Traffic Characteristics and Queueing Behaviour of Discrete-time On-off Sources.

Abstract

In this paper, some results are presented from an attempt to study — in a discrete-time setting — the phenomenon of long-range dependence. For a well-known source model, traffic characteristics such as the power spectral density and the index of dispersion for counts are analyzed. Based on these characteristics, the distinction between short-range and long-range dependence is touched upon. The traffic generated by a superposition of sources is also studied, whereby the case of an infinite number of sources gets special attention. In this, the discrete-time version of the M/G/∞ queue model is called upon. Finally, some results pertaining to the queueing behavior of such traffic are discussed.

Full text

Traffic
characteristics
and
queueing
behavior
of
discrete-
time
on-off
sources
K.
Laevens
and
H.
Bruneel
1
SMACS Research
Group,
University
of
Ghent
Sint-Pietersnieuwstraat
41, B-9000
Ghent,
Belgium
Email: kl,hb@lci.rug.ac.
be
Abstract
In this paper, some results are presented from an attempt to
study -in a discrete-time setting -the phenomenon of long-range
dependence. For a well-known source model, traffic characteristics
such as the power spectral density and the index of dispersion for
counts are analyzed. Based on these characteristics, the distinc-
tion between short-range and long-range dependence
is
touched
upon. The traffic generated by a superposition of sources
is
also
studied, whereby the case of an infinite number of sources gets spe-
cial attention.
In
this, the discrete-time version of the
M/G/oo
queue model
is
called upon. Finally, some results pertaining
to
the queueing behavior of such traffic are discussed.
1
Introduction
Since
the
notions
'long-range
dependence'
and
'self-similarity'
[1,
10,
16]
have
been
brought
to
the
attention
of
the
teletraffic community,
a
tremendous
amount
of
research effort
has
been
spent
on
the
sub-
ject.
Issues
of
importance
hereby
are
traffic analysis
and
modelling,
and
queueing
analysis. Fluid-flow models
[2,
8,
9,
12]
and
the
related
fractional
Brownian
storage
[13, 14, 18], combined
with
elements
oflarge-
deviations theory,
seem
to
have
been
the
most
successful modelling ap-
proaches yet,
be
it
sometimes
at
the
cost
of
complex
mathematics.
1
The
authors wish
to
thank
the
Belgian Federal Office for Scientific, Technical
and
Cultural Affairs (DWTC)
and
the
Flemish Fund for Scientific Research (FWO-
Vlaanderen) for
support
of this research.
©
The original version of this chapter was revised: The copyright line was incorrect. This has
been corrected. The Erratum to this chapter is available at DOI:
10.1007/978-0-387-35353-1_28
IFIP International Federation for Information Processing
D. Kouvatsos (ed.), Performance Analysis of ATM Networks
2000

What
we
present here, are some results from
an
attempt
to
study
these notions
in
a discrete-time setting. As a source model,
we
opted
for
the
well-known on-off source. Long-range dependence is expected
to
emerge when heavy-tailed distributions for e.g.
the
durations
of
the
on-
periods come into play. These heavy-tailed distributions typically lead
to
probability generating functions -one of
the
basic tools of
our
analysis -
having a
branch
point
at
z =
1.
This
branch point affects
the
use
that
is
made of residue theory
and
urges us
to
reconsider some results obtained
for 'light-tailed' distributions.
This
work is
part
of ongoing research
and
additional
study
is required
to
fill
in
the
remaining gaps
and
to
provide
a more solid
mathematical
framework.
The
paper
is
structured
as follows.
In
the
next section,
the
on-off
source model is introduced.
In
Section
3,
traffic characteristics such
as
the
autocovariance function
and
the
index of dispersion for counts
are analyzed.
The
difference between short-
and
long-range dependent
sources is examined
in
Section
4.
In
Section
5,
the
superposition of
N sources is considered, whereby
the
case N
--+
oo
receives
the
most
attention.
It
leads
to
a discrete-time
M/G/oo
queue model, recently
also
studied
in e.g.
[15,
19].
In
Section
6,
two possible approaches
to
the
analysis
of
the
queueing
behavior-
a Benes approach
and
a slot-to-slot
approach-
are discussed. Conclusions are drawn in Section
7.
2
The
source
model
An on-off source
alternates
between two
states
:
the
on-state -wherein
one cell is generated
per
slot -
and
the
off-state -wherein no cells are
generated, as shown in Figure
1.
I I I I I I I I I I I I I I I I I I I I I I
+---
A(z)
B(z)
___.
A(z)
___.
Figure
1:
A discrete-time on-off source.
The
durations, expressed in numbers
of
slots, of
the
visits
to
the
on-state -called
the
on-periods -are iid
random
variables (rv's) char-
acterized by
the
probability density function (pdf) a(n) =
Pr[TA
=
n]
209

