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A short tutorial on network calculus. I. Fundamental bounds in communication networks

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ASHORTTUTORIALONNETWORKCALCULUSI:FUNDAMENTALBOUNDSINCOMMUNICATIONNETWORKS
Jean-Yves Le Boudec and Patrick Thiran
ICA-DSC, EPFLCH-1015 Lausanne, Switzerland
http://icawww.epfl.ch
ABSTRACT
Network Calculus is a collection of results based on Min-Plusalgebra, which applies todeterministic queuing systemsfound in communication networks. It can be used for exam-ple to understand the computations for delays used in theIETF guaranteed service, why re-shaping delays can be ig-nored in shapers or spacer-controllers, a common model forschedulers, etc. This short tutorial presents the basic resultsof network calculus and their application to some fundamen-tal performance bounds in communication networks.
1. INTRODUCTION
Network Calculus is a set of recent developments which pro-vide a deep insight into ﬂow problems encountered in net-working. The foundation of network calculus lies in themathematical theory of dioids, and in particular, the Min-Plus dioid (also called Min-Plus algebra). With network cal-culus, we are able to understand some fundamental proper-ties of integrated services networks, of window ﬂow control,of scheduling and of buffer or delay dimensioning. Thesetwo companion papers [1] are a very short introduction tothis theory.Network calculus can be viewed as the system theory thatapplies to computer networks. The main difference with tra-ditional system theory, as the one which was so successfullyapplied to design electronic circuits, is that here we consideranother algebra, where the operations are changed as fol-lows: addition becomes computation of the minimum, mul-tiplication becomes addition.Let us illustrate this difference with an example. Consider avery simple circuit, such as the RC cell represented in Fig-ure 1. If the input signal is the voltage
x
(
t
)
2
R
, then theoutput
y
(
t
)
2
R
of this simple circuit is the convolutionof
x
by the impulse response of this circuit, which is here
h
(
t
)=exp(
?
t=RC
)
=RC
for
t
0
:
y
(
t
)=(
h
x
)(
t
)=
Z
t
0
h
(
t
?
s
)
x
(
s
)
ds:
Consider now a node of a communication network, whichis idealized as a (greedy) shaper. A (greedy) shaper is a de-vice that forces an input ﬂow
x
(
t
)
to have an output
y
(
t
)
that conforms to a given set of rates according to a trafﬁcenveloppe
(the shaping curve), at the expense of possi-bly delaying bits in the buffer. Here the input and output‘signals’ are cumulative ﬂow, deﬁned as the number of bitsseen on the data ﬂow in time interval
0
;t
]
. These functionsare non-decreasing with time
t
. We will denote by
G
theset of non-negative wide-sense increasing functions and by
F
denote the set of wide-sense increasing functions (or se-quences) such that
f
(
t
)=0
for
t<
0
. Parameter
t
can becontinuous or discrete. We will see in this paper that
x
and
y
are linked by the relation
y
(
t
)=(
x
)(
t
)=inf
s
:0
s
t
f
(
t
?
s
)+
x
(
s
)
g
:
(1)This relation deﬁnes the min-plus convolution between
and
x
.
x(t)
σ
x(t) y(t)R C
+ +--
y(t)
Figure 1: Traditional system theory for an elementary circuit(top) and min-plus system theory for a shaper (bottom).This paper reviews the basic concepts of network calculus,namely the way we characterize the ‘signals’ (i.e. the ﬂows)via arrival curves (Section 2) and the ‘system’ (e.g., the net-work node) via a service curve (Section 3). These tools willenable us to derive some deterministic performance boundson quantities such delays and backlogs (Section 4), whichare deﬁned as follows, for a lossless system with input ﬂow
x
(
t
)
and output ﬂow
y
(
t
)
:
Deﬁnition 1 (Backlog and Delay)
The
backlog
at time
t
is
x
(
t
)
?
y
(
t
)
, the
virtual delay
at time
t
is
d
(
t
)=inf
f
0:
x
(
t
)
y
(
t
+
)
g
:
The backlog is the amount of bits that are held inside thesystem; if the system is a single buffer, it is the queue length.In contrast, if the system is more complex, then the backlogis the number of bits “in transit”, assuming that we can ob-serve input and output simultaneously. The virtual delay attime
t
is the delay that would be experienced by a bit arriv-ing at time
t
if all bits received before it are served before it.If we plot
x
(
t
)
and
y
(
t
)
versus
t
, the backlog is the verticaldeviation between these two curves. The virtual delay is thehorizontal deviation.We will conclude the paper with ‘the linear time-invariantsystem’ of communication network: the shaper. The inter-ested reader is also referred to the pioneering work of Cruz[5], Chang [3], Agrawal and Rajan[4].
