State-dependent M/G/ Type Queueng Analyss for Congeston Control n Data Networks Etan Altman, Kostya Avrachenkov, Chad Barakat 2 INRIA 24, route des Lucoles { B.P. 93, 692 Sopha Antpols, France fetan.altman,k.avrachenkov,chad.barakatg@sopha.nra.fr Rudesndo Nu~nez Queja 3 CWI P.O. Box 9479, 9 GB Amsterdam, The Netherlands sndo@cw.nl ABSTRACT We study a TCP-lke lnear-ncrease multplcatve-decrease ow control mechansm. We consder congeston sgnals that arrve n batches accordng to a Posson process. We focus on the case when the transmsson rate cannot exceed a certan maxmum value. The dstrbuton of the transmsson rate n steady state as well as ts moments are determned. Our model s partcularly useful to study the behavor of TCP, the congeston control mechansm n the Internet. Burstness of packet losses s captured by allowng congeston sgnals to arrve n batches. By a smple transformaton, the problem can be reformulated n terms of an equvalent M/G/ queue, where the transmsson rate n the orgnal model corresponds to the workload n the `dual' queue. The servce tmes n the queueng model are not..d., and they depend on the workload n the system. 2 Mathematcs Subject Classcaton: 6K25, 68M2, 9B8, 9B22. Keywords & Phrases: state-dependent queue, congeston control n data networks, lnear ncrease and multplcatve decrease, transmsson control protocol (TCP). Note: The work of R. Nu~nez Queja s part of the PNA 2. project `Qualty n Future Networks' of the Telematcs Insttute. Supported wth a CNET France-Telecom grantonow control n Hgh Speed Networks. 2 Fnanced by an RNRT (French Natonal Research Network n Telecommuncatons) \Constellatons" project on satellte communcatons. 3 Vsted INRIA Sopha Antpols from February untl June (2) wth support from the Netherlands Organzaton for Scentc Research (NWO).
Introducton 2 Introducton In today's hgh speed telecommuncaton networks, a large part of the trac s able to adapt ts rate to the congeston condtons n the network. Congeston control s typcally desgned so as to allow the transmsson rate to ncrease lnearly n tme n the absence of congeston sgnals, whereas when congeston s detected, the rate decreases by a multplcatve factor. Ths s both the case of the Avalable Bt Rate (ABR) servce category n ATM[] (see denton and use of RDF and RIF) as well as the Transmsson Control Protocol (TCP) n the Internet envronment[9, 2]. Congeston s detected by the source through sgnals. In case of ABR, the congeston sgnals are RM (Resource Management) cells that have been marked due to congeston nformaton n some swtch along the path of the connecton. In case of the Internet, the congeston sgnals are packet losses that are detected by the source ether through the expraton of a retransmsson tmer, or through some negatve acknowledgement mechansm (three duplcate ACKs [2]). There s also a proposal to add some explct congeston sgnalng to the Internet (the ECN proposal [6]). The performance evaluaton of congeston control mechansms s an mportant ssue for network and protocol desgn. Ths evaluaton requres a descrpton of tmes between the arrvals of consecutve congeston sgnals. Expermentatons over the Internet [4, 4] have shown that on long dstance connectons, the Posson assumpton about the tmes between congeston sgnals s qute reasonable. Ths happens when the throughput of the studed connecton s small compared to the exogenous trac, and when the number of hops on the path s large so that the superposton of the packet drops n routers leads to exponental tmes between congeston sgnals. For local area networks, we notced that the congeston sgnals may arrve n bursts [4]. However, the tmes between bursts correspond well to the Posson assumpton. For ths reason, we consder the case when congeston sgnals arrve n batches accordng to a Posson process. Batches contan a random number of congeston sgnals and each such sgnal causes the dvson of the transmsson rate by some constant. In the sequel, we also refer to a batch of congeston sgnals as a loss event. We focus on the case when a certan lmtaton on the transmsson rate exsts. We determne the exact expresson of the throughput under such a lmtaton. In the lterature, only smplstc approxmatons have been proposed [3, 8] so far. We study two possble scenaros that lead to such a lmtaton: () Peak Rate lmtaton: the lmtaton s not due to congeston n the network but rather to some external agreement. In that case, when the transmsson rate reaches a certan level M, t remans constant untl a loss event appears. For example n case of TCP, the wndow cannot exceed the buer space avalable at the recever [2]. In the ABR servce of ATM, the transmsson rate cannot exceed the Peak Cell Rate mposed by the contract between the user and the network. It s expected that such lmtatons on the transmsson rate wll become more and more mportant as the capacty and the speed of the lnks n the network grow, snce t s then more lkely that connectons reach ther maxmum peak rate before congeston n the network occurs. Of course, ths s not the case f peak rates ncrease n proporton wth the speed of network lnks. () Congeston lmtaton: the lmtaton on the transmsson rate s due to congeston n the network that occurs whenever the nput rate reaches a level M. In that case we shall have an extra batch of congeston sgnals when the level M s attaned whch also causes a
Introducton 3 decrease of the transmsson rate by a random factor. A typcal example of such lmtaton s the avalable bandwdth n the network. Another example s the reserved bandwdth n a Derentated Servces network [8] n cases where packets exceedng the reserved bandwdth are dropped rather than njected nto the network as low prorty packets [2]. In the partcular case n whch the batches contan a sngle congeston sgnal, the peak rate lmtaton model reduces to the one studed n [4], who already attempted at computng the rst two moments of the transmsson rate. A remarkable observaton s done n that reference showng that the ow control can be reformulated n terms of an equvalent M/G/ queue, where the transmsson rate s translated nto the workload of the queue. The congeston sgnals correspond to customers arrvng at the queue accordng to a Posson process. The servce tmes n the `dual' queueng model are not..d., and they depend on the workload n the system. Ths transformaton s also vald n our more general settng, except that n our model wth congeston lmtaton, there s an addtonal arrval n the equvalent queueng model (n addton to the Posson arrval stream) that occurs whenever the queue emptes. We solve the Kolmogorov equatons and obtan the exact probablty dstrbuton as well as the moments of the transmsson rate (of the wndow n case of TCP) for both problems. In dong so, we correct an error n [4]. We brey menton some related results. Queueng analyss wth servce tmes that depend on the workload or on the queue length have been also consdered n [2,, 6, 9]. Our model s a specal case of the one studed n [9], where an mplct characterzaton of the steady state dstrbuton s obtaned (closed-form expressons were obtaned for specal cases that do not cover our model). In [] an asymptotc approxmaton s used for solvng statedependent GI/G/ queues n whch both nter-arrval tmes, servce requrements and the servce rate may depend on the workload. The peak rate lmtaton model s a specal case of the model wth a general statonary and ergodc arrval process studed n [4]. For that model only bounds on the throughput were obtaned. Exact expressons for the throughput were obtaned there for the case n whch no lmtaton on the transmsson rate exsts (see also [3, 3, 5, 8]). The paper s structured as follows. In Secton 2 we descrbe a general model of ow control wth lmtaton on the transmsson rate and we provde a prelmnary analyss. The two cases of peak rate lmtaton and congeston lmtaton are descrbed separately n Sectons 2. and 2.2. It s shown that a specal case of the model wth congeston lmtaton reduces to that of the model wth peak rate lmtaton. Therefore, n the followng we rst focus on the case of peak rate lmtaton. In Secton 3 we show that the model s dual to an M/G/ queueng model wth servces that depend on the total workload n the system. We then derve the moments and the dstrbuton of the transmsson rate n Sectons 4 and 5 n terms of the probablty that the transmsson rate s at ts maxmum value. Ths quantty can be determned usng that the dstrbuton functon s non negatve, but n order to derve a computatonally tractable expresson for t, we pursue an alternatve approach n Secton 6. The results are speced n Secton 7 for an mportant partcular case, that of one congeston sgnal per batch and a reducton factor of 2. Ths case corresponds to long dstance TCP connectons n today's Internet where the congeston sgnals do not cluster sgncantly. In Secton 8 we present the analyss for a more general model of congeston lmtaton than the In a prvate communcaton, the authors of [4] announced to replace the draft.