(n
=
1,
2,
...
) or
the
associated probability generating function (pgf)
+oo
A(z) =
E[z'TA]
= L a(n)zn
n=l
Likewise,
the
durations
of
the
off-periods are iid rv's characterized by
the
pdf
b(n) =
Pr[TB
= n] or
the
pgf
B(z)
= E[zrs]. Hereby,
TA
and
TB
were used
to
denote
the
duration
of a generic on- or off-period
respectively. Durations
of
on-
and
off-periods are mutually independent,
their
mean
values equal
E[rA]
= A'(1)
and
E[rB] = B'(1)
respectively. Unless otherwise stated, these values are assumed
to
be
finite. As a consequence,
the
stationary
version
of
the
process, which is
of
interest here, exists. Variances are given by
Var[rA] = A"(1) +
A'(1)-
A'(1) 2
and
a1
= Var[rB] = B"(1) +
B'(1)-
B'(1)2
and
can
be
either finite or infinite.
Two distributions
that
are frequently encountered in
the
rest of
the
analysis, are
the
distributions
of
the
remaining durations
rA.
or
r'B
of
the
on- or off-period
to
which a randomly chosen slot belongs (not counting
the
arbitrary
slot itself).
It
can be shown, see e.g.
[3],
that,
for on-
periods, this distribution is given by
a*(n) =
Pr[rA.
=
n]
=
Pr[TA
> n]/E[rA] n =
0,
1,
...
and
mutatis
mutandis for
the
off-periods.
The
associated pgf's take
the
form
*
A(z)-
1
A (z) = A1
(1)(z-
1)
and
*
B(z)-
1
B (z) =
B'(1)(z-
1)
The
number of cells generated by
the
source during slot k, either 0 or
1,
will be denoted by
qk.
The
average of
qk
can be expressed as
210

and
its variance as
In
the
next section, some further characteristics of
the
traffic process
qk
are derived.
3 Traffic
characteristics
3.1
The
power
spectral
density
The
Fourier-transform
+oo
S(f)
= L
C(m)ei27rmf
m=-oo
of
the
autocovariance function
C(m) =
E[(qo
-
.\)(qm
-.\)]
is
known as
the
power spectral density of
the
traffic process. Studies such
as
[5]
or
[11]
come
to
the conclusion
that
the
power spectral density
at
the
low frequencies has a serious impact on
the
queueing behavior of
the traffic.
The
power spectral density of long-range dependent traffic
behaves totally different
at
these frequencies
than
that
of short-range
dependent traffic.
We
return
on this in Section
4.
In
Appendix
A,
it
is
shown
that
{1)
whereby
with
A(z)
-1
B(z)-
1
[A'{1)
+ B'{1)]{z-
1)
P(z)
= A1
(1)(z-
1)
B'{1){z-
1)
A(z)B(z)-
1
211

Given A(z)
and
B(z),
S(f)
is
easily evaluated.
As
such,
C(m)
can
be calculated by numerical transform inversion, as outlined in e.g.
[4].
Further, it follows from residue theory
that
C ( m) = Res [ a 2
z=O
If
the
only singularities of Q(z) are poles outside the unit disk, say
Zi,
the
above can be rewritten as
C(m)
[a
2
For m large, one can
then
retain only
the
dominant contribution
and
obtain
an
approximation for C(m).
In
that
case,
the
decay of
the
auto-
covariance will
be
dominated by a geometric term, i.e., the process will
be
short-range dependent, as
to
be
discussed later on. For long-range
dependent processes, (one of)
the
generating functions A(z) or
B(z)
will
have a branch point
at
z =
1,
due
to
the 'heavy tail' of
the
distribution
involved. Then,
the
result from residue theory should be reformulated
and
a non-geometric
term
will dominate
the
decay of the autocovariance
function.
A quantity of interest
in
Section 4 is
the
so-called 'DC-component' of
the
power spectral density, given by
(2)
3.2
Index
of
dispersion
for
counts
Another well-known traffic characteristic,
the
index of dispersion for
counts, was discussed in e.g.
[6].
It
is
defined as
IDC(m) = Var[ql + ... +
qm]
E[ql + · · · +
qm]
and
is
related
to
the
autocovariance function
C(m)
by
[6]
IDC(m) = l
(1-
C(k)
k=-m
(3)
In
this respect,
both
traffic characteristics convey the same information.
212

Consider
the
double
transform
+oo
J(z,
t)
=
L
tm
E[zq
1
+
...
+qm]
m=l
of
the
number
of arrivals
during
m consecutive slots. One
can
show
that,
for
the
on-off source model used here,
1
(A'(1)zt
B'(1)t
J(z,
t)
=
A'(1)
+
B'(1)
1-
zt
+
as explained in
Appendix
B.
_
A'(1)B'(1)t2(z-
1)
2
A*(zt)B*(t))
(1-
zt)(1-
t)(1-
A(zt)B(t))
Taking
the
first-order
partial
derivative
with
respect
to
z
at
z
=
1,
one finds
a
I
+oo
m
1 A'(1)t
8z
J(z,
t)
z=l
=
I-1
t
E[ql
+ ... +
qm]
=
A'(1)
+
B'(1) .
(1
-
t)2
from which follows
the
obvious result
Taking
the
second-order
partial
derivative
with
respect
to
z
at
z
=
1,
one finds
after
some
further
manipulation
+oo
m
2
1
+
t-
2tP(t)
L t
Var[ql
+
...
+
qm]
=
a
t ( )
3
1-t
m=l
(4)
From
this, one
can
derive
the
result
(5)
As for
C(m),
transform
inversion,
be
it numerical or based on residue
theory, of
equation
(4) or (5),
then
yields Var[p1
+ ... +
Pm]
or
IDC(m).
(For this,
equation
(
4)
seems more appropriate, since
it
is
of a simpler
nature.)
Of
course,
the
same problems
with
transform
inversion as for
the
power
spectral
density mentioned above will emerge when long-range
dependent
processes are considered.
213

