We prove that the buffer bound in [CFZF98] can be improved, by using an extension of the so-called network calculus bounds. We also show that the delay bound in [CFZF98] is simply the sum of worst case queuing delays at all nodes along the path of a connection

Hebuterne, Gerard

Le Boudec, Jean-Yves

We prove that the buffer bound in [CFZF98] can be improved, by using a modification of the proofs in the original paper [CFZF98], together with so-called network calculus bounds. We also show that the delay bound in [CFZF98] is the sum of worst case queuing delays at all nodes along the path of a connection.LCALCA

Hébuterne, G.

Infoscience - École polytechnique fédérale de Lausanne

Comment on \A Deterministic Approach to theEnd-to-end Analysis of Packet Flows in ConnectionOriented Networks"Jean-Yves Le BoudecGerard HebuterneAbstractWe prove that the buer bound in [CFZF98] can be improved, by using a modicationof the proofs in the original paper [CFZF98], together with so-called network calculusbounds. We also show that the delay bound in [CFZF98] is the sum of worst case queuingdelays at all nodes along the path of a connection.Contents1 Introduction 12 Improving the bounds in [CFZF98] 23 Concluding Remarks 5A Appendix: Proof of Theorem 2.2 61 IntroductionIn [CFZF98], the authors consider a network of discrete time, First In First Out (FIFO)queues. They assume that the network uses a connection oriented paradigm, and thatpackets (called \cells") all have the same size (as is the case with ATM). In particular,it is assumed that all cells belonging to one connection follow the same path, establishedat connection setup. In this context, the words \connection" and \ow" have the samemeaning. It is further assumed that a connection is spaced at the source by at leastthe Route Interference Number (RIN) of the connection. The RIN of connection r isdened as the number of occurences of other connections joining the path of r. Thisassumption is called the source rate condition; it is shown in [CFZF98] that under thisset of assumptions: the network is stable (namely, queue lengths remain bounded);1 the delay for any connection is bounded by its RIN; the maximum buer required at a queue with I input links and Niconnections oninput link i is bounded by max1iI(N  Ni), with N =PIi=1Ni.In this contribution, we show that, under the same assumptions, it is possible to improvethe buer bound to min1iI(N Ni) instead of max1iI(N Ni). This can be achievedusing a small variation of network calculus bounds [Cru95, Bou98, Cha98, RR96], togetherwith a modication of the main proofs in [CFZF98]. This also implies some improvementsfor the delay bound.Essentially the same result was found independently by H. Zhang in [Zhn99], usinga dierent approach, based on a detailed analysis of worst case delays. See also someconcluding remarks in Section 3. In [Zhn99], Zhang also analyzes the tightness of thebound.In Section 2 we give the new bounds. Section 2 rely on a number of lemmas, whichare given in appendix.Throughout the paper we adopt a discrete time model, as in [CFZF98], and assumethat all packets have a the same size, equal to one unit of data.2 Improving the bounds in [CFZF98]Our starting point is a number of results which we collectively refer to as \network cal-culus" [Cru95, Bou98, Cha98, RR96]. These results give deterministic bounds on buerand delay, assuming input processes are limited by some arrival curves, and the serviceelement oer some form of service guarantee. We say that a ow admits a function (t)as arrival curve if the number of cells that can be observed on the ow during any intervalof duration t is  (t). We also say that a node e oers to a ow r a \strict service curve"(t) if, during any time interval of length t, for which the backlog of connection r at nodee is positive, the number of cells of ow r which are output by the node is  (t). Thebacklog for ow r at node e at some time instant is dened as the number of cells of ow rwhich have entered node e and did not depart yet. The strict service curve property wasdened for example in [LM96] and is an abstraction of the generalized processor sharingconcepts introduced in [KG94]. The following theorem is a new variant of classical resultsin [Cru95, Bou98, Cha98, RR96].Theorem 2.1. Consider a node which receives an input connection, with a buer largeenough to avoid discarding data. Assume that the node oers a strict service curve  andthat the input connection has an arrival curve . Assume that (u0)  (u0) for someu0> 0. Then the maximum buer occupancy is bounded by sup0u<u0((u)  (u))The theorem says that, for the computation of a buer bound, it is sucient to considertime intervals less than u0. The idea is that the busy period duration is less than u0.Proof: The proof is similar to network calculus bounds in [Cru95, Bou98, Cha98,RR96]. Call x(t) [resp. x(t)] the input [resp. output] function. This is traditionallydened as the cumulative number of cells observed on the input [resp. output] ow overthe time interval [0; t]. Consider a given time t at which the buer is not empty, and call sthe last time instant before t at which the buer was empty. Then, from the strict servicecurve property, we havex(t)  x(s) + (t  s) = x(s) + (t  s)2Thus the buer size at time t satisesb(t) = x(t)  x(t)  x(t)  x(s)  (t  s)  (t  s)  (t  s)Now if t   s  u0, then there exists a time t0= s + u0, with s + 1  t0 t such thatb(t0) = 0. This contradicts the denition of s. Thus we can assume that t  s < u0.Now we proceed with a property which generalizes the results in [CFZF98], and whichwill be required for improving the buer bound.Theorem 2.2. With the assumptions in [CFZF98], consider a given link e and a subsetS of m connections which use that link. Let n be a lower bound on the number of routeinterferences that any connection in the subset will encounter after this link. Then overany time interval of duration m+ n, the number of cells belonging to S that leave link eis bounded by m.An equivalent way to formulate the theorem is to say that, if we call  the minimumarrival curve for the aggregate of the m ows on link e, then we have(m+ n)  m (1)Before giving the proof, let us mention the following. For m = 1, the theorem meansthat the spacing between cell departures from link e is at least n + 1, where n is the"remaining" route interference number for the connection, past link e. In general, n isless than the RIN of the connection. The dierence n RIN is due to cell delay variationaccumulated in the buers along the path. In other words, a connection gains cell delayvariation along its path, and the cell delay variation is bounded by the route interferencenumber consumed along the path. This result (namely for m = 1) derives immediatelyfrom Theorem 1 in [CFZF98]However the theorem is more powerful. It gives a bound on the number of arrivals foran aggregate ow. It can easily be seen that the bound is not a simple consequence ofbounds for individual connections; indeed, the bound so obtained would be (1+n)  m,instead of (m+n)  m. In contrast, the bound in the theorem takes into account globalinteractions between connections.Proof: Remember that we have assumed a connection oriented network; thus everycell (= packet) belongs to one connection, and every connection uses one path. Consideran arbitrary cell c, and call r the connection it belongs to; as a shortcut, we also say thatcell c \is in S" if r is in S.Consider now a xed time interval (s; t] = [s + 1; t] with t = s +m + n. Call A theset of all cells in S which leave the link during (s; t]. Note that connections in S mayinterfere at several dierent links, but since they all end up using link e, there is alwaysone last link before or at e at which they interfere. We call this link the merging pointof the two connections. We use the classical denition of busy period used in queuingtheory, namely, a time interval during which the backlog for the ow at the node is alwayspositive. For two cells c and d in S, and for some link f , we say that d 4 c if c and dare in the same busy period at the merging points for the connections of c and d (seeFigure 1). We will use the binary relation 4 as follows. By Lemma 1 in [CFZF98], thedelay for a cell c in S due to interferences in S at the merging point is bounded by thenumber of cells d in S satisfying d 4 c.3efhs  + 1gC e l l  c 2C e l l  dt  -  k  C e l l  c 1t   =   s  +  m  +  n   h g  f eC e l l  c 2C e l l  c 1C e l l  dgA 1A 0AFigure 1: A time-space diagram illustrating the denitions of d 4 c1and c14 c2(all cells shownare in S). Time ows downwards. Rectangles illustrate busy periods. c1; c2are in A1while d is inA0. The merging point for the ows of d and c1is link g (not g0).Our denition of d 4 c is very close to the concept of delay chain used in [CFZF98].More precisely, d 4 c at some link f is equivalent to saying that cells c and d are in thesame delay chain at link f , that link f is the merging point for the connections of c andd, and that d reached link f before c.We now dene the set A0as follows; we say that some cell d is in A0if and only if d isin S, d leaves link e before or at time s, and there exists a sequence of cells c0= d; c1; :::; ckall in S, such that ci 14 cifor i = 1; :::; k, and ck2 A. The denition of A0is similar tothat of the superchain for a given path in [CFZF98]; however it diers in that it does notapply to one specic path, and that the cells in the chain are restricted to be in S.Call k = #A0, where the # sign indicates the number of elements in a set (thus k isthe number of connections that are in A0). Note that it is possible that A0is empty, inwhich case k = 0. Also call A1the subset of A made of those cells that leave the link in(s; t  k]. Since the link rate is one cell per time unit, we have:(#A)  k + (#A1) (2)The main idea of the proof is that(#A1)  m  k (3)which together with Equation (2) will prove the result. Equation (3) follows from Lemma A.1which shows that there can be at most one cell per connection in A0[A1.We now come to our main result. Consistent with the