2 Flow control wth rate lmtaton: models and prelmnary analyss 4 one n Secton 2.2. In the general case the model does not reduce to the peak rate lmtaton model. The model wth peak rate lmtaton s valdated n Secton 9. By means of numercal examples, Secton 9. llustrates that our results lend themselves for computaton of the wndow (or, transmsson rate) dstrbuton and densty functons. In Secton 9.2 we compare the results of Secton 7 to measurements from long dstance TCP connectons. Secton draws conclusons from the obtaned results and ndcates drectons of further research. Fnally, the Appendx dsplays techncal results needed n the mathematcal analyss. 2 Flow control wth rate lmtaton: models and prelmnary analyss In ths secton we present our model for the rate evoluton of the ow control mechansm. In the sequel we adopt the usual termnology for TCP, the well known wndow-based congeston control protocol of the Internet: we shall work wth the wndow sze rather than the transmsson rate. The transmsson rate of a wndow-based ow control mechansm s at any moment equal to the wndow sze dvded by the round-trp tme (RTT) of the connecton. Let M denote the maxmum wndow sze. The lmtaton on the wndow sze s ether due to a peak rate lmtaton or to a congeston lmtaton. In the followng we explan the smlartes and the derences between the models n the two cases. Whle no congeston sgnal s receved and the wndow s smaller than M, the wndow of the protocol ncreases lnearly at rate >. In case of TCP, ==(brt T ) where b s the number of data packets covered by anack (usually 2, see e.g.[8, 2]). We assume that batches contanng a random number of congeston sgnals arrve accordng to an ndependent Posson process. We denote the szes (.e., the numbers of congeston sgnals) of consecutve batches by N ;N 2 ;N 3 ;:::, and we assume that these consttute an..d. sequence. The sze of an arbtrary batch s genercally denoted by N = d N k. The Posson process and the sequence N k, k =; 2;:::, are ndependent of each other and ndependent of the past evoluton of the wndow. For each congeston sgnal receved, the wndow s dvded by a factor > whch saxed parameter. That s, f an arrvng batch contans N = n congeston sgnals, the wndow s multplcatvely decreased by a factor,n. Immedately after the multplcatve decrease, the wndow restarts ts lnear ncrease. In case of peak rate lmtaton, the wndow stays constant at M when ths maxmum level s reached untl the next congeston sgnal s receved. In case of congeston lmtaton, mmedately upon reachng M, a congeston sgnal s receved and the wndow s decreased. We present the two cases separately n Sectons 2. and 2.2, showng how the analyss of a partcular case of the congeston lmtaton model reduces to that of the peak rate lmtaton model. In Secton 8 we consder a more general model of congeston lmtaton. Frst we ntroduce some further common notaton. We denote the p.g.f. (probablty generatng functon) of the dstrbuton of N by Q(z) :=E z N =: X n= z n q n ; jzj : (2.) Note that the peak rate lmtaton model wth = 2 and q = reduces to the model studed n [4], where congeston sgnals appear accordng to a Posson process and where the wndow
2 Flow control wth rate lmtaton: models and prelmnary analyss 5 s dvded by two upon every congeston sgnal occurrence. By consderng a general model, we am to account for a wde range of ow control mechansms other than TCP and for future enhancements to TCP congeston control. Let us denote the wndow sze at tme t by W (t) 2 (;M]. We have the followng stablty result whch follows from Theorem n [4]: Theorem 2. There exsts a statonary process W (t) such that W (t) converges to W (t) n dstrbuton for any ntal state. Moreover, we have P-a.s. lm sup jw (s), W (s)j =: (2.2) t st Note that (2.2) mples that the statonary dstrbuton of W (t) s unque. For x 2 (;M], denote the (tme-average) dstrbuton functon by F (x) := lm T T Z T t= P fw (t) xg dt: (2.3) It follows from Theorem 2. that ths lmt s ndependent of W () and concdes wth the statonary dstrbuton of W (t). We rst assume that F (x) s contnuous n x 2 (;M) (n the case of peak rate lmtaton t s clear from physcal consderatons that F (x) has an atom at x = M). Under ths assumpton we nd a functon F (x) whch s an equlbrum dstrbuton for the wndow sze and, hence, from ts unqueness t follows that t s the desred dstrbuton. Instead of F (x) t wll be convenent towork wth the complementary dstrbuton functon F (x) =, F (x) =P fw >xg ; x 2 (;M]: To derentate between the cases of peak rate lmtaton and congeston lmtaton, n the latter case we attach a superscrpt cl to the symbols ntroduced above, e.g., the dstrbuton functon s denoted by F cl (x). Next we treat the two cases separately. We show how the analyss of a specal case of the model wth congeston lmtaton reduces to that of the model wth peak rate lmtaton. The analyss of the general congeston lmtaton model s presented n Secton 8. 