From (5), one
can
also derive
that
as follows from equations (3)
and
(2), see
[6].
3.3
A
special
case
A source whereby transitions from one
state
to
the
other
form a two-
state
Markov chain, is a special case
of
the
more general on-off source
model considered here.
It
is
obtained
when
the
distributions for
the
durations
of
the
on-
and
off-periods are geometric
with
mean
1/a
and
1/
fJ
respectively, i.e., when
A(z)
=
az
1-
(1-
a)z
and
B(z) =
fJz
1 -
(1
-
fJ)z
The
transition
probabilities from
the
on-
to
the
off-state
and
vice versa
are
then
given by a
and
fJ
respectively. Traffic characteristics for
this
specific model can
be
calculated using
standard
techniques from Markov
chain analysis -see e.g.
[7].
One finds, amongst others,
and
2 (
28
( 1 -
om))
V ar[ql +
...
+
qm]
= a m + 1 - o m -1 - o
Hereby o =
1-a-fJ
is one of
the
two eigenvalues
of
the
transition
matrix
governing
the
Markov chain,
the
other
being
1.
It
has been verified
that
these results
are
in
full agreement
with
the
ones obtained
in
the
previous
subsections for
the
more general model. For instance,
1/
o is
the
only
pole
of
P(t),
which is now given by
P(t)
=
1-
o
1-
Ot
Transform inversion is
then
quite straightforward
and
leads
to
the
above
expressions.
214

4
Short-range
versus long-range
dependence
By definition
[10],
long-range dependence
is
present when
+oo
L C(m) =
S(O)
= +oo
m=-oo
Recalling equation (2),
we
see
that
this will
be
the
case when, for in-
stance, = Var[rA]
is
infinite (given
E[rA]
is finite),
the
so-called
'infinite variance syndrome'
[10].
This condition implies
that
A(z) has
a singularity
at
z =
1,
that
cannot
be
a pole since A(1) =
1,
but
is
a
branch point. As a consequence, the distribution a(n) of which A(z)
is
the
pgf, will
not
decay geometrically,
but
hyperbolically, i.e., a(n) will
have a heavy tail.
Well-known heavy-tailed distributions of a continuous random vari-
able are e.g.
the
Pareto-distribution
[16].
In
search of a versatile heavy-
tailed distribution for a discrete-time random variable,
we
focussed on
a distribution based on the hypergeometric function
+oo
r(a
+ n)r(,B +
n)r(T)
n
F(a,
,6;
'Yi
z) =
r(a)r(,B)r(T
+ n)n! z
To
be
more specific,
we
used a generating function of
the
form
A(
4 )
F(a,
,6;
-y;
z)
a,
tJi
-y;
z = z
F(
4. . 1)
a,
tJ,
-y,
The
resulting distribution seems versatile
in
the
sense
that
its
pgf
is
based on a well-studied function for which numerical procedures are
available
[17]
and
that
it has three (real-valued) parameters, which can
be fitted
to
yield e.g. a given mean
and
tail decay.
The
pgf has a branch
point
at
z = 1
and
the
tail of
the
distribution decays hyperbolically as
( a )
"'
r(-y-
a)r(T-
,6)
-(-y-a-,8+1)
a a,tJ;'Y;n
"'r(a)r(,B)r(-y-a-,B)n
(6)
for n »
1.
The
variance
is
infinite
and
long-range dependence will result
whenever 1 <
'Y-
a-
,6
2.
(The lower
bound
is
required for the mean
to
be
finite.)
Throughout this paper, three different distributions for
the
on-periods
will
be
used for illustrative purposes, as summarized in Table
1:
a light-
tailed geometric distribution (A), a heavy-tailed distribution (B) of
the
215

