Thông tin tài liệu
17
NETWORK DESIGN AND CONTROL
USING ON=OFF AND MULTILEVEL
SOURCE TRAFFIC MODELS WITH
HEAVY-TAILED DISTRIBUTIONS
N. G. DUFFIELD AND W. W HITT
AT&T Labs±Research, Florham Park, NJ 07392
17.1 INTRODUCTION
In order to help design and control the emerging high-speed communication
networks, we want source traf®c models (also called offered load models or
bandwidth demand models) that can be both realistically ®t to data and successfully
analyzed. Many recent traf®c measurements have shown that network traf®c is quite
complex, exhibiting phenomena such as heavy-tailed probability distributions, long-
range dependence, and self similarity; for example, see Ca
Â
ceres et al. [7], Leland
et al. [23], Paxson and Floyd [24], and Crovella and Bestavros [10].
In fact, the heavy-tailed distributions may be the cause of all these phenomena,
because they tend to cause long-range dependence and (asymptotic) self-similarity.
For example, the input and buffer content processes associated with an on=off source
exhibit long-range dependence when the on and off times have heavy-tailed
probability distributions; for example, see Section 17.9. Heavy-tailed distributions
are known to cause self-similarity in models of (asymptotically) aggregated traf®c;
see Willinger et al. [27].
In this chapter we propose a way to analyze the performance of a network with
multiple on= off sources and more general multilevel sources in which the on-time,
off-time, and level-holding-time distributions are allowed to have heavy tails. To do
Self-Similar Network Traf®c and Performance Evaluation, Edited by Kihong Park and Walter Willinger
ISBN 0-471-31974-0 Copyright # 2000 by John Wiley & Sons, Inc.
421
Self-Similar Network Traf®c and Performance Evaluation, Edited by Kihong Park and Walter Willinger
Copyright # 2000 by John Wiley & Sons, Inc.
Print ISBN 0-471-31974-0 Electronic ISBN 0-471-20644-X
so we must go be beyond the familiar Markovian analysis. To achieve the required
analyzability with this added model complexity, we propose a simpli®ed kind of
analysis. In particular, we avoid the customary queueing detail (and its focus on
buffer content and over¯ow) and instead concentrate on the instantaneous offered
load. We describe the probability that aggregate demand (the input rate from a
collection of sources) exceeds capacity (the maximum possible output rate) at any
time. Focusing on the probability that aggregate demand exceeds capacity is
tantamount to considering a bufferless model, which we believe is often justi®ed.
By also considering the probability that aggregate demand exceeds other levels, we
provide a quite ¯exible performance characterization. This approach also can
generate approximations describing loss and delay with ®nite capacity; for example,
see Duf®eld and Whitt [14], Section 5. To a large extent, the present chapter is a
review of our recent work [14, 15], to which we refer the reader for additional
discussion. In Duf®eld et al. [16] the model is extended to include a nonhomoge-
neous Poisson connection arrival process. Then each active connection may generate
traf®c according to one of the source traf®c models presented here. It is signi®cant
that we are able to obtain useful descriptions of the offered load in the nonstationary
context.
17.2 A GENERAL SOURCE MODEL
Motivation for considering on=off and multilevel models as source models comes
from traces of frame sizes generated by certain video encoders; for example, see
Grasse et al. [19]. Shifts between levels in mean frame size appear to arise from
scene changes in the video, with the distribution of scene durations heavy-tailed.
Indeed, the expectation that scene durations will have heavy-tailed distributions is
one of the motivations behind the renegotiated constant bit rate (RCBR) proposal of
Grossglauser et al. [20].