2. Flow control wth peak rate lmtaton Wth peak rate lmtaton, when the wndow reaches the maxmum level M, t stays there untl the next congeston sgnal s receved. In Secton 3 below we show that the wndow sze process W (t) can be related to the workload of an M/G/ queue (see also [4]). The workload of ths state-dependent M/G/ queue can be seen to be a Markov process (e.g., see []), and hence the wndow sze evoluton W (t) s a Markov process as well. Wth ths n mnd, we derve a steady-state Kolmogorov equaton for F (x) = PfW > xg whch s the bass to our analyss. We use the followng up and down crossng argument: Assume that the process s n equlbrum and consder a level x 2 (;M). Whenever the wndow sze ncreases from less than or equal to x to more than x we say that an up crossng of the level x has occurred. Smlarly, f the wndow sze decreases from more than x to less than or equal
2 Flow control wth rate lmtaton: models and prelmnary analyss 6 to x we say that a down crossng of the level x has occurred. Let [t; t +] be a small tme nterval, where t s a determnstc tme moment. When the process s n equlbrum, the probablty of up-crossng (, )P fx, <W xg + o() s equal to the probablty of down-crossng X n= q n P fx <W mn( n x; M)g + o(): After equatng these, we pass #. Snce we assumed that F (x) = P fw xg s contnuous for x < M (see Remark 5. for a justcaton of ths assumpton), we conclude that the dervatve off (x) exsts and s contnuous for all x except at x = M,n, when q n >. For x 2 (;M)nfM,n g n=;2;::: we obtan the followng steady-state Kolmogorov equaton d X dx P fw xg = q n P fx <W mn( n x; M)g ; or, equvalently,, d F (x) = dx F n= (x), X n= q n F (mn( n x; M)) : (2.4) From ths derental equaton we shall determne F (x), x 2 (;M), n terms of the probablty P M := P fw = Mg =, F (M,) =F (M,): In Secton 4 we rst use (2.4) to determne the moments of the wndow sze dstrbuton n terms of P M. Then we nd the dstrbuton functon tself n Secton 5. The unknown P M s then determned usng the fact that F (x) s a complementary probablty dstrbuton functon (F () = ). However, the expresson obtaned for P M n ths way, does not lend tself for computatonal purposes. Therefore we show an elegant alternatve to determne P M n Secton 6, whch leads to an ecent and numercally stable algorthm for computatons. 2.2 Congeston lmtaton: a specal case When the maxmum wndow sze M s due to congeston lmtaton, mmedately upon reachng the level M a batch of congeston sgnals s generated. In ths secton we study the case when the sze of such a batch has the same dstrbuton as the random varable N. In Secton 8 we present the analyss of the more general case when the number of congeston sgnals that result from reachng M has a derent dstrbuton than N. Smlarly as n Secton 2., we can derve the followng derental equaton for F cl (x), <x<m:, d dx F cl (x)= F cl (x), +gp X N n= q n F cl (mn( n x; M)) ln(m), ln(x) ln() ; (2.5)
2 Flow control wth rate lmtaton: models and prelmnary analyss 7 wth, g :=, d dy F cl (y) : y=m, The addtonal term, compared to (2.4), comes from the fact that a down crossng of the level x may be due to the fact that the level M s reached and that the rate s decreased by a factor,n wth,n M x. Note that f F (x) s the unque complementary dstrbuton functon satsfyng (2.4) then F cl (x) := F (x), P M, P M ; <x<m; (2.6) s the unque complementary dstrbuton functon satsfyng (2.5). Ths follows mmedately by substtutng (2.6) nto (2.5). Ths relaton has a smple geometrc nterpretaton. Usng the fact that the Posson process s memoryless, f we consder the model wth peak rate lmtaton only at moments when the wndow s less than M (.e., we cut out all perods where the wndow equals M), what we get s dentcal to the model wth congeston lmtaton. Thus, we can concentrate on ndng the dstrbuton functon F (x) for the peak rate lmtaton model and then use (2.6) or the equvalent: F cl (x) = F (x), P M : In partcular, the moments of the wndow sze n the two models are related by: E W cl k = E W k, P M M k, P M : (2.7) In Secton 4 below we derve a recursve relaton for E W k. Combned wth (2.7), ths gves a recurson on Eh, W cl k whch we report at ths pont for completeness: k h, E W ke W cl k, cl = (, Q(,k )), P M, P M M k : (2.8) Remark 2. We emphasze that n the congeston lmtaton model, the quantty P M has no clear nterpretaton. In Secton 6 we use the nterpretaton of ths quantty n the peak rate lmtaton model to compute t. If we were to analyze the congeston lmtaton model wthout usng (2.6), then from (2.5) we could express F cl (x) usng the same technques as n Secton 5 n terms of g nstead of P M. Note that these two constants are related: g = P M, P M ; (2.9) The constant g can be determned usng that F cl (x) s a complementary probablty dstrbuton, see (5.8) below. Snce from the analyss of Secton 6 we obtan a more tractable expresson for P M (see Remark 5.2 for a related dscusson), we wll not further dwell on ths approach.