form
(6)
with
infinite variance
and
a
third
distribution (C), also
of
the
form (6), with finite variance
but
infinite
third
moment. Including
the
latter
distribution will allow us
to
distinguish between 'long-range de-
pendent'
features
and
features originating from a 'heavy tail'. All
three
distributions have
mean
E[r
A]
= 100.0 slots.
variance tail behavior
A <+oo
-n
rv
Zo
B =+oo
rv
n-2.5
c <
+oo
rv
n-3.5
Table
1:
Three
different distributions
In
Figure
2,
where these distributions were plotted,
the
slow decay
of
the
tails
of
distributions (B)
and
(C) clearly shows.
In
Figure 3, a
log-log plot
of
the
complementary cummulative distribution, this is even
more apparent.
0
200
400
600
800
1000
l.OE+O
-f-----t----t-----j----+-----j
l.OE-2
l.OE-4
l.OE-6
A
B
c
Figure
2:
Tail behavior, logPr[TA =
n]
ver-
sus
n,
for various types of distributions.
Power spectral densities
of
sources with on-periods as introduced
above, are shown in Figure
4.
A geometrically distributed off-period,
with
mean 25.0 slots, was assumed for all cases, yielding a traffic in-
tensity
of
0.8 Erlang.
It
is known
[10]
that,
for long-range dependent
216

l.OE+O
l.OE+2
1.0E+4
l.OE+6
l.OE+O
-t------==-+------t-----1
l.OE-3
i.OE-6
A
l.OE-9
l_
Figure
3:
Tail behavior, log
Pr[TA
>
n]
ver-
sus log
n,
for various types of distributions.
sources,
S(f),.....,
f-v
or logS(!),.....,
-vlogf,
when
f--+
0, while for
short-range
dependent
sources, log
S(f)
,.....,
log S(O).
Both
types
of
be-
havior are clearly distinguishable in Figure
4.
Note
that
while
the
tail
of
distribution
(C) is also hyperbolic
and
thus
'heavy', it decays
too
fast
to
yield long-range dependence
in
the
strict
sense. Corresponding
autocovariance functions,
obtained
numerically, are shown
in
Figure
5.
Evidence for long-range dependence is also present
in
the
sample
traces presented
in
Figure
6.
The
figure was
obtained
by aggregating
the
traffic over various timescales (1,10,100,
...
10
6
slots respectively).
The
traffic was generated by a superposition (see Section
5)
of 5 iid sources
and
the
total
traffic intensity is 0.8 Erlang.
In
traffic of
type
(B), large
fluctuations occur over large time-scales, while for traffic
oftype
(A)
and
(C), fluctuations quickly die
out
as
the
time
scale increases.
5
Superposition
5.1 N
sources
Traffic characteristics of a superposition of
N
identical
and
independent
sources, are easily derived from those of a single source. Assume
the
N
sources generate
the
aggregate traffic
stream
Pk·
The
mean
total
arrival
217

B
c
1.0E+2
l.OE+l
1------+------+---\--+
1.0E+O
-2
l.OE-1
Figure
4:
Power spectral density,
logS(!)
versus log
f,
for various types of source.
0.20
0.15
0.10
0.05
0.00
0
20
40
60
80
100
Figure
5:
Autocovariance function, G(m)
versus m, for various types of sources.
rate
is
then
>..r
=
E[pk]
= N
>...
Other
traffic characteristics are given by
218

1!,1--------
1
!li
IW
.....,.
___
..._..._........,_
ill
bl
..
J
..
Ll.
LJ.
L
JwJ
,uJI
Figure 6: Aggregated traffic traces for source types
A,
B
and
C (left
to
right) .
e.g.
219

and
From
the
latter
follows easily
Var[p1
+ ... +
Pm]
= NVar[ql + ... +
qm]
l.OE+3
1.0E+2
1.0E+1
1--------
!------+-----+---"'--+
1.0E+0
l.OE-5
1.0E-4
-2
l.OE-1
Figure
7:
Power spectral density, log
S(f)
versus log
f,
for a superposition of sources
of type
A.
(7)
For illustrative purposes, Figures 7
and
8 show the power spectral
density for a superposition of
1,
2,
5
and
an
infinite number of sources.
(The
latter
case
is
treated
in
more detail below.) Figure 7 is for a short-
range dependent source of type (A), Figure 8 for a long-range dependent
source of type (B).
The
total
arrival
rate
was kept constant
at
0.8 Erlang
by varying
the
mean duration of
the
off-periods.
5.2
N
---+
+oo
An interesting case
is
that
whereby
the
number of superpositioned sources
grows infinitely,
with
given
>..r
and
A(z). As illustrated by Figures 7
and
8,
traffic characteristics quickly approach their limiting values as
220