Our approach is interesting for on=off and multilevel source models, but with
little extra effort we can treat a wider class. The general model we consider has two
components. The bandwidth demand for each source as a function of time,
fBt: t ! 0g, is represented as the sum of two stochastic processes: (1) a macro-
scopic (longer-time-scale) level process fLt: t ! 0g and (2) a microscopic (shorter-
time-scale) within-level variation process fW t: t ! 0g, that is,
BtLtW t; t ! 0: 17:1
We let the macroscopic level process fLt: t ! 0g be a semi-Markov process (SMP)
as in CË inlar [9, Chap. 10]; that is, the level process is constant except for jumps, with
the jump transitions governed by a Markov process, while the level holding times
(times between jumps) are allowed to have general distributions depending on the
originating level and the next level. Given a transition from level j to level k, the
holding time in level j has cumulative distribution function (cdf) F
jk
. Conditional on
the sequence of successive levels, the holding times are mutually independent. To
422 NETWORK DESIGN USING HEAVY-TAILED DISTRIBUTIONS
obtain models compatible with traf®c measurements cited earlier, we allow the
holding-time cdf 's F
jk
to have heavy tails.
We assume that the within-level variation process fW t: t ! 0g is a zero-mean
piecewise-stationary process. During each holding-time interval in a level, the
within-level variation process is an independent segment of a zero-mean stationary
process, with the distribution of each segment being allowed to depend on the level.
We allow the distribution of the stationary process segment to depend on the level,
because it is natural for the variation about any level to vary from level to level.
We will require only a limited characterization of the within-level variation
process; it turns out that the ®ne structure of the within-level variation process plays
no role in our analysis. Indeed, that is one of our main conclusions. In several
examples of processes that we envisage modeling by these methods, there will only
be the level process. First, the level process may be some smoothed functional of a
raw bandwidth process. This is the case with algorithms for smoothing stored video
by converting into piecewise constant rate segments in some optimal manner subject
to buffering and delay constraints; see Salehi et al. [25]. With such smoothing, the
input rate will directly be a level process as we have de®ned it. Alternatively, the
level process may stem from rate reservation over the period between level-shifts,
rather than the bandwidth actually used. This would be the case for RCBR
previously mentioned. In this situation we act as if the reservation level is the
actual demand, and thus again have a level process.
A key to being able to analyze the system with such complex sources represented
by our traf®c model is exploiting asymptotics associated with multiplexing a large
number of sources. The ever-increasing network bandwidth implies that more and
more sources will be able to be multiplexed. This gain is generally possible, even in
the presence of heavy-tailed distributions and more general long-range dependence;
for example, see Duf®eld [12, 13] for demonstration of the multiplexing gains
available for long-range dependent traf®c in shared buffers. As the scale increases,
describing the detailed behavior of all sources become prohibitively dif®cult, but
fortunately it becomes easier to describe the aggregate, because the large numbers
produce statistical regularity. As the size increases, the aggregate demand can be
well described by laws of large numbers, central limit theorems, and large deviation
principles.
We have in mind two problems: ®rst, we want to do capacity planning and,
second, we want to do real-time connection-admission control and congestion
control. In both cases, we want to determine whether any candidate capacity is
adequate to meet the aggregate demand associated with a set of sources. In both
cases, we represent the aggregate demand simply as the sum of the bandwidth
requirements of all sources. In forming this sum, we regard the bandwidth processes
of the different sources as probabilistically independent.
The performance analysis for capacity planning is coarser, involving a longer
time scale, so that it may be appropriate to do a steady-state analysis. However, when
we consider connection-admission control and congestion control, we suggest
focusing on a shorter time scale. We are still concerned with the relatively long
time scale of connections, or scene times in video, instead of the shorter time scales
17.2 A GENERAL SOURCE MODEL 423
of cells or bursts, but admission control and congestion control are suf®ciently short-
term that we propose focusing on the transient behavior of the aggregate demand
process. In fact, even for capacity planning the transient analysis plays an important
role. The transient analysis determines how long it takes to recover from rare
congestion events. One application we have in mind is that of networks carrying
rate-adaptive traf®c. In this case the bandwidth process could represent the ideal
demand of a source, even though it is able to function when allocated somewhat less
bandwidth. So from the point of view of quality, excursion of aggregate bandwidth
demand above available supply may be acceptable in the short-term, but one would
want to dimension the link so that such excursions are suf®ciently short-lived. In this
or other contexts, if the recovery time from overload is relatively long, then we may
elect to provide extra capacity (or reduce demand) so that overload becomes less
likely. However, we do not focus speci®cally on actual design and control here; see
Duf®eld and Whitt [14] for some speci®c examples. Our main contribution here is to
show how the transient analysis for design and control can be done.