3 The dual queueng model 8 3 The dual queueng model Before proceedng wth determnng the moments and the dstrbuton of the wndow sze, we brey show how the problem can be related to an M/G/ queueng problem wth servce dependng on the system workload, see also [4]. Frst we concentrate on peak rate lmtaton, below we comment on congeston lmtaton. Dene U(t) = M, W (t) : (3.) I.e., U(t) s obtaned by `ppng' W (t) around a horzontal lne and then scalng by a factor =. In partcular, the area between W (t) and the maxmum wndow sze M (Fgure ) corresponds to the area below U(t). Note that U(t) resembles the evoluton n tme of the workload (or the vrtual watng tme) n a queueng system. A wndow equal to M corresponds to an empty queueng system. The lnear ncrease n workload between arrvals of congeston sgnals corresponds to the decrease n workload due to servce n the queueng model. The arrval of a batch of congeston sgnals n our model corresponds to an arrval to the queue. The reducton of the wndow upon a loss event corresponds to the ncrease n workload upon arrval n the equvalent queueng model. Gven that the amount by whch the wndow s reduced depends on the current value of the wndow (and of course on the number of congeston sgnals n the batch), the servce tme n the dual queueng model s dependent on the current workload there. We conclude that the dual model behaves ndeed as an M/G/ queue (nnte buer capacty, one server and Posson arrvals wth ntensty ) wth state-dependent servce requrements. If U n s the workload seen by arrval n n the M/G/ queue, then ts servce tme x n s equal to M x n =, U n :, ; Nn where N n s the number of congeston sgnals n the nth batch of congeston sgnals n the orgnal model. Instead of drectly workng wth the congeston control model as we do n ths paper, one could analyze the queueng model and swtch back to the ow control problem by usng Equaton (3.). In partcular, E W k = E (M, U) k, P fw xg =, P fu (M, x)=g for x < M and P M s equal to the fracton of tme that the dual queue s empty. In the case of congeston lmtaton, the only derence n the dual queueng model s that we have an addtonal arrval once the system becomes empty. Thus, the arrval process s the sum of a Posson process of ntensty and another process that depends on the workload of the system (t generates an mmedate arrval when the queue becomes empty). The denton of the servce tmes n the dual queue and the transformaton back to the ow control problem reman the same. 4 Moments of the wndow sze dstrbuton Now focus on the model wth peak rate lmtaton (the results obtaned can also drectly be used for the specal case of congeston lmtaton descrbed n Secton 2.2). In ths secton we study the moments of the wndow sze. The k-th moment of the transmsson rate can be
4 Moments of the wndow sze dstrbuton 9 smply obtaned by dvdng the k-th moment of the wndow sze by (RT T ) k. Of partcular nterest s the expectaton of the transmsson rate whch concdes wth the throughput of the transfer or the tme average of the transmsson rate. Let X denote the throughput. We have Z T X = lm X(t)dt = E [W ] T T RT T : (4.) Dene for Re() the LST (Laplace-Steltjes Transform) of the wndow sze dstrbuton by ^f() = Z M + x= e,x df (x): Takng LTs (Laplace Transforms) n (2.4) leads to: ^f(), PM e,m =, ^f(), X n=,n, q ^f(,n ) n,n : Note that (4.2) holds n partcular for M =,.e., no lmtaton on the wndow sze, n whch case P M =. Usng E W k M k, k =; 2;:::,wemay wrte X (,) h k E W k ;, ^f(,n ),n ^f() = + = X k= k= k (,,n ) k E hw k+ : (k + ) (4.2) Substtutng ths n (4.2), usng the absolute convergence of the doubly-nnte seres to nterchange the order of summaton and equatng the coecents of equal powers of we get, for k =; 2;:::, E hw k = k, E W k,, P M M k, (, Q(,k )) ; (4.3) from whch the moments of the wndow sze dstrbuton can be obtaned recursvely. partcular we nd for k =; 2: E [W ]= (, P M) (, Q(, )) ; (4.4) E W 2 2, (, P M ), P M M, Q(, ) = : (4.5) 2 (, Q(, )) (, Q(,2 )) These rst two moments can also be obtaned usng drect arguments, see Remarks 4. and 4.2 below. Such arguments were also used by Msra et al. [4] for the case = 2 and N. However, n ther analyss an error appears whch results n an addtonal equaton besdes In
4 Moments of the wndow sze dstrbuton W(t) -N γ M W W t Fgure : Area assocated wth a sngle loss (4.4) and (4.5) from whch they determne an ncorrect expresson for the probablty P M (see Remark 4.2). Remark 4. The mean wndow sze can be obtaned by consderng the mean drft. The upward drft of the wndow sze s gven by P fw <Mg and the downward drft equals E [W ],, E,N. Equatng these gves (4.4). We can further derve E W 2 applyng an argument smlar to Lttle's law aswas done by Msra et al. [4] for the case =2and N. The man dea s sketched n the followng. For the dual queueng model descrbed n Secton 3, we can equate the mean workload E [U] wth tmes the mean area below U(t) `nduced by a sngle arrval' (use that Posson arrvals see tme averages: PASTA). Back n the orgnal model, the `mean surface' of the area above W (t) n Fgure equals M, E [W ]. We nd an alternatve expresson for ths area by determnng the surface of the area `nduced' by a loss event. Ths s also depcted n Fgure. Suppose a loss occurs at wndow sze W and the wndow s reduced by a factor,n. We can assocate wth ths loss an area above the curve (the surface of the larger trangle mnus that of the smaller one) equal to, M,,N W 2, 2 (M, W )2 : 2 Because of PASTA and the fact that N s ndependent ofw, the expectaton of the surface of ths area s 2,, Q(,2 ), E W 2, 2M, Q(, ), E [W ] : The rate at whch losses occur s, and so: M, E [W ]= 2 Together wth (4.4) ths ndeed gves (4.5).,, Q(,2 ), E W 2, 2M, Q(, ), E [W ] : (4.6)