network model in [CFZF98],a network nodes is modelled as a collection of output buers, with no contention otherthan at the output buers. Every buer is associated with one unidirectional link which4it feeds. Every link has one origin node and one end node. We say that a link f is incidentto link e if the origin node of link e is the destination node of link f . In general, a linkhas several incident links.Theorem 2.3. Consider some arbitrary link e with I incident links, and call Nithenumber of connections that use link e and arrive on the ith incident link, i  I. Also callN =PIi=1Ni. With the assumptions in [CFZF98], the amount of data in the buer isbounded by min1iI(N  Ni) (instead of max1iI(N  Ni) as in [CFZF98])Proof: We apply Theorem 2.2 to the set of connections that arrive at e on link i, withm = Niand n = N . The maximum number of cells that can arrive during t time slots atlink e via the ith incident link is thus limited by i(t) = min(t;Ni), for t  N . We nowapply Theorem 2.1 to  =PIi=1iand the strict service curve (t) = t. We can assumewithout loss of generality that N1 N2 :::  NI. The function     is continuousand has a derivative at all points except the Ni's. The derivative changes its sign at NI(=max1iI(Ni)) thus the maximum of    is at NIand its value is N  NI.The delay bound Lastly, let us discuss the delay bound. Call j(r; e) the link bywhich connection r arrives at node e. From Theorem 2.3, the delay experienced by a cellof connection r arriving at node e is bounded bymin1iI(e)(N(e)  Ni(e))  N(e)  Nj(r;e)(e) (4)Here we have denoted with I(e) the number of incident links at node e, Ni(e) the numberof connections arriving at node e on link i, and N(e) =PI(e)i=1Ni(e). Now N(e) Nj(e) isthe number of route interferences for connection r at node e. Also write e 2 r to expressthat node e is on the path of connection r. The end-to-end delay for connection r is thusbounded by(r) =Xe2r(N(e)  Nj(r;e)(e)) (5)which is precisely the RIN of connection r.This result is already in the original paper [CFZF98]. However, we should mentionhere rstly that, contrary to what might be interpreted from [CFZF98], the end-to-enddelay bound is the sum of the local, independent delay bounds at every node. Secondly, abetter bound can be directly obtained by using the left-hand side in Equation (4) insteadof the right-hand side. This gives the following bound for the end-to-end delay:0(r) =Xe such that e2rmini such that 1iI(e)(N(e) Ni(e))Namely, the end-to-end delay is bounded by the sum of the minimum numbers of route in-terferences for all connections at all nodes along the path of a connection. For asymmetriccases, this is less than the RIN of the connection as given in Equation (5).3 Concluding RemarksEssentially the same result was found independently by H. Zhang in [Zhn99], using adierent approach, not based on network calculus. In our approach, we show an interme-diate result (Theorem 2.2), which gives a property of the arrival function for an aggregate5number of connections. We believe that this direction could be used to analyze general-izations of the original problem in [CFZF98], in particular, if we consider more generalgeneral source rate conditions.References[Bou98] J.-Y. Le Boudec. Application of network calculus to guaranteed service net-works. IEEE Transactions on Information Theory, 44:1087{1096, May 1998.[CFZF98] I. Chlamtac, A. Farago, H. Zhang, and A. Fumagalli. A deterministic approachto the end-to-end analysis of packet ows in connection oriented networks.IEEE/ACM transactions on networking, (6)4:422{431, 08 1998.[Cha98] C.S. Chang. On deterministic trac regulation and service guarantee: A sys-tematic approach by ltering. IEEE Transactions on Information Theory,44:1096{1107, August 1998.[Cru95] R.L. Cruz. Quality of service guarantees in virtual circuit switched networks.IEEE JSAC, pages 1048{1056, August 1995.[KG94] Parekh A. K. and Gallager R. G. A generalized processor sharing approachto ow control in integrated services networks: The multiple node case.IEEE/ACM Trans. Networking, vol 2-2, pages 137{150, April 1994.[LM96] Cruz R. L. and Okino C. M. Service guarantees for window ow control. In34th Allerton Conf of Comm., Cont., and Comp. Monticello, IL, Oct 1996.[RR96] Agrawal R. and Rajan R. Performance bounds for guaranteed and adaptiveservices, December 1996. IBM Technical Report RC 20649.