The remainder of this chapter is devoted to showing how to do transient analysis
with the source traf®c model. We suggest focusing on the future time-dependent
mean conditional on the present state. The present state of each level process
consists of the level and age (elapsed holding time in that level). Because of the
anticipated large number of sources, the actual bandwidth process should be closely
approximated by its mean, by the law of large numbers (LLN). As in Duf®eld and
Whitt [14], the conditional mean can be thought of as a deterministic ¯uid
approximation; for example, see Chen and Mandelbaum [8]. Since the within-
level variation process has mean zero, the within-level variation process has no effect
on this conditional mean. Hence, the conditional mean of the aggregate bandwidth
process is just the sum of the conditional means of the component level processes.
Unlike the more elementary M =G=I model considered in Duf®eld and Whitt [14],
however, the conditional mean here is not available in closed form.
In order to rapidly compute the time-dependent conditional mean aggregate
demand, we exploit numerical inversion of Laplace transforms. It follows quite
directly from the classical theory of semi-Markov process that explicit expressions
can be given for the Laplace transform of the conditional mean. More recently, it has
been shown that numerical inversion can be an effective algorithm; see Abate et al.
[1].
For related discussions of transient analysis, design and control, see Chapters 13,
16, and 18 in this volume.
17.3 OUTLINE OF THE CHAPTER
The rest of this chapter is organized as follows. In Section 17.4, we show that the
Laplace transform of the mean of the transient conditional aggregate demand can be
expressed concisely. This is the main enabling result for the remainder of the chapter.
The conditional mean itself can be very ef®ciently computed by numerically
inverting its Laplace transform. To carry out the inversion, we use the Fourier-
424 NETWORK DESIGN USING HEAVY-TAILED DISTRIBUTIONS
series method in Abate and Whitt [2] (the algorithm Euler exploiting Euler
summation), although alternative methods could be used. The inversion algorithm
is remarkably fast; computation for each time point corresponds simply to a sum of
50 terms. We provide numerical examples in Examples 17.6.2 and 17.8.1. Example
17.8.1 is of special interest, because the level-holding-time distribution there is
Pareto.
In Section 17.5 we show that in some cases we can avoid the inversion entirely
and treat much larger models. We can avoid the inversion if we can assume that the
level holding times are relatively long compared to the times of interest for control.
Then we can apply a single-transition approximation, which amounts to assuming
that the Markov chain is absorbing after one transition. Then the conditional mean is
directly expressible in terms of the level-holding-time distributions. Alternatively, we
can perform a two-transition approximation, which only involves one-dimensional
convolution integrals.
In Section 17.6 we describe the value of having more detailed state information,
speci®cally the current ages of levels. With heavy-tailed distributions, a large elapsed
holding time means that a large remaining holding time is very likely; for examples
see Duf®eld and Whitt [14, Section 8] for background, and Harchol-Balter and
Downey [21] for an application in another setting.
In Section 17.7 we turn to applications to capacity planning. The idea is to
approximate the probability of an excursion in demand using Chernoff bounds and
other large deviation approximations, then chart its recovery to a target acceptable
level using the results on transience. Interestingly, the time to recover from
excursions suf®ciently close to the target level depends on the level durations
essentially only through their mean. Correspondingly, the conditional mean demand
relaxes linearly from its excursion, at least approximately so, for suf®ciently small
times. If the chance for a larger excursion is negligible (as determined by the large
deviation approximation mentioned) then this simple description may suf®ce. An
example is given in Section 17.8.