5 Wndow sze dstrbuton functon Remark 4.2 For a specal case of our model, yet another way s pursued n [4] to derve (4.4) and (4.5). However, there, the nal result s ncorrect due to a small error n an ntermedate step. Denng P M (t) :=P fw (t) =Mg, the rst two moments of W (t) satsfy: d dt E [W (t)] =,,, Q(, ) E [W (t)] + (, P M (t)) ; d dt E W (t) 2 =,,, Q(,2 ) E W (t) 2 +2 (E [W (t)], MP M (t)) : In steady state we have E [W (t)] E [W ], E W (t) 2 E W 2 and P M (t) P M. Substtuton nto (4.7) gves (4.4) and substtuton nto (4.7) gves: =,,, Q(,2 ) E W 2 +2 (E [W ], P M M) : (4.7) The latter s a lnear combnaton of (4.4) and (4.6) and, hence, leads to (4.5). For the case = 2 and N, the formula gven n [4] for E W 2 (below Formula (4) n that reference) ders from (4.7) by a factor, =,=RT T. Ths resulted n a thrd (ncorrect) equaton whch s ndependent of (4.4) and (4.5) from whch P M was determned smultaneously wth E [W ] and E W 2. In Secton 6 we show how P M can be determned correctly and computed ecently. 5 Wndow sze dstrbuton functon In ths secton we determne the cumulatve dstrbuton functon of the wndow sze explctly. The dstrbuton of the transmsson rate can be smply obtaned by rescalng the wndow axs by /RTT. We start wth the case where M <, provdng an expresson of the dstrbuton functon n the ntervals [M= k ;M= k, ] wth k =; 2;:::. Then, for the case M =, we gve an expresson of the dstrbuton for any x> as an nnte sum of exponentals. 5. Wndow dstrbuton for nte M For M= x<m, Equaton (2.4) reduces to: hence,, d F (x) =F (x); dx F (x) =P M e (M,x) ; M x<m: (5.) To nd the entre dstrbuton we ntroduce, for k =; 2; 3;:::, F k (x) :=F (x); M k x< M : (5.2) k,
5 Wndow sze dstrbuton functon 2 Equaton (2.4) can now be wrtten as: X d dx F k(x) =, F k(x)+ k, n= Snce F (x) s contnuous for <x<m we have: q n F k,n ( n x): (5.3) F k ( M )=F k,( M ); k =2; 3;:::: (5.4) k, k, F k s recursvely gven by F k (x) = F k, (M= k, )e, e, x Z M= k, u=x M k,,x X e u k, n= q n F k,n ( n u)du: We conclude from the above recurson that a soluton to (5.3) and (5.4) has the followng form F k (x) =P M kx = c (k) e,,x ; k =; 2; ::: (5.5) To determne the coecents c (k), we substtute (5.5) nto (5.3) and change the order of P summaton n the last term k, q n= nf k,n ( n x): P M kx = =, P M c (k) (,, )e,,x )= kx = c (k) e,,x + P M " k, X X = n= q n c (k,n),n+ # e, x By equatng the terms wth the same exponents, we get the followng recursve formula c (k) + =, X n= q n c (k,n),n+ ; =; :::; k, : (5.6) Once the coecents c (k) ; = 2; :::; k are computed, the coecent c (k) can be determned from (5.4): c (k) =e M k, " k, X = c (k,) e,,km, kx =2 c (k) e,,k M = # : (5.7) Note that to compute the coecents c (k), we do not need P M. Hence, usng that F (x) s a complementary dstrbuton functon, P M s then determned by: kx =F () = lm F k(m= k, )=P M lm c (k) e, M=k, : (5.8) k k
5 Wndow sze dstrbuton functon 3 However, ths relaton s not sutable to compute P M, see Remark 5.2 below. Remark 5. Wth (5.2) and (5.5) we have found an equlbrum dstrbuton functon F (x) satsfyng (2.4). By Thm. 2. t s the unque soluton and, hence, the assumpton that F (x) s contnuous for x<m s justed. Remark 5.2 Recurson (5.6) s sutable to determne the dstrbuton functon on an nterval M= k x M when k s not too large. For large k the recurson may become nstable, snce t nvolves subtracton of numbers of the same order. Therefore (5.8) s not sutable to compute P M. In Secton 6 below we derve an alternatve expresson for P M, whch leads to anumercally stable and ecent algorthm to compute P M. Remark 5.3 One can alternatvely show that the functons F k (x) are of the form (5.5) usng Laplace Transform technques. Note that by means of the derental equatons (5.3), these functons can be extended beyond the ntervals [M= k ;M= k, ] to the whole real lne. Of course, outsde the ntervals [M= k ;M= k, ] the functons F k (x) may (and wll) be unequal to F. One may take Laplace Transforms n (5.3), solve the resultng recurson on k, and nvert the transforms after applyng partal fracton expanson. Ths approach s used n Sectons 5.2 and 6. 5.2 Wndow dstrbuton for nnte M In ths case, the results derved n the prevous subsecton cannot be appled mmedately by lettng M go to nnty. However, we can derve the LST of the wndow sze dstrbuton by smlar arguments as before. When M =, (4.2) becomes " ^f() =, ^f() X, ^f(,n ) q n n= # ; or, equvalently, ^f() = + X n= q n ^f(,n ): (5.9) Substtutng the above equaton repeatedly nto tself l tmes, applyng partal fracton expanson at each step, and then takng l, we conclude that ^f() can be expressed as follows: ^f() = X = c, + ; (5.) for certan coecents c (ths s formally justed later). To determne the constants c ; = ; ; :::, we substtute (5.) nto (5.9) and equate coecents multplyng the terms =( +
6 The probablty of maxmum wndow sze 4 ). Ths leads to the recursve formula c = c, X k= q k c,k c ; whch determnes the ratos c =c (t s for ths reason that both sdes contan a factor =c ). The coecent c follows from ^f() =, P = c =: c =,( + X = c c ), : (5.) Inverson of (5.) back nto the tme doman gves: F (x) =C + X = c e, x ; (5.2) wth C = (because F () = ). Note that the above seres s absolutely convergent for any value of x 2 [; ). Thus, t s the (unque) soluton to (2.4) when M =. For the case of no wndow sze lmtaton and N ; (q = ), F (x) was already obtaned n [7]. 6 The probablty of maxmum wndow sze In Sectons 4 and 5 we determned the wndow sze dstrbuton and ts moments n terms of P M. In ths secton we derve an expresson for P M from whch t can be computed ecently. For ths we ntroduce the random varable T (x) whch s the tme untl the wndow sze returns to the value x, startng just after a loss event occurs wth the wndow sze beng equal to x 2 (;M]. We denote ts expectaton by E(x) :=E [T (x)], x 2 (;M]. Then, from elementary renewal theory, P M := P fw = Mg = = = + E(M) : (6.) We now proceed to nd the functon E(x). Atypcal evoluton of the wndow sze s depcted n Fgure 2. For smplcty n the gure only losses havng N = are depcted and the tmes to recover from losses are partly cut out of the pcture (denoted by the shaded areas). Suppose for the moment that the ntal loss (at the level x) was such that N = n (n the gure n = ). Let T n (x) be the tme to get back at level x condtonal on N = n and we further wrte E n (x) :=E [T n (x)] := E [T (x)jn = n]. Note that E(x) = X n= q n E n (x): (6.2) If no losses occur durng the tme T n (x) then T n (x) =(,,n )x=,.e., the wndow sze x s reached n a straght lne from the startng pont at,n x (n the gure, x). Each tme a loss occurs at a level y 2 (,n x; x) t takes T (y) tme unts to get back at the level y. Because of the memoryless property of the Posson process, f we take out the shaded areas n Fgure