2. ARRIVAL CURVES
To provide guarantees to data ﬂows requires some speciﬁcsupport in the network; as a counterpart, the trafﬁc sent bysources needs to be limited. With integrated services net-works (ATM or the integrated services internet), this is doneby using the concept of arrival curve, deﬁned below.
Deﬁnition 2 (Arrival Curve)
Given a wide-sense increas-ing function
deﬁned for
t
0
(namely
2F
), we saythat a ﬂow
x
is constrained by
if and only if for all
s
t
:
x
(
t
)
?
x
(
s
)
(
t
?
s
)
:
Note that this is equivalent to imposing that for all
t
0
x
(
t
)
inf
0
s
t
f
(
t
?
s
)+
x
(
s
)
g
=(
x
)(
t
)
(2)The simplest arrival curve is
(
t
)=
Rt
. Then the constraintmeans that, on any time window of width
, the number of bits for the ﬂow is limited by
R
. We say in that case thatthe ﬂow is peak rate limited. This occurs if we know that theﬂow is arriving on a link whose physical bit rate is limitedby
R
bits/sec. A ﬂow where the only constraint is a limiton the peak rate is often (improperly) called a “constant bitrate” (CBR) ﬂow.Moregenerally, because oftheirrelationshipwithleakybuck-ets, we will often use
afﬁne
arrival curves
r;b
, deﬁned by:
r;b
(
t
)=
rt
+
b
for
t>
0
and
0
otherwise. Having
r;b
as an arrival curve allows a source to send
b
bits at once,but not more than
r
bits/s over the long run. Parameters
b
and
r
are called the burst tolerance (in units of data) and therate (in units of data per time unit). The Integrated servicesframework of the Internet (Intserv) uses arrival curves, suchas
(
t
)=min
f
M
+
pt;rt
+
b
g
=
p;M
(
t
)
^
r;b
(
t
)
where
M
is interpreted as the maximum packet size,
p
as thepeak rate,
b
as the burst tolerance, and
r
as the sustainablerate Figure 2. Notation
^
stands for minimum or inﬁmum.In Intserv jargon, the 4-uple
(
p;M;r;b
)
is also called a T-SPEC (trafﬁc speciﬁcation). ATM uses similar curves.
TRtPrMb dmax
β
(t)
α
(t)wmax
Figure 2: Arrival curve
for ATM VBR and for Intservﬂows, rate-latency service curve
and vertical and horizon-tal devaitions between both curves.One can always replace an arrival curve
by itssub-additiveclosure, which is deﬁned as
=inf
f
0
;;
;:::;
(
n
)
;:::
g
where
(
n
)
=
:::
(
n
times) and
0
is the “impulse”function deﬁned by
0
(
t
)=
1
for
t>
0
and
0
(0)=0
).One can show indeed that
x
x
if and only if
x
x
. If
(0)=0
and
is sub-additive (meaning that forall
s;t
0
,
(
s
+
t
)
(
s
)+
(
t
)
), then
=
. As anexample, one can check that
r;b
=
r;b
.Finally, it is possible to compute from measurements of agiven ﬂow
x
(
t
)
itsminimal arrival curve, which is
(
x
x
)(
t
)
where
denotes the min-plus deconvolution operator de-ﬁned by
(
x
)(
t
)=sup
u
0
f
x
(
t
+
u
)
?
(
u
)
g
;
(3)for a given function
2F
. Note that if
x;
2F
, then
(
x
)
2F
but in general
(
x
)
=
2F
(it belongs to
G
). One can check however that
(
x
x
)
2F
. Let us alsomention that the name deconvolution is justiﬁed by the factthat for any
x;y;z
2F
,
x
y
z
if and only if
x
z
y
.
3. SERVICE CURVES
We have seen that one ﬁrst principle in integrated servicesnetworks is to put arrival curve constraints on ﬂows. In or-der to provide reservations, network nodes in return needto offer some guarantees to ﬂows. This is done by packetschedulers. The details of packet scheduling are abstractedusing the concept of service curve, which we introduce inthis section.