In Section 17.9 we show how long-range dependence in the level process arises
through heavy-tailed level-holding-time distributions. Finally, we draw conclusions
in Section 17.10.
17.4 TRANSIENT ANALYSIS
17.4.1 Approximation by the Conditional Mean Bandwidth
Throughout this chapter, the state information on which we condition will be either
the current level of each source or the current level and age (current time) in that
level of each source. No state from the within-level variation process is assumed.
Conditional on that state information, we can compute the probability that each
source will be in each possible level at any time in the future, from which we can
calculate the conditional mean and variance of the aggregate required bandwidth by
adding.
17.4 TRANSIENT ANALYSIS 425
The Lindberg±Feller central limit theorem (CLT) for non-identically-distributed
summands can be applied to generate a normal approximation characterized by the
conditional mean and conditional variance; see Feller [18, p. 262]. For the normal
approximation to be appropriate, we should check that the aggregate is not
dominated by only a few sources.
Let Bt denote the (random) aggregate required bandwidth at time t, and let I 0
denote the (known deterministic) state information at time 0. Let BtjI 0
represent a random variable with the conditional distribution of Bt given the
information I0. By the CLT, the normalized random variable
BtjI0À EBtjI0
VarBtjI0
p
17:2
is approximately normally distributed with mean 0 and variance 1 when the number
of sources is suitably large.
Since the conditional mean alone tends to be very descriptive, we use the
approximation
BtjI0 % EBtjI0; 17:3
which can be justi®ed by a (weaker) law of large numbers instead of the CLT. We
will show that the conditional mean in Eq. (17.3) can be ef®ciently computed, so that
it can be used for real-time control. From Eq. (17.2), we see that the error in the
approximation (17.3) is approximately characterized by the conditional standard
deviation
VarBtjI0
p
. We also will show how to compute this conditional
standard deviation, although the required computation is more dif®cult. If there are n
sources that have roughly equal rates, then the conditional standard deviation will be
O
n
p
, while the conditional mean is On.
Given that our approximation is the conditional mean, and given that our state
information does not include the state of the within-level variation process, the
within-level variation process plays no role because it has zero mean. Let i index the
source. Since the required bandwidths need not have integer values, we index the
level by the integer j; 1 j J
i
, and indicate the associated required bandwidths in
the level by b
i
j
. Hence, instead of Eq. (17.1), the required bandwidth for source i can
be expressed as
B
i
tb
i
L
i
t
W
L
i
t
t; t ! 0: 17:4
Let P
i
jk
tjx be the probability that the source-i level process is in level k at time t
given that time 0 it was in level j and had been so for a period x (i.e., the age or
elapsed level holding time at time 0 is x). If j j
1
; ; j
n
and x x
1
; ; x
n
are
the vectors of levels and ages of the n source level processes at time 0, then the state
426 NETWORK DESIGN USING HEAVY-TAILED DISTRIBUTIONS
information is I0j; xj
1
; ; j
n
; x
1
; ; x
n
and the conditional aggregate
mean is
EBtjI0 Mtjj; x
P
n
i1
P
J
i
k
i
1
P
i
j
i
k
i
tjx
i
b
i
k
i
: 17:5
From Eq. (17.5), we see that we need to compute the conditional distribution of
the level, that is, the probabilities P
i
jk
tjx, for each source i. However, we can ®nd
relatively simple expressions for the Laplace transform of P
i
jk
tjx with respect to
time because the level process of each source has been assumed to be a semi-Markov
process.
We now consider a single source and assume that its required bandwidth process
is a semi-Markov process (SMP). (We now have no within-level variation process.)