6 The probablty of maxmum wndow sze 5 x W(t) y 2 y x/ γ y y 2 T( ) T( ) t Fgure 2: TCP wndow 2 and concatenate the non-shaded areas then the cut ponts (where the shaded areas used to be) form a Posson process on the straght lne from,n x to x. Thus f the cut ponts are gven by y ;y 2 ;:::;y m (n the gure m = 2) then E n (x) = (,,n )x + E(y )+E(y 2 )++ E(y m ): Snce the loss process s a Posson process, the mean number of cut ponts s (,,n )x= and the poston of each of the ponts y j s unformly dstrbuted over the nterval (,n x; x), see for nstance [22, Thm..2.5]. Hence, E n (x) = (,,n )x = (,,n )x + (,,n )x + Z x Usng (2.) and (6.2) we now arrve at E(x) = (, Q(, ))x + y=,n x X n= q n Z x Z x y=,n x E(y) (,,n )x dy E(y)dy: (6.3) y=,n x E(y)dy: (6.4) Although n the nte-wndow case (M < ) the above ntegral equaton has only meanng for <x M, tswell dened for all x>. In the followng we solve the ntegral equaton for all x>. Frst we note that t has a unque soluton, see Appendx A.. Dene the LT (Laplace Transform) of E(x): ^e() := Z x= e,x E(x)dx:
6 The probablty of maxmum wndow sze 6 In Appendx A.2 t s shown that ^e() < for > =. Hence, for large enough and usng that the q n and E(x) are non-negatve tonterchange the order of ntegraton and summaton (twce), we can rewrte (6.4) as: ^e() =, Q(, ) + X 2 ^e(), q n^e( n ) n= : Ths gves ^e() =,, Q(, ), X n= q n^e( n ) : (6.5) Substtutng ths equaton repeatedly nto tself, applyng partal fracton expanson at each step and usng that ^e( k ) # ask leads us to the followng canddate soluton: ^e() =, Q(, ) X = e, ; (6.6) where the e are constants to be determned. Ths representaton wll be justed by showng that t leads us to the (unque) solutons to (6.5) and (6.4). Substtutng (6.6) nto (6.5) and equatng the coecents multplyng the terms =(, ) leads to: e e = e =,, @ + X n= X n=,n e,n q n ; =; 2; 3;:::; (6.7) e X,n e j =e q n j+n, j= A, : (6.8) We note that the ratos e =e are non negatve and can be computed recursvely from (6.7). Then the normalzng constant e > can be computed from (6.8). From (6.7) t can be shown (by nducton on ) that e, e ; =; 2;:::; (6.9).e., the e decay exponentally fast n as. Therefore the rght hand sde of (6.6) certanly converges for > = and, from ts constructon, (6.6) s the soluton to (6.5). By partal fracton (6.6) can be rewrtten as: ^e() =, Q(, ) X e,, =, : (6.) = Invertng ths LT gves: E(x) =, Q(, ) X = e e, (=)x, : (6.)
7 Specal case: sngle congeston sgnals and =2 7 Usng ths n (6.) we have = P M = = + E(M) = +, X, Q(, ) Note that because of (6.9) and e, (=)M,, (=)M; ; P M can be computed ecently from (6.2). =, e e, (=)M, : (6.2) Remark 6. In partcular cases we can nd the coecents e explctly. For nstance, when the reducton of the TCP wndow s always by a constant factor,.e., N (hence, q = and Q(z) z). Note that wth =2wehave TCP's most common wndow decrease factor (see Secton 7 for more specc results n that case). From (6.7, 6.8) we get e e = e = + Q, (, ; =; 2;:::; (6.3) j=,j ) X =, e e, : (6.4) In ths case we could have obtaned these coecents also n a drect way, wthout usng (6.6), see Appendx A.3. There t s also shown that n ths case X = e =,, ; and, hence, from (6.2): P M =,,, X = e e, (=)M, : (6.5) 7 Specal case: sngle congeston sgnals and = 2 In ths secton we specfy our results for the mportant partcular case of TCP ow control wth only one dvson of the wndow by a factor 2 at loss events. I.e., we take = 2 and N (q =, and q n = ;n = 2; 3; :::) n the model wth peak rate lmtaton, see [4] for a smlar model. In Secton 9.2 we compare the results from ths partcular case of our model to measurements from the Internet. We worked wth long dstance connectons where congeston sgnals rarely appear n batches. From (4.4) and (4.5) we obtan the expressons for the rst two moments of the wndow sze dstrbuton: E[W ]= 2 (, P M):
8 General congeston lmtaton model 8 E[W 2 ]= 8[2(, P M ), P M M] 3 2 ; where P M s gven by (6.5) wth = 2. The throughput of TCP can be obtaned from Equaton (4.). The dstrbuton functon tself or the complementary dstrbuton functon F (x) s computed successvely on the ntervals [M=2 k ;M=2 k, ];k =; 2; ::: usng (5.5) wth = 2. Recurson (5.6) reduces to c (k) = c(k,) + ; =; :::; k, ;, 2 and c (k) s gven by (5.7). When M =, the dstrbuton functon s gven by (5.2) wth c =, 2 c,; =; 2;:::; and c s gven by (5.). 8 General congeston lmtaton model In the case of congeston lmtaton t seems unrealstc to assume that the number of congeston sgnals n the batch that s generated when reachng the maxmum transmsson rate has the same dstrbuton as the sze of the batches generated by the Posson process. Let us therefore assume that the number of congeston sgnals that result from reachng the maxmum transmsson rate s dstrbuted as the non negatve dscrete random varable N (M ) havng p.g.f. Q (M ) (z) = X n= z n q (M ) n jzj : Instead of (2.5) we then have for <x<m: where, d dx F cl (x) = F cl (x), H(x) := P b (M ) :=, N (M ) X n= ln(m), ln(x) ln() d dy F cl (y) : y=m, q n F cl (mn( n x; M)) ; + b (M ) H(x); (8.) Note that H(x) s a non negatve, non decreasng step functon of the varable x, constant on the ntervals,k M <x,k+ M, k =; 2;:::, wth H(M) =: n o H(x) =h k := P N (M ) k ;,k M x<,k+ M:
8 General congeston lmtaton model 9 8. The moments Smlar to Secton 4 we nd the followng recurson on the moments after takng Laplace Transforms n (8.): k ke E W cl = h, W cl k,, b (M ) M k,, Q (M ) (,k ) (, Q(,k )) Note that f Q (M ) (z) =Q(z) ths recurson ndeed reduces to (2.8). 8.2 The dstrbuton functon Denng, for k =; 2;:::, F cl k (x) :=F cl (x); M k x< M k, ; we nd by the same technques as n Secton 5: F cl (x) = b (M ) e (M,x), F cl k (x) =,b (M ) h k + Ths leads to: F cl k (x) =b (M ) The coecents d (k) k, d (k) = h k + d (k) = and for k>:, e, x Z M= k, ; F cl k,(m= k, )+b (M ) h k e u=x X k,,d (k) + = are gven by: X n=,, d (k) =,d (k,) X k, + n= X e u k, n= d (k) e,, x q n d (k,n) ; k>;, X n= k,2 + q n F cl k,n( n u)du: q n d (k,n),n ; k>>; X = q n e M=k,n, d (k,) e M=k, + h k e M=k, : X k,n, d (k,n), = d (k,n) : M k,,x,,+n e, M=k,,2n Note that the d (k) are all non negatve, but that the sgns of the d (k) for > alternate. :
9 Model valdaton 2 8.3 The constant b (M ) Note that, smlar to (5.8) we nd from F cl () = : = lm b (M ) k,d(k) X k, + = d (k) e, M=k, : (8.2) However, for computatonal purposes, we agan prefer to translate the model nto a peak rate lmtaton type of model. Therefore, consder a peak rate lmtaton model n whch congeston sgnal batches arrve accordng to a Posson process wth rate but, derent from the model n Secton 2., wth the dstrbuton of the batch sze dependng on whether the transmsson rate s below or at the maxmum level M. The batch sze has p.g.f. Q(z) f the rate s below M, otherwse t has p.g.f. Q (M ) (z). Smlar to (2.9) we have: b (M ) = P (M ) M, P (M ; ) M where P (M ) M s the probablty of beng at the maxmum transmsson rate M. For <x<m (as we shall see t s convenent not to nclude x = M), let the functons E(x) and E n (x), n = ; 2;:::, be dened as n Secton 6. Note that as long as the process s below the maxmum level M, t behaves exactly as the ordnary peak rate lmtaton model of Secton 2.. Therefore, for < x < M the functons E(x) and E n (x) are exactly as we found n Secton 6, see (6.) and (6.3). Ths s not true for x = M and to avod confuson we wrte E (M ) (M) nstead of E(M) for the return tme to level M n the present model. Of course, P (M ) M = +E (M ) (M) : Smlar to (6.2) we have: E (M ) (M) = X n= q n (M ) E n (M): And usng (6.3) and (6.) we nd: E (M ) (M) =,, Q (M ) " (, ) M, X,, Q(, ) +, Q(, ) 9 Model valdaton X = e X n= = e # q n (M ) e, (=)M, e,,n (=)M : In ths secton we compare measurements from long dstance and long lfe TCP connectons wth the results of Secton 7 (N ; = 2, peak rate lmtaton). Comparson of real measurements wth the model wth clustered (batch) arrvals of congeston sgnals s a topc of current research, see also Secton.
9 Model valdaton 2 Due to the large number of hops and the multplexng of exogenous trac n network routers, the Posson loss process assumpton s expected to hold on long dstance connectons [4]. Our TCP recevers mplement the delay ACK mechansms and our TCP senders ncrease ther wndow n the congeston avodance mode by approxmately one packet every wndow's worth of ACKs. Thus, we take equal to =(2RT T ) [8]. Frst, we show theoretcally how the wndow sze s dstrbuted n the statonary regme. Second, we compare our results to measurements from the Internet. 9. Numercal results Consder the case of a long TCP connecton wth packets of sze 46 bytes and a constant RTT of one second. Usng the results of Secton 7, we plot the cumulatve dstrbuton functon F (x) of the wndow sze and ts probablty densty functon f(x) for a range of values for the loss ntensty (or rather, for the mean nter-loss tme s = =). Two values of M are consdered. Frst, we let the congeston wndow be lmted by a recever wndow of 32 kbytes. Then we consder the case where the wndow s not lmted and therefore contnues to grow lnearly untl a loss occurs. The numercal results are presented n Fgures 3, 4, 5 and 6. For the case M = 32kbytes, we computed the dstrbuton functon successvely for the ntervals [M=2;M]; [M=4;M=2] and so on. When computng P M we truncate the nnte seres n (6.5). In the case of an unlmted wndow, we also truncate the nnte seres n (5.2). As dscussed prevously, these nnte seres converge fast. We choose the number of terms of these seres large enough to get a neglgble error. When M = 32 kbytes, the dscontnutes of F (x) and f(x) at x = M and x = M=2, respectvely, are clearly llustrated n Fgures 3 and 4. The dscontnutes are most notceable for large nter-loss tmes. The dscontnuty off (x) s also depcted n Fgure 4 by plottng a pulse for f(x) at x = M such that ts area s equal to P M. When M =, the densty functon exhbts nether pulses nor dscontnutes (Fgure 6). 9.2 Expermental results Our expermental testbed conssts of a long lfe and long dstance TCP connecton between INRIA Sopha Antpols (France) and Mchgan State Unversty (US). The TCP connecton s fed at INRIA by an nnte amount of data. The New Reno verson of TCP [7] s used for data transfer. We change the socket buer at the recever n order to account for derent values of M. We consdered three values of M: 32, 48 and 64 kbytes. For every value of M, we ran the TCP connecton for approxmately one hour and we regstered the trace of the connecton usng the tcpdump tool of LBNL [2]. We developed a tool that analyzes the trace of the connecton and that detects the tmes at whch the wndow s reduced. Ths tool gves also the average RTT of the connecton and the statstcs of the wndow per RTT. We plotted for the three values of M, the dstrbuton of the wndow sze from measurements and that gven by our model. The results are plotted n Fgures 7, 8 and 9. When M s small, we observe a good match between the measured dstrbuton and the one resultng from our model. For larger values of M, the derence between the two ncreases. In partcular, as M ncreases, the measured probablty densty concentrates around the average wndow sze. Ths devaton can be explaned from the measured nter-loss tme dstrbuton.