Deﬁnition 3 (Service Curve)
Consider a system
S
and a ﬂow through
S
with input and ouptut function
x
and
y
. Wesay that
S
offers to the ﬂow a
service curve
if and only if for all
t
0
, there exists some
t
0
0
, with
t
0
t
, suchthat
y
(
t
)
?
x
(
t
0
)
(
t
?
t
0
)
:
Again, we can recast this deﬁnition as
y
(
t
)
inf
0
s
t
f
(
t
?
s
)+
x
(
s
)
g
=(
x
)(
t
)
(4)Let us consider a few examples. A simple one is a GPS(Generalized Processor Sharing) node which, by offering aservice curve
(
t
)=
Rt
, guarantees that each ﬂow is servedat least at rate
R
bits/s during a busy period.A second example is a guaranteed delay node. Here the onlyinformation we have about the network node is that the max-imum delay for thebits of a given ﬂow
x
is bounded by someﬁxed value
T
, and that the bits of the ﬂow are served in ﬁrstin, ﬁrst out order. This is used with a family of schedulerscalled “earliest deadline ﬁrst” (EDF), and can be translatedas
y
(
t
)
x
(
t
?
T
)
for all
t
T
. Using the “impulse”function
T
deﬁned by
T
(
t
)=0
if
0
t
T
and
T
(
t
)=+
1
if
t>T
, wehavethat
(
x
T
)(
t
)=
x
(
t
?
T
)
.We have therefore shown that a guaranteed delay node offersa service curve
=
T
.As alast example, the IETFassumes that RSVP routers offera service curve of the form
R;T
(
t
)=
R
t
?
T
]
+
=
R
(
t
?
T
)
if
t>T
0
otherwiseas shown on Figure 2. We call this curve the rate-latencyservice curve.Finally, let us mention the following result, which is well-known in traditional system theory, and which is easy to es-tablish in network calculus:
Theorem 1 (Concatenation of Nodes)
Assume a ﬂow tra-verses systems
S
1
and
S
2
in sequence. Assume that
S
i
offersa service curve of
i
,
i
=1
;
2
to the ﬂow. Then the concate-nation of the two systems offers a service curve of
1
2
to the ﬂow.
As an example, consider two nodes offering each a rate-latency service curve
R
i
;T
i
,
i
=1
;
2
, as is commonly as-sumed with Intserv. A simple computation gives
R
1
;T
1
R
2
;T
2
=
R
1
^
R
2
;T
1
+
T
2
:
(5)Thus concatenating RSVP routers amounts to adding the la-tency components and taking the minimum of the rates.We are now also able to give another interpretation of therate-latencyservicecurvemodel. Wecancomputethat
R;T
=
T
R;
0
; thus we can view a node offering a rate-latencyservicecurveastheconcatenation of aguaranteed delaynode,with delay
T
and a CBR or GPS node with rate
R
.
4. THREE FUNDAMENTAL BOUNDS
In this section we see the main simple network calculus re-sults. They are all bounds for lossless systems with serviceguarantees [4]. The proofs are straightforward applicationsof the deﬁnitions of service and arrival curves.The ﬁrst theorem says that the backlog is bounded by thevertical deviation between the arrival and service curves:
Theorem 2 (Backlog Bound)
Assume a ﬂow, constrained by arrival curve
, traverses a system that offers a servicecurve
. The backlog
x
(
t
)
?
y
(
t
)
for all
t
satisﬁes:
x
(
t
)
?
y
(
t
)
sup
s
0
f
(
s
)
?
(
s
)
g
We now use the concept of horizontal deviation, which is alittle complex, but is supported by the following intuition.Call
(
s
)=inf
f
0:
(
s
)
(
s
+
)
g
. From Deﬁ-nition 1,
(
s
)
is the virtual delay for a hypothetical systemwhich would have
as input and
as output, assuming thatsuch a system exists (namely, assuming that (
). Let
h
(
;
)
be the supremum of all values of
(
s
)
. The secondtheorem gives a bound on delay for the general case.
Theorem 3 (Delay Bound)
Assume a ﬂow, constrained byarrival curve
, traverses asystemthat offersaservice curveof
. Thevirtual delay
d
(
t
)
forall
t
satisﬁes:
d
(
t
)
h
(
;
)
.
Theorem 4 (Output Flow)
Assume a ﬂow, constrained byarrival curve
, traverses asystemthat offersaservice curveof
. The output ﬂow is constrained by the arrival curve
=
.
As a ﬁrst application of the previous results, consider a VBRﬂow, deﬁned by TSPEC
(
M;p;r;b
)
(hence
(
t
)=
f
M
+
pt
g^f
rt
+
b
g
) and served in one node which guarantees aservicecurve equal totherate-latencyfunction
(
t
)=
R
t
?
T
]
+
. This example is the standard model used in Intserv
(Figure 2). Let us apply Theorems 2 and 3. Assume that
R
r
namely the reserved rate is as large as the sustainablerate of the ﬂow. The buffer required for the ﬂow is boundedby
w
max
=
b
+
r
max
b
?