[Zhn99] Hongbiao Zhnag. A note on deterministic end-to-end delay analysis in connec-tion oriented networks. In Proc of IEEE ICC'99, Vancouver, pp 1223{1227,1999.A Appendix: Proof of Theorem 2.2We use a discrete time model, and assume that all propagation times are 0. The proof inthis appendix can easily be modied to incorporate propagation times, but we prefer toleave this to the reader as this complicates a notation which is already complex enough.The main technical result is the following lemma. It is an extension of the \excludedsuperchain" lemma in [CFZF98].Lemma A.1. With the notation in Theorem 2.2, there is at most one cell per connectionin A0[A1.Proof: Firstly, we prove that there cannot be two cells of the same connection in A0.We proceed by contradiction. Assume that d; d02 A0belong to the same connection andwere emitted in that order. There exists c0= d; c1; :::; ckin S such that c04 c1; c14c2; :::; ck 14 ck, with ck2 A1. Call fithe merging point for the connections of ci 1and ci. We show now that cell d0must be owing on the common subpath after cell ck.Indeed, otherwise, from the FIFO property, it would have reached fkbefore ck. Assumeit has reached fkafter ck 1; then it would belong to a super chain from d to d0. This is6ehs  + 1f  1c  k - 1c e l l  dt  -  k  c 1t   =   s  +  m  +  n   f  k - 1 f  kc  kd  A 1A 0AFigure 2: Two cells of same connection cannot be in A0ehs  + 1f  1C e l l  dA 1t  -  k  c 1t   =   s  +  m  +  n   d   1s  + tA 0AFigure 3: There cannot be two cells of the same connection in A1.impossible from Lemma 2 in [CFZF98]. Thus d0reaches fkafter ck 1. By recursion, thisshows that d0must have reached f1before c0= d which is a contradiction. Thus cell d0must be owing on the common subpath after cell ck. Now this contradicts the facts thatd02 A0and ck2 A1.Secondly, we show that there cannot be two cells of the same connection in A1. Letd be a cell in A1. Call s +  , with   1 the time at which d leaves the link. FromLemma 1 in [CFZF98] the delay experienced by d along its path is bounded by thenumber of interference units experienced by d. Now let us make the distinction betweenan interference unit which is due to a connection in S and which occurs at the mergingpoint with the connection of d (call r the number of such interference units experiencedby c), and other interference units experienced by d (call r0their number). For exampleon Figure 3, the interference of c1at g0is counted in r0, whereas the interferences of c1atg and c2at f are counted in r. The delay experienced by d between its source and linke is thus bounded by r + r0. Now if c is an interfering cell counted in r, we have c 4 d.Thus either c 2 A0or c 2 A1. In the latter case, by the FIFO property, c must leave e7before d . Thusr  k +    1 (6)Also call R0the number of route interferences for the connection of d due to connectionsthat are either not in S or are in S but are not at the merging point with the connectionof d. We have thus r0 R0. Call t0the departure time for cell d at its source. From theabove we havet0= s+    r   r0 s  k + 1 R0Call t1the emission time for the next cell, say d1, following d at its source. By theassumptions in the theorem, the total number of route interferences for the connection ofd is at least (m  1) +R0+ n, thust1 t0+m+R0+ ncombining the two previous equations we have:t1 s+m+ n  k + 1 = t+ k + 1Thus, from the denition of A1, d1is not in A1, which proves that there cannot be twocells of the same connection in A1.Thirdly, we prove that if d 2 A0and d02 A1, then necessarily d and d0belongto dierent connections. Consider some d 2 A0, and a sequence c0= d 4 c1; c14c2; :::; ck 14 ck, with all cells in S and ck2 A1. Call fithe merging point for ci 1andci. See Figure 4. Call b the arrival time of d at f1and s +  , with   1 the departuretime for ckat link e. Dene rias the number of interference units in the busy periodat fiwhere ci 1and ciinterfere, plus those experienced by cell cibetween fiand fi+1,excluding fi+1, due to cells in S at their merging point with the connection of d. Followingthe same reasoning as in the proof of Lemma 2 in [CFZF98] we have:s+    b kXi=1(ri+ r0i) (7)Call t(d); t(d0) the departure times of cells d and d0at their (common) source. Wehave similarly:b  t(d)  r0+ r00(8)where r0[resp. r00] is the number of interferences units for cell d on its path from thesource to f1, excluding f1, due to cells in S at their merging point with the connection ofd [resp. due to other cells].Now since the spacing between cells at the source is at least the route interferencenumber for the path of d and d0, we have:t(d0)  t(d)  m+ n+R0(9)where R0is the number of route interferences for the path of d, due to connections not inS at the merging point with the connection of d. ThuskXi=0r0i R0(10)8es  + 1f  1t  -  k  t   =   s  +  m  +  n   f  ic  kdc  ic  1f  kbt ( d )d   t ( d    )r i  + r      i  i n t e r f e r e n c e sA 1A 0AFigure 4: There cannot be two cells of the same connection in A0[ A1.Combining Equations (7), (8), (9) and 10, we get:t(d0)  s+  +m+ n kXi=0riNow every interference counted in ricorresponds to one cell  in S with  4 ciforsome i. Thus either  2 A0or  2 A. By the FIFO property, in the latter case  mustleave link e before ck. ThusPki=0ri k+  1. Thus t(d0)  s+m+n k+1 = t k+1thus d0is not in A1.Combining the three arguments, we nd that there is at most one cell per connectionin A0[A1.9

Comment on 'A Deterministic Approach to the End-to-end Analysis of Packet Flows in Connection Oriented Networks'

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Infoscience - École polytechnique fédérale de Lausanne

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