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

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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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