We now omit the superscript i. Let Lt and Bt be the level and required bandwidth,
respectively, at time t as in Eq. (17.4). The process fLt: t ! 0g is assumed to be an
SMP, while the process fBt: t ! 0g is a function of an SMP, that is Btb
Lt
,
where b
j
is the required bandwidth in level j.Ifb
j
T b
k
for j T k, then fBt: t ! 0g
itself is an SMP, but if b
j
b
k
for some j T k, then in general fBt: t ! 0g is not an
SMP.
17.4.2 Laplace Transform Analysis
Let At be the age of the level holding time at time t. We are interested in calculating
P
jk
tjxPLtkjL0j; A0x17:6
as a function of j; k; x, and t. The state information at time 0 is the pair j; x. Let P
be the transition matrix of the discrete-time Markov chain governing level transitions
and let F
jk
t be the holding-time cdf given that there is a transition from level j to
level k. For simplicity, we assume that F
c
jk
t1 À F
jk
t > 0 for all j; k, and t,so
that all positive x can be level holding times. Let Ptjx be the matrix with elements
P
jk
tjx and let
^
Psjx be the Laplace transform (LT) of Ptjx, that is, the matrix with
elements that are the Laplace transforms of P
jk
tjx with respect to time:
^
P
jk
sjx
I
0
e
Àst
P
jk
tjx dt: 17:7
We can obtain a convenient explicit expression for
^
Psjx. For this purpose, let G
j
be
the holding-time cdf in level j, unconditional on the next level, that is,
G
j
x
P
k
P
jk
F
jk
x: 17:8
17.4 TRANSIENT ANALYSIS 427
For any cdf G, let G
c
be the complementary cdf, that is, G
c
x1 À Gx. Also, let
H
jk
tjx
P
jk
F
jk
t x
G
c
j
x
and G
j
tjx
P
k
H
jk
tjx17:9
for G
j
in Eq. (17.8). Then let
^
h
jk
sjx and
^
g
j
sjx be the associated Laplace±Stieltjes
transforms (LSTs):
^
h
jk
sjx
I
0
e
Àst
dH
jk
tjx and
^
g
j
sjx
I
0
e
Àst
dG
j
tjx: 17:10
Let
^
hsjx be the matrix with elements
^
h
jk
sjx. Let
^
qs be the matrix with elements
^
q
jk
s, where
Q
jk
tP
jk
F
jk
t and
^
q
jk
s
I
0
e
Àst
dQ
jk
t: 17:11
Let
^
Dsjx and
^
Ds be the diagonal matrices with diagonal elements
^
D
jj
sjx1 À
^
g
j
sjx=s;
^
D
jj
s1 À
^
g
j
s=s; 17:12
where
^
g
j
s is the LST of the cdf G
j
in Eq. (17.8).