5
2
l.OE+3
l.OE+2
I.OE+l
l.OE+O
-2
I.OE-1
Figure
8:
Power spectral density, log
S(f)
versus log f, for a superposition
of
sources
of
type
B.
the
number
of
sources increases. For
the
power spectral density,
we
find
by
taking
a limit
whereby
A*(z) -1
Q(oo)(z) = z
z-
1
(8)
Concerning
the
total
number
of arrivals in m consecutive slot, see Section
3.2, derivation of
through
a limiting procedure seems more cumbersome. For e.g.
the
variance
of
that
number,
on
the
contrary,
we
easily
obtain
.
+oo
m 1 +
t-
2tA*(t)
hm
L t Var[p1 + ... +
Pm]
=
>.r
(1 )3 t
N-+oom=l
-t
221

through
equations
(4)
and
(7). Numerical or approximate transform
inversion,
then
again yields C(m)
or
Var[p1
+ ... +
Pml·
By observing, however,
that
the
number of arrivals in a slot is equiv-
alent
with
the
number
of
customers in a discrete-time GJ-G-oo queue,
the
equivalent
of
the
continuous time
MIG
I
oo
queue, some more results
can
be
obtained. Recently this model has been studied
in
e.g. [15, 19].
One can show
that
the
numbers
of
newly arriving 'customers' in
each slot, become iid rv's with a Poisson distribution, with mean
>.
* =
>.riA'(1)
and
pgf
exp{>.*(z-
1)}
The
service times
of
the
customers are,
of
course, also iid
rv's
with
pgf A(z),
the
on-time distribution. By analyzing this equivalent queue
model on a slot-to-slot basis,
it
is possible to derive e.g.
that
+oo
C(m)
=
>.*
L
Pr[TA
>
k]
k=m
Note
that
this is
in
full agreement with equation
(8)
derived above.
This expression illustrates once more
that
light-tailed on-periods lead to
short-range dependence, since
+oo
Pr[TA
=
m]
rv
z()m::::}
C(m)
rv
z()m::::}
L:
C(m) <
+oo
m=-oo
On
the
other
hand, for heavy-tailed on-periods one has
+oo
Pr[TA
=
m]
rv
m-q::::} C(m)
rv
m-(q-
2)::::}
L:
C(m) =
+oo
m=-oo
when 2 < q
:::;
3.
Infinite variances for
the
on-periods thus lead
to
long-
range dependence. Note, however,
that
also for q >
3,
as for traffic
of
type
(C),
the
autocovariance function may decay quite slowly, i.e.,
correlation may
extend
over long time periods.
It
does not, however,
lead
to
long-range dependence in
the
strict sense.
One can further derive
that
{
+oo
E[zPI+···+Pm]
= exp
>.*
Pr[TA
>
k](zm-
1)
+
>.*};
Pr[TA
>
k](zk-
1)
+
>.*};
Pr[TA
>
k](m-
k)zk(z-
1)}
222

The
first two
sums
in
the
RHS
represent
the
contribution
of
'old' sources,
i.e., source
that
were
already
active
prior
to
slot
1.
The
last
sum
repre-
sents
the
contribution
of
sources
that
started
generating
cells
during
slot
1
or
later.
Taking
derivatives
and
performing some algebra, one
obtains
m-1
Var[p1 + ... +
Pm]
= m2
AT-
A*
L
Pr[TA
>
k](m-
k)(m-
k-
1)
k=O
in
agreement
with
the
result
obtained
through
a limiting procedure.
Form=
1,
the
above
pgf
reduces
to
E[zP1] =
exp
{A*(z-
1)A'(1)} = exp
{)..T(z-
1)}
The
distribution
of
the
number
of
active sources or, equivalently,
the
total
number
of
cells
generated
in
a
random
slot, is
thus
Poisson
and
function
of
the
load
AT
only.
This
'marginal'
distribution
is a
rather
smooth
distribution
and
is
in
no way influenced
by
the
exact
form
of
the
distribution
of
the
on-periods.
The
latter
does, however,
strongly
affect
the
correlation
structure
of
process.
An
interesting
property
of
the
GJ-G-oo arrival processes is
that
the
aggregation
of
two
or
more such processes is
again
of
that
type.
This
is a consequence
of
the
fact
that
the
arrival process
of
new
'customers'
is Poisson.
The
'parameters'
of
the
aggregated GJ-G-oo
arrival
process
are given
by
A*=
Ai
+
...
+AN
and
A(
) = AiA1(z) + · · · + Al\rAN(z)
z '* '*
"I+
... +
"N
From
this,
it
is easily seen
that
the
tail
of
the
aggregated message
length
will
be
dominated
by
the
heaviest
tale
of
the
constituent
message lengths.
In
other
words, long-range
dependent
processes will
dominate
over
short-
range
dependent
ones.
6
Queueing
Two
approaches
seem promising
to
analyze
the
queueing
behavior
of
traffic
of
the
type
described
above
when
it
is fed into a single-server
223

system.
The
first is based
on
the
Benes formula,
the
second
on
a slot-
to-slot approach. At
the
time, however,
it
is by no means clear whether
these approaches will lead
to
'practical' results, such as
an
approximate
formula for
the
tail
of
the
distribution
of
the
system contents.
l.OE+4
l.OE+5
l.OE+6
LOE+7
l.OE+O
+---+--+--+-+++++f----+----+-<-+-H-++t-->--+-+++++H
LOE-1
l.OE-21
Figure
9:
log Pr[u >
n]
versus log
n,
simu-
lations for GI-G-oo traffic
of
type
B.
Simulation results, shown
in
Figures 9
and
10, for a GJ-G-oo arrival
process
of
type
B
and
C respectively
with
intensity 0.8 Erlang, give
an
indication of
the
magnitude
of
the
queues -denoted by
the
variable u -
that
can
build
up. For instance, from Figure 9,
we
learn
that
for
the
long-
range
dependent
case,
the
queue exceeds
the
order of
10
5 cells during
10% of
the
time. For
the
other
case,
the
magnitude of
the
queue is
about
a
hundred
times smaller,
but
still very large. Although
the
simulations
are
too
crude
to
draw detailed conclusions,
the
figures already point
towards a hyperbolic decay
of
the
queue contents
(a
straight
line
in
a
log-log plot).
6.1
The
Benes
approach
The
system
contents -observed
at
the
beginning
of
a slot
and
denoted
uk
for slot k -is governed by
the
equation
224

