from (37) the value fd is a
weak-dependent on the number of chan-
nels in the route - V and of the traf-
fic in the route - A • when rr=const.
897
Thus. for calculation characteristics
of large capacity link systems the ef-
fective accessibility factor 'd can be
accepted as an invariant.
We will illustrate the behaviour
of 'd by example of link system from
[2]. Let the given link system have un-
equal group sizes and nonuniform offe-
red traff~cs per group. Total offered
traffic A=248.92 erl. The system struc-
ture is:
5 8 5 5 5 5 8 5
50 80 80 50
Tab. 1 shows effective accessibility
factor 'd and simulation results for
probability P of loss for various valu-
es of offered traffic and number of
channel in the route.
No of Number Traffic P
'droute cf chan- offered
nels in to the
the route route
1 25 14.94 0.008± 0.58
0.002
2 25 29.87 0.251± 0.60
0.002
3 50 29.87 0.0004 0.58
±
0.0003
4 50 87.12 0.452± 0.60
0.001
5 100 87.12 0.025± 0.59
0.001
Tab. 1 Effective accessibility factor
versus offered traffic. number
of channel and probability of
loss.
The shown properties of effective ac-
cessibil~ty factor Id give us the pos-
sibility to essentially accelerate the
route d1mensioning.
ACKNOWLEDGEMENT
The authors wish to thank P. Kahn
(University of Stuttgart. Germany) and
8. 898
U. Herzog (Friedrich-Alexander-Univer-
sitat. Erlangen-NQrnberg. Germany) for
many valuable discussions.
REFERENCES
[1 ]
[2 ]
Ershova E.B .• Ershov V.A .• Digital
Systems for Information Distributi-
on (Radio and Communication Publ.
1983 in Russian).
Bazlen D.• Kampe G.• Lotze A •• On
the Influence of Hunting Mode and
Link Wiring .on the Loss of L~
System (7th ITC. Stockholm. 1973)
pp. 232/1-232/12.
[3] Pedersen Ole A •• An Effective
Availability Theory with Appli-
cations (Vol. COM-23. No 8. August
1975) pp. 798-803.
[4] Fo~tet R.M .• Grangean C.H .• Study
of Congestion in Loss System
(4th ITC. London. 1964).
[5] Gimp~lson L.A .• Analysis of Mixtu-
res of Wide and Narrow-band Traffic
(IEEE Trans. on Comm.• Technology.
Vol. 13. No 3. 1965).
[6] Conradt G.• Buchheister A •• Consi-
deration on Loss Probability of
Multi-slot connections (11th ITC
Tokyo. 1985) pp. 4.4B-2-1 -
4.4B-2-7.
![892
group of sources, requir~ for
service ( channels (k=1,u;(=1,k);
Nk - number of sources in k-th group
of sources of traffic (k=1,u).
2.2. The Analitical Model of System
Let us assume that u traffics with
parameters a
k
, k=1,u are offered to
an EIG hav~ V channel and characte-
rised by accessibility d. A call from
k-th group of sources requiring for
service ( channel is offered with pro-
bability wk,i' Evidently, u~. Farther.
we will name call. requir~ for ser-
'vice ( channels an (-channel call.
Operation of system can be described
by Marcovian process
,X _{ (1). (2) ( 2 ) (u ) (u )}
, - x 1 .x 1 .x2
: ••• :x 1 •... xu
(J)
!where Xi - number of (-channel calls
'from J-th group of sources served by
system. Let S be a set of states of
process:
'We define two sets of neighbour states
for each state X: X+. X-.
x:{x+ ={x(1):"'X(J~1 ... ;x (~?.,x(u)}}J.i 1 i 1 u
i~-lxl
x={x- ={x(1): ... x(J~1 ... :x(U~ ...• x(u)}1
J.i 1 ( 1 u r...
x(J )~O
(
rherefore it is possible to write the
' follow~ linear equation system:
[~ ~a W (N _:(J;& +
J=1 (=1 J J.t J i Ixl.i
U J] (1)(I) u ~ (J)
+I IT (x (4 " Px=I IT (x "+1)P + +
J=1(=1 J. J.i ( XJ (
J=1(=1 •
u J (J)
+I Ia w (N -x +1)~ P
J J.i J i Ixl-1.i xJ
-. i
J= 1 i= 1
where l=min(u. v-IXI). The conditional
probability of transition from state
with Ixl busy channels to sta~: e with
IXI+i busy channels due to offer~ an
i-channel call is
Ixl+i-1
~ = r-r ~(J)
Ixl. i J=lxl
cl, d
where ~(J)=1- CJ/Cv '
(2 )
Solv~ this system yields
probability of state
stationary
Ixl-1
[
(1) (u) (u) nx
1
: ... :x
1
..... x
u
]=[0] ~(k) x
k=O
[
w a ]xiJ)
u u J J.i J
,rT NJ I nn T
J .( (3)
J (J) (J)
J=1 (N _, x ) J=1i=1 Xi
J i~1 i
With the condition that the sum of all
probability of states is unity one ob-
tains
Ixl-1 u
[0]-1=~ n~(k)rT
8
.&=0 J=1
(j)
x mUJ [W;;~:J]:I
(J)
J=1 i=1 Xi 1
NJ. J
x
J (J)
-2(N
J Xi )
i=1
' ( 4)](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)