9 Model valdaton 22 Cumulatve Dstrbuton Functon.8.6.4.2 Average nter-loss tme (s) 5 5 2 25 3 35 4 45 5 5 5 35 3 25 2 Wndow sze (kbytes) Fgure 3: Lmted recever wndow: M = 32 kbytes Probablty Densty Functon.5.45.4.35.3.25.2.5..5 Average nter-loss tme (s) 5 45 4 35 3 25 2 5 5 5 5 2 25 3 35 Wndow sze (kbytes) Fgure 4: Lmted recever wndow: M = 32 kbytes
9 Model valdaton 23 Cumulatve Dstrbuton Functon.8.6.4.2 5 5 225 Average nter-loss tme (s) 3 354 45 5 2 4 6 2 8 TCP transmsson rate (kbps) Fgure 5: Unlmted recever wndow: M = Probablty Densty Functon.35.3.25.2.5..5 5 5 225 Average nter-loss tme (s) 3 354 45 5 2 4 8 6 TCP transmsson rate (kbps) Fgure 6: Unlmted recever wndow: M =
9 Model valdaton 24.2 Cumulatve Dstrbuton Functon Model Expermentaton.8.6.4.2 5 5 2 25 3 TCP wndow (bytes).2 Fgure 7: Recever wndow M = 32 kbytes Cumulatve Dstrbuton Functon Model Expermentaton.8.6.4.2 5 5 2 25 3 35 4 45 TCP wndow (bytes).2 Fgure 8: Recever wndow M = 48 kbytes Cumulatve Dstrbuton Functon Model Expermentaton.8.6.4.2 2 3 4 5 6 TCP wndow (bytes) Fgure 9: Recever wndow M = 64 kbytes
Conclusons and future research 25 In Fgure, we plot ths dstrbuton for M = 32. Ths dstrbuton s n agreement wth an exponental law, resultng n a good match between the model and the measurements. Fgures and 2 show the measured dstrbutons for the other two values of M. We observe that the loss process s no longer Posson, but closer to a determnstc process. Small nterloss tmes are less frequent as M ncreases and the tal of the dstrbuton also becomes less mportant (although t stll looks lke the tal of an exponental dstrbuton). Ths results n a degradaton of the correspondence between our model and the measurements. One explanaton of the devaton of the loss process from a Posson process for larger values of M s the followng. A true Posson loss process mples that the tme untl the next loss event s ndependent of the past. Ths s the case when the congeston of the network s domnated by the exogenous trac and not dependent on the measured connecton. I.e., t s the case when the measured connecton's share of the avalable bandwdth on the path s small compared to that of the exogenous trac. A small M lmts the bandwdth share of our connecton and lmts ts mpact on the network, resultng n a loss process close to Posson. However for large M, the measured connecton acheves a larger share and thus contrbutes more to the congeston of network routers. When t reduces ts wndow, the state of the network changes and becomes under-loaded. For a certan amount of tme the occurrence of a new congeston s less lkely. When the network s agan more heavly loaded the next congeston sgnal s lkely to appear soon. Ths explans why we observe a low densty for small nter-loss tmes (Fgures 2), then a peak n the mddle followed by an exponental decay. In summary, our model leads to accurate results when the tmes between losses are exponentally dstrbuted. However, n stuatons where the congeston n the network s largely due to the TCP connecton under consderaton, the loss process s close to a determnstc process and a smple heurstc as that proposed n [3, 8] can be used to approxmate the acheved throughput. Conclusons and future research We studed addtve-ncrease multplcatve-decrease ow control mechansms under the assumpton that congeston sgnals arrve n batches accordng to a Posson process. As hghlghted n [4], the model can be reformulated as an M/G/ queung problem wth servce tme dependent on system workload. We tred to keep the model as general as possble n order to account for a wde range of congeston control strateges. We derved closed form expressons for the moments as well as the dstrbuton of the transmsson rate. For the case of sngle congeston sgnals, we compared our results to measurements from TCP connectons over the Internet. From our experments, we concluded that our model wth sngle congeston sgnals leads to accurate results when the tmes between losses are close to beng exponentally dstrbuted. Currently, we are workng on the valdaton of our model wth clustered congeston sgnals. Internet measurements have shown that on some paths (especally short dstance ones) the loss process exhbts a hgh degree of burstness. We also study the extenson of the analyss
Conclusons and future research 26 3 Hstogram of nter loss tme (rwnd = 32 kbytes) 25 2 5 5 2 3 4 5 6 9 Fgure : Case of M =32kbytes Hstogram of nter loss tmes (rwnd = 48 kbytes) 8 7 6 5 4 3 2 2 3 4 5 6 7 2 Fgure : Case of M =48kbytes Hstogram of nter loss tmes (rwnd = 64 kbytes) 8 6 4 2 5 5 5 2 25 Fgure 2: Case of M =64kbytes
Conclusons and future research 27 to more general nter-loss processes, n partcular to Markov Modulated Posson Processes. Acknowledgement The authors would lke to thank V.M. Abramov for comments that mproved the presentaton of the paper, as well as the department of Mathematcs of Mchgan State Unversty and n partcular L.B. Fredovch for provdng us wth a computer account for our Internet experments. Appendx A. Unqueness of E(x) If E(x) e s a second soluton to (6.4) then D(x) :=E(x), E(x) e satses: X hence, D(x) = jd(x)j n= X n= Z x q n Z x y= y=,n x q n Z x y=,n x jd(y)jdy: Dene the functon h(x), x, by Z x y= jd(y)jdy =: e x h(x); D(y)dy; x ; jd(y)jdy jd(x)j = e x h(x)+e x d dx h(x): Substtuton nto (A.) gves (A.) d h(x) ; dx Obvously, from ts denton above, h(x) and h() =, hence, h(x) =for all x. Ths proves that D(x). A.2 Exstence of ^e() Usng that E(y) s non negatve for y t follows from (6.4) that Wrtng Z x E(x) (, Q(, ))x y= + E(y)dy =: e x h(x); Z x y= E(y)dy:
Conclusons and future research 28 gves hence, d dx h(x) (, Q(, ))x e, x ; h(x), Q(, ) Therefore ^e() < for > =. 2, e, x, xe, x : A.3 Drect dervaton of P M for sngle congeston sgnals Consder the case where N. We show a drect approach to nd the coecents e wthout usng the `canddate' (6.6). We shall need the followng denttes (whch hold for >): kx = + X =,2 Q j= (,,j ), = Q k, m= (, m ) Q n= (,,n ) Equaton (6.5) becomes:,, ^e() =,, ^e() ; X k=,k Q k l= (, l ) ; and substtutng ths equaton repeatedly nto tself we nd: ^e() =,, X k= (,, ) k Q k j= (j, ) : (A.2) = : (A.3) (A.4) Note that the nnte sum converges absolutely for all 6= and 6=,j =, for j = ; ; 2;:::, because the k-th term s of the order of,k2 =2 when k. By partal fracton expanson we have: ^e() = (,, ) X k= =, +(,, ),k 2 6 4, + X k=,k Qk kx j=;j6= (,, 3 ) j, 7 5 =,, Qk kx j=;j6= (,, ) j, =,, : (A.5) (By conventon the empty product equals.) Usng the absolute convergence of the nnte seres we have: E(x) =, +,, X k=,k kx @ = k Y j=;j6= (, j, ) A, x exp : (A.6)
Conclusons and future research 29 Usng (A.2) we obtan the coecents e n (6.3) and (6.4). If we use (A.3) we can also show that n ths case X = e =,, ; and, hence, from (6.2) we obtan (6.5).
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