1. 0
+4
1.0
+5
I
.OE
- 1
1.0
- 2
1
t
r
I .0
-3
Figure
10:
logPr[u
>
n]
versus
logn,
sim-
ulations for G I-G-oo traffic of
type
C.
The
'Benes result'
[12,
18]
for this system reads
Uk+1
=
max(pk
+
Pk-1
+ ... +
Pk-l
-
l)
l>O
from which one
obtains
Pr[uk+1
>
m]
=
+oo
L
Pr[pk
+
Pk-1
+ ... +
Pk-l
>
m
+
l\uk-l
=
O]Pr[uk-l
=
0]
(9)
l=O
This
result is appealing, since it provides a formula for
Pr[uk+
1
>
m]
irrespective
of
the
precise
nature
of
the
arrival process. Also , a simi-
lar expression
can
be
derived for systems
with
service capacity larger
than
1 or variable service capacity. Intriguing questions are
what
the
link
is
between
this
general result
and
the
general observation made in
e.g.
[5,
11]
concerning
the
impact
of
the
power spectral density
at
low
frequencies,
and
how, for instance,
the
index of dispersion of
the
traffic
process relates
to
the
probabilities in
the
RHS of
the
above formula.
Th
e event
uk-l
= 0 implies -
at
least -
that
no sources were active
in
the
slot preceding slot k -l .
This
latter
observation is sufficient to
determine
the
future
evolution of
the
process.
On
e can show
that,
for
225

the
G I -G-oo model,
E[zPk+Pk-1
+·
..
+Pk-lluk-l =
O]
=
exp
{A*
t,
Pr[TA > n](l +
1-
n)zn(z-
1)}
Introducing residues in (9),
we
get
the
following expression
+oo +oo
[
Pr[uk+l >
m]
= L
Pr[uk-l
=
0]
L Res z-(k+l)
l=O
k=m+l+l
exp
{A*
t,
Pr[TA > n](l +
1-
n)zn(z-
1)}]
z=O
(10)
For a system in equilibrium,
Pr[uk-l
=
0)
is given by 1 - p, a well-
known result from queueing theory. Hereby, p
is
the
load
of
the
system
and
equals Ar.
It
remains
to
be
determined
if
replacing residues
around
z = 0 by residues
around
the
other
singularities (poles or branches)
of
the
function involved, will lead
to
'practical' results, or
if
an
accurate
numerical transform inversion is feasible.
6.2
A
second
approach
The
queueing of discrete-time on-off sources was studied by a slot-to-
slot approach in e.g.
[20]
for a finite
number
of sources (with geometric
off-times),
and
in e.g.
[21)
for a infinite
number
of
sources.
The
model
in
the
latter
paper
is more general
than
the
model of Section 5.2,
in
that
the
number
of
new sources becoming active during a slot can have
an
arbitrary
distribution.
The
special case
of
a Poisson distribution
then
leads
to
the
GJ-G-oo arrival process considered here.
It
is noteworthy
that
the
Poisson
distribution
has a number
of
properties which simplify
the
analysis
and
results
to
some extent.
The
analysis can proceed as follows. Consider
the
joint
pgf
of uk,
the
number
of
cells in
the
system,
and
of
Vi,k,
the
numbers
of
messages in
the
equivalent GJ-G-oo model which still contain i cells, i.e., which will
still generate a single arrival
per
slot during
the
i slots
to
come. One
can
then
establish
the
following recurrence relation
pk+l (z,
XI,
X2,
...
) = E[zuk+l
...
]
=
z-
1 exp
{A*
E
a(k)(xk-
1)}
226