Theorem 17.4.1. The transient probabilities for a single SMP source have the
matrix of Laplace transforms
^
Psjx
^
Dsjx
^
hsjx
^
Psj0; 17:13
where
^
Psj0I À
^
qs
À1
^
Ds: 17:14
Proof. In the time domain, condition on the ®rst transition. For j T k,
P
jk
tjx
P
l
t
0
dH
jl
ujxP
lk
t À uj0;
so that
^
P
jk
sjx
P
l
^
h
jl
sjx
^
P
lk
sj0;
428 NETWORK DESIGN USING HEAVY-TAILED DISTRIBUTIONS
while
P
jj
tjxG
c
j
tjx
P
l
t
0
dH
jl
ujxP
lj
t À uj0;
so that
^
P
jj
sjx
1 À
^
g
j
sjx
s
P
l
h
jl
sjx
^
P
lj
sj0:
Hence, Eq. (17.13) holds. However, Ptj0 satis®es a Markov renewal equation, as in
CË inlar [9, Section 10.3]; that is, for j T k,
P
jk
tj0
P
l
t
0
dQ
jl
uP
lk
t À uj0;
and
P
jj
tj0G
c
j
t
P
l
I
0
dQ
jl
uP
lj
t Àuj0;
so that
Ptj0DtQtÃPtj0
where à denotes convolution, and Eq. (17.14) holds. j
To compute the LT
^
Psj0, we only need the LSTs
^
f
jk
s and
^
g
j
s associated with
the basic holding-time cdf 's F
jk
and G
j
. Abate and Whitt [3±5] give special attention
to heavy-tail probability densities whose Laplace transforms can be computed and,
thus, inverted. However, to compute
^
Psjx, we also need to compute
^
Dsjx and
^
hsjx, which require computing the LSTs of the conditional cdf's H
jk
tjx and
G
j
tjx in Eq. (17.9). In general, even if we know the LST of a cdf, we do not
necesssarily know the LST of an associated conditional cdf. However, in special
cases, the LSTs of conditional cdf 's are easy to obtain because the cdf 's inherit their
structure upon conditioning. For example, this is true for phase-type, hyperexpo-
nential and Pareto distributions; Duf®eld and Whitt [15, Section 4]. Moreover, other
cdf's can be approximated by hyperexponential or phase-type cdf's; see Asmussen et
al. [6] and Feldman and Whitt [17].
If the number of levels is not too large, then it will not be dif®cult to compute the
required matrix inverse I À qs
À1
for all required s. Note that, because of the
probability structure, the inverse is well de®ned for all complex s with Res > 0.
To illustrate with an important simple example, we next give the explicit formula
for an on=off source. Suppose that there are two states with transition probabilities
17.4 TRANSIENT ANALYSIS 429
P
12
P
21
1 and holding time cdf's G
1
and G
2
. From Eq. (17.9) or by direct
calculation,
^
Psj0
^
P
11
sj0
^
P
12
sj0
^
P
21
sj0
^
P
22
sj0
!
I À
^
qs
À1
^
Ds
1
s1 À
^
g
1
s
^
g
2
s
1 À
^
g
1
s
^
g
1
s1 À
^
g
2
s
^
g
2
s1 À
^
g
1
s 1 À
^
g
2
s
: 17:15
Suppose that the levels are labeled so that the initial level is 1. Note that all
transitions from level 1 are to level 2. Hence, when considering the matrix
^
hsjx in
Eq. (17.10), it suf®ces to consider only the element
^
h
12
sjx. Since
H
c
12
tjxG
c
1
tjx
G
c
1
t x
G
c
1
x
; 17:16
then
^
h
12
sjx
^
g
1
sjx
I
0
e
Àst
dG
1
tjx: 17:17
Since P
11
tjx1 À P
12
tjx, it suf®ces to calculate only P
12
tjx. Hence, in this
context
^
P
12
sjx
^
g
1
sjx1 À
^
g
2
s
s1 À
^
g
1
s
^
g
2
s
: 17:18
We now determine the mean, second moment, and variance of the bandwidth
process of a general multilevel source as a function of time, ignoring the within-level
variation process. It is elementary that
m
j
tjxEBtjL0j; A0x
P
k
P
jk
tjxb
k
; 17:19
s
j
tjxEBt
2
jL0j; A0x
P
k
P
jk
tjxb
2
k
; 17:20
v
j
tjxVarBtjL0j; A0xs
j
tjxÀm
j
tjx
2
: 17:21
We can calculate m
j
tjx and s
j
tjx by single inversions of their Laplace transforms,
using
^
m
j
sjx
I
0
e
Àst
m
j
tjx dt
P
k
P
jk
sjxb
k
430 NETWORK DESIGN USING HEAVY-TAILED DISTRIBUTIONS
Ngày đăng: 21/01/2014, 23:20
Xem thêm: Tài liệu Mạng lưới giao thông và đánh giá hiệu suất P17 ppt, Tài liệu Mạng lưới giao thông và đánh giá hiệu suất P17 ppt