M p
?
r ;T
The maximum delay for the ﬂow is bounded by
d
max
=
M
+
b
?
M p
?
r
(
p
?
R
)
+
R
+
T:
We can also apply Theorem 4 and ﬁnd an arrival curve
for the output ﬂow.Asa second application, let us show how these bounds, com-bined withTheorem1, allowustounderstand aphenomenonknown in the Insterv community as “PayBursts Only Once”.Consider the concatenation of two nodes offering each arate-latency service curve
R
i
;T
i
,
i
=1
;
2
, as is commonlyassumed with Intserv. Assume the fresh input is constrainedby
r;b
. Assume that
r<R
1
and
r<R
2
. We are inter-ested in the delay bound, which we know isa worst case. Letus compare the results obtained by applying Theorem 3 (i)to the network service curve (5), resulting in a delay bound
D
0
; (ii) iteratively to every node, resulting in two individualbounds
D
1
and
D
2
.(i) The delay bound
D
0
can be computed by application of Theorem 3:
D
0
=
b R
1
^
R
2
+
T
1
+
T
2
:
(ii) Now apply the second method. A bound on the delay atnode 1 is (Theorem 3):
D
1
=
b=R
1
+
T
1
. The output of the ﬁrst node is constrained by by
(
t
)=
b
+
rt
+
rT
1
,because of Theorem 4. A bound on the delay at the secondbuffer is therefore
D
2
=(
b
+
rT
1
)
=R
2
+
T
2
. Consequently,
D
1
+
D
2
=
b R
1
+
b
+
rT
1
R
2
+
T
1
+
T
2
It is easy to see that
D
0
<D
1
+
D
2
. In other words, thebounds obtained by considering the global service curve arebetter than the bounds obtained by considering every bufferin isolation.
5. GREEDY SHAPERS
We call
policer
with curve
a device that counts the bits ar-riving on an input ﬂow and decides which bits conform withan arrival curve of
. We call
shaper
, with shaping curve
,a bit processing device that forces its output to have
as ar-rival curve. We call
greedy shaper
a shaper which delays theinput bits in a buffer, whenever sending a bit would violatethe constraint
, but outputs them as soon as possible.With ATM and sometimes with Intserv, trafﬁc sent over oneconnection, orﬂow, ispolicedatthenetworkboundary. Polic-ing is performed in order to guarantee that users do not sendmore thanspeciﬁed by thecontract of theconnection. Trafﬁcin excess is either discarded, or marked with a low priorityfor loss in the case of ATM, or passed as best effort trafﬁc inthe case of Intserv. In the latter case, with IPv4, there is nomarking mechanism, so it is necessary for each router alongthe path of the ﬂow to perform the policing function again.Policing devices inside the network are normally buffered,they are thus shapers. Shaping is also often needed becausethe output of a buffer normally does not conform any morewith the trafﬁc contract speciﬁed at the input.The main result on greedy shapers is the following.
Theorem 5 (Input/output characterization of greedy shapers)
Consider agreedy shaper withshaping curve
. Assumethat the shaper buffer is empty at time
0
, and that it is is largeenough so that there is no data loss. For an input ﬂow
x
, theoutput
y
is given by
y
=
x
(6)
where
is the sub-additive closure of
.
A simple proof of this theorem will be given in [1]. Remem-ber that if
is sub-additive and
(0)=0
,
=
. An im-mediate consequence of this theorem is that a greedy shaperoffers to the incoming ﬂow a service curve equal to
. Theinput-output characterization of greedy shapers
y
=
x
is however much stronger than the service curve property.
6. REFERENCES
[1] J. Y. Le Boudec, P. Thiran, S. Giordano, ‘A short tu-torial on Network Calculus II: Min-plus system the-ory applied to Communication Networks’,
Proc. IS-CAS’2000
, Geneva, May 2000.[2] J.-Y. Le Boudec, ‘Application of Network CalculusTo Guaranteed Service Networks,’
IEEE Trans. In- formation Theory
, vol 44(3), May 1998.[3] C.S.Chang. ‘Aﬁlteringtheory for deterministictraf-ﬁc regulation’, in
Proceedings Infocom’97
, Kobe,Japan, April 1997.[4] R. Agrawal, R. L. Cruz, C. Okino and R. Rajan,‘Performance Bounds for Flow Control Protocols’,
IEEE Trans. on Networking
, vol 7(3), pp 310–323,June 99.[5] R. L. Cruz. ‘Quality of service guarantees in vir-tual circuit switched networks’,
IEEE Journal onSelected Areas in Communication
, pp. 1048–1056,August 1995.

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