The
pgf
Pk(O,x
1
,x
2,
..
. )
can
easily
be
obtained
by observing
that
the
queue being
empty
at
the
beginning
of
a slot implies no messages arrived
during
the
previous slot.
This
implies
that
the
only messages
in
the
GI-
G-oo queue are new messages.
This
straightforwardly leads
to
Pk(O,
x1, x2,
...
)
=
Pr[uk
=OJ
exp {
>.*
E
a(k)(xk-
1)}
We will
not
go
further
into
the
details of
the
analysis here,
but
it
is
possible
to
derive from this
an
expression for
the
mean buffer contents
in
regime.
It
is
This
expression
can
also
be
found from
that
in
[20]
by a limiting proce-
dure, or from
that
in
[21]
by
assuming a Poisson arrival process for new
messages.
The
formula contains
the
variance of
the
durations
of
the
on-periods
and
becomes infinite for heavy-tailed on-time distributions
having infinite variance.
We believe -
but
couldn't
prove yet -
that,
in general,
Pr[TA
=
m]
"'m-q:::::}
Pr[u
>
m]
"'m-(q-
2)
(12)
while
(13)
Similar observations have
been
made for fluid-flow models
[2].
Of
course,
in
order
for a result like (12) or (13)
to
be
of
'practical' value, one should
also
be
able
to
derive
the
constant of proportionality, i.e.,
the
'intercept'
of
the
curve
m-(q-
2
).
In
e.g.
[21]
it was assumed
that
the
dominating
singularity of
pgf
of
the
system contents is
an
isolated pole, somewhere
in
the
interval (1,
+oo)
of
the
real line,
what
then
leads
to
geometric tail
decay. However,
this
assumption is no longer valid when heavy-tailed
on-time
distributions
are involved, since
the
corresponding pgf's have a
branchpoint
at
z
=
1.
It
remains
to
be
studied
how this approach has
to
be
modified
to
deal
with
that
case.
227

Both
approaches presented here,
the
Benes approach
and
the
slot-
to-slot approach, are
naturally
related, since
both
pertain
to
the
same
model.
The
connection becomes
apparent
when one recurses
the
relation
(11) (infinitely) many times,
and
sets all
Xi
equal
to
one.
This
yields
E[zuk+1] = lim [ E
Pr[uk-l
=
O]z-(!+1)
M--+oo l=O
· exp
{>.*to
Pr[TA > n](l +
1-
n)zn(z-
1)}
+z-(M+l)
exp
{>-*
Pr[TA >
n](M-
n)zn(z-
1)}
n ( 2 3 M M +1 M + 1
)]
·
rk-M
z, z,
Z , Z ,
•.•
, z , z , z ,
...
Comparing
this
with
equation (10), one easily recognizes
the
terms
they
have
in
common. However, establishing how
they
converge exactly, still
requires
further
study.
7
Conclusions
A
number
of
results were presented concerning traffic characteristics
and
queueing behavior
of
discrete-time on-off sources. At various instances,
the
distinction between short-range
and
long-range dependent traffic was
touched upon. Some key issues remain unsolved, and, as such, create
challenging areas for future research.
In
its
strictest
sense, long-range dependence is present when e.g.
the
on-period
distribution
of
the
sources has
an
infinite variance.
This
leads
to
an
infinite 'DC-component' in
the
power spectral density, a
system
contents having infinite mean, etcetera. However,
to
the
authors'
opin-
ion,
the
distinction between geometric
tail
decay
and
hyperbolic
tail
de-
cay is as
important
as
that
between short-
and
long-range dependence.
A relation like (12) shows
that
the
system contents
can
still have a slowly
decaying tail, even when
the
distributions of
the
sources have a finite
variance
and
when, as such,
the
system contents has a finite mean. Also
for
this
type
of
'short-range
dependent'
traffic, a tremendous
amount
of
buffering might
be
needed
in
order
to
avoid cell loss, or, in
other
words,
to
allow for feasible statistical multiplexing.
228

Appendix
A
In
this
appendix,
an
expressions is derived for
, +oo
S(J) = L C(m)(p1rmj
m=-oo
whereby
C(m)
was defined as
E[(qo-
.X)(qm-
.X)].
Since
there
is
at
most
one
arrival
per
slot, one
has
E[qoqm]
= Pr[(qo = l)&(qm =
1)].
This
can
be
further
expressed
as
+oo
L Pr[qm = ll(qo =
l)&(TA.
= k)]Pr[(qo =
l)&(TA.
=
k)]
(14)
k=O
whereby
7*
denotes
the
number
of
remaining
slots
of
the
on-
or
off-
period
in
which
slot
0 falls (not
counting
slot 0 itself).
The
factor
Pr[(qo = l)&(T* =
k)]
=
Pr[T*
=
klqo
= l]Pr[qo =
1]
can
be
ex-
pressed
as
a*
( k)
.X.
Recall -see section 2 -
that
the
pgf
associated
with
the
distribution
a*(k) is A*(z) =
[A(z)-
1]/[A'(l)(z-
1)].
The
probability
Pr[qm = ll(qo =
l)&(TA.
=
k)]
is given
by
Pr[qm = ll(qo =
l)&(TA.
=
k)]
= {
pl
[
=liB]
m
kk
(15)
r qm-k 1 . m >
whereby
B1
was
used
to
denote
the
event
that
slot 1, i.e.,
the
slot
just
after
slot
0, is
the
first slot
of
an
off-period.
By
considering all possible
values for
the
durations
of
that
off-period,
with
proper
weights, one finds
Pr[qm = liB1] =
+oo
L b(k)
{I(m
k)
· 0 +
I(m
> k) · Pr[qm-k = liA1]}
(16)
k=l
Similar
as above,
A1
denotes
the
event
that
slot 1 is
the
first slot
of
an
on-period. J(.)
denotes
the
indicator
function.
The
probability
Pr[qm =
liA1] can, likewise,
be
expressed as
Pr[qm = liA1] =
+oo
L a(k)
{I(m
k)
· 1 +
I(m
>
k)
· Pr[qm-k = liB1]}
(17)
k=l
229

Introducing z-transforms
in
equations
{16)
and
{17),
and
performing
some algebra, one can show
that
+oo
m
A(z)-
1
B(z)
X8(z) =
Pr[qm
=
11B1]z
= z z _ 1 · 1 _
A(z)B(z)
and
+oo
m A(z) -1 1
XA(z) =
Pr[qm
=
11A1]z
= z z _1 · 1 _
A(z)B(z)
Returning
to
equations
{14)
and
{15), one obtains
+oo
( A*(z) 1 )
L
E[qoqm]zm
=A
z z _ + A*(z)XB(z)
m=1
and, after some further manipulation,
whereby a2 = V
ar[qk]
= .\(1 -
.X)
and
A(z)-
1
B(z)-
1 [A'(1) +
B'(1)](z-
1)
P(z)
= A1
(1)(z-
1)
·
B'(1)(z-
1)
·
A(z)B(z)-
1
Finally, one obtains equation
(1)
Appendix
B
In
this appendix,
an
expression is derived for
+oo
J(z,t)
= L tmE[zqt+
...
+qm]
m=1
The
derivations are quite similar
to
those
in
appendix
A.
Starting
point
is
the
expression
A'{1)
B'{1)
J(z, t) = A'(1) +
B'(
1) JA(z, t) + A'(1) +
B'(
1) JB(z, t)
{18)
230

whereby
and
It
is easily shown
that
and
As
in
the
previous appendix,
B1
and
A1
denote
the
events
that
slot 1 is
the
first slot
of
an
off-
or
an
on-period respectively.
One has
E[zqt+
...
+qmiBl]
=
+oo
L b(k) {
I(m::;
k)
·1m+
I(m
> k) ·1k E[zq1+
...
+qm-kiA1l}
and
k=l
E[zq1
+
...
+qm
IA1]
=
+oo
L a(k) {
I(m::;
k) · zm +
I(m
> k) · zk E[zq1+
...
+qm-k
IB
1
J}
k=l
Introducing a z-transform (in variable t)
in
the
above equations
and
performing some straightforward algebra, one obtains
+oo
KA(z,t)
= L tmE[zqt+···+qmiAl]
m=l
= 1 (
A(zt)-
1
A{
)
B(t)-
1)
1-A(zt)B(t)
zt
zt-1
+
ztt
t-1
231

and
+oo
KB (z, t) = L
tm
E[zq
1
+
...
+qm
IB1]
m=l
= t t
zt
1 (
B(t)-
1 B
A(zt)-
1)
1-
A(zt)B(t)
t-
1 + ( )
zt-
1
Further,
and
JA(z,t) = ztA*(zt) -1
+A*(zt)KB(z,t)
zt
-1
zt
tA*(zt)(B(t)-
1)(z-
1)
= -1
--zt
+
...,.-(
z-t
)....:,.(
1------'-:-A'-:-(
z.....,t
)_B..:.,.(
t...,...,..))
B*(t)-
1
JB(z, t) = t t _ 1 + B*(t)KA(z, t)
t
tB*(t)(A(zt)-
1)(z-
1)
=
--
-
__
__:_.:_;__....:....,...:.,____:_..:..,_..,---.:.,.-.,..,-
1-
t
(zt-
1)(t-
1)(1-
A(zt)B(t))
Inserting these expressions in equation (18), one finally obtains
1
(A'(1)zt
B'(1)t
J(z, t) = A'(1) + B'(1)
1-
zt
+ 1"=T
References
_ A'(1)B'(1)t2(z
-1)
2
A*(zt)B*(t))
(1-
zt)(1-
t)(1-
A(zt)B(t))
[1]
J. Beran, R. Sherman,
M.
Taqqu
and
W. Willinger, "Long-range
dependence in variable-bit-rate video traffic",
IEEE
Transactions
on Communications
43
(1995) 1566-1579.
[2]
O.J. Boxma,
"Fluid
queues and regular variation", Proceedings of
PERFORMANCE'96 (Lausanne, October 1996) Performance Eval-
uation 27&28 (1996) 699-712.
[3]
H.
Bruneel and B.G. Kim, Discrete-time models for communication
systems including ATM, Kluwer Academic Publishers (1993).
232

[4]
G.L.
Choudhury
en W.
Whitt,
"Computing distributions
and
mo-
ments
in
polling models by numerical transform inversion", Perfor-
mance Evaluation
25
(1996) 267-292.
[5]
R.
Griinenfelder
and
S.
Robert, "Which arrival law parameters
are decisive for queueing system performance", Proceedings ITC-14
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