Temporal Correlations of Local Network Losses

T emp oral Correlations of Lo cal Netw ork Loss es A. S . Stepanenko, 1 C. C. Constantinou, 1 I. V. Y urk evich, 2 and I. V. Lerner 2 1 Scho ol o f Engine ering, Univer sity of Birmingham, Edgb aston, Bi rmingham, B15 2TT, UK 2 Scho ol of Physics and Astr onomy, University of Birmingham, Edgb aston, Bi rmingham, B15 2TT, UK (Dated: N o vem b er 10, 2018) W e introduce a contin uu m mod el d escribing data losses in a single no de of a pack et- switched netw ork (like the Internet) whic h preserves the d iscrete nature of th e data loss pro cess. By c on- struction , the mo d el has critical b ehavior w ith a sharp transition from exp onentially sma ll to fi nite losses with increasing data arriv al rate. W e show that suc h a mod el ex hibits strong fluctu ations in the loss rate at th e critical p oint and non-Marko vian p ow er-law correlations in time, in spite of the Mark o v ian character of the data arriv al pro cess. The con t inuum mod el allow s for rather general incoming data p ac ket distributions and can b e naturally generalized to consider th e buffer server idleness statistics. P A CS n umbers: 64.60.Ht, 05.70.Jk, 89.20.Hh, 89 . 75.Hc I. INTRO D UCTION Complex netw orks underpin many diverse areas of sci- ence. They manifest themselves in relatio nships b etw een net work top ology and functional or ganization of complex neuron structures [1, 2], interacting organic molecules de- scribing metabolic activity in living cells [3], multi-species fo o d webs [4, 5], n umerous asp ects of so cial netw or ks [6, 7, 8, 9], and the co nnectivity and o per ation of the In- ternet [10, 11, 1 2]. New mo dels o f netw ork top ology s uch as scale- free [13] or s mall-world [14] have b een found to be surpr isingly go o d at des cribing r eal-world struc tur es. A conseq uenc e of the r ealisation that complex netw o rks describ e universal prop erties of many such pro blems ha s resulted in extensive resea rch activity by the physics com- m unit y in the pas t decade (see Refs. [15, 16] for rev iews). A problem o f particula r significance in many applica- tion domains is the resilie ncy of complex ne tw orks to the random or sele ctive remov al o f no des o r links. F o r example, the loss of connectivity in sca le-free netw orks [10, 17, 18, 1 9, 20] has implicatio ns on the tolera nce of the Internet to proto col or equipment failures. Typically , the site or bond disorder a cts as an input which makes them very general and applicable to a wide v a r iety of net works. More recently there has b een an increasing realization that netw or k breakdowns can not only res ult from the ph ysical loss of connectivity , but can arise due to the lo ss of data tr a ffic in the netw o rk (i.e. c ongestion) [21, 22]. How ever, only a few dynamical mo dels of tr affic in net- works hav e b een considered to date [11, 2 4, 25, 26]. In the case of comm unica tion net works the excessive loading of even a single no de ca n give r ise to c ascades of fa ilures arising from tra ffic congestion and c o nsequently isolate large parts of the netw ork [27]. T o desc r ib e the oper - ational failur e ar ising due to cong estion at a par ticular net work no de, one needs to acco un t for distinct featur e s of the dynamica lly ‘r andom’ data traffic which is the rea- son for such a breakdown. In this pap er we mo del da ta lo sses in a single no de of a pack et-switched netw o rk like the Internet. There ar e t wo distinct featur es which m ust b e preserved in this case: the discrete character of data pr o pagation and the p o s sibility of data ov erflow in a sing le no de. In the pack et-switched net work data is divided into pa ck ets which are routed from source to destination via a set of interconnected no des (router s ). A t each no de pa ck ets ar e queued in a memory buffer befor e being servic e d , i.e. forwarded to the next no de (there are s eparate buffers for inco ming a nd outgoing pa ck ets but we neglect this for the s ake of s im- plicit y). Due to the finite capacity of memory buffers a nd the sto chastic nature o f da ta traffic, any buffer can be- come overflo wn which r esults in pack ets b eing disc ar de d . W e fo c us on a contin uum description of the discrete pro cess o f da ta packet loss . Such a co nt in uum mo del represents a simplification tha t preserves the salient fea - tures of the data lo ss mechanism, while at the same time it can b e more ea sily embedded in a la rger mo del descr ib- ing data pack et lo sses in a lar ge netw o rk. The co ntin uum description allows us to o vercome inevitable difficulties in incorp or ating realistic distributions of incoming traffic int o a dis crete-time class of mo de ls , lik e one we intro- duced ea rlier [23]. On the co n trary , the contin uum mo del can ea sily incorp or ate a completely ge neral distribution of pack et lengths and inter-arr iv al times, b oth essential in mo deling data los s in finite-sized buffers. W e introduce a mo del where noticeable da ta losse s in a single memor y buffer sta rt when the av e r age r ate of r an- dom packet a rriv als a pproaches the serv ic e rate. Under this condition the mo del has a built-in sharp tra nsition from free flow to lossy b ehavior with a sizeable fraction of a r riving pa ck ets b eing dropp ed. A sha r p o nset of net- work cong e stion is familiar to everyone using the Internet and w as num erically confirmed in different mo dels [2 8]. Here we s tress that such a conges tio n or iginating from a single no de is characterized b y s tr ong critical fluctuations of the data loss in the vicinity of the built-in transitio n. In particular, we will s how that a Marko vian input pro- cess can g ive rise to long-ra nge temp ora l correla tions of data losses that are strong ly non-Markovian in the crit- ical regime. In the co n text of the Internet, this mea ns that when excessive da ta losses s tart it is mor e pr obable 2 that they p ers ist for a while, thus impa c ting on netw ork op eration. As we will discuss later in this pa p e r, this non- Marko vian b ehavior has a pro found e ffect o n the o per a- tion of current Internet proto c o ls, s uch as the T ranspor t Control P roto col (TCP), that dicta te how users e x pe r i- ence the netw or k o p eration. While da ta los s is natural and inevitable due to data ov erflow, we show that loss rate statistics turn out to b e highly nontrivial in the rea listic cas e of a finite buffer, where at the critical p oint the magnitude of fluctuations can exceed the av era g e v alue. The fluctuatio ns still obey the central limit theorem but only in the unrealistically long time limit. The impor tance of fluctuations in s ome int ermediate r e gime is a definitive featur e of mesosc opic ph ysics, a lb eit the reasons for this are absolutely different (note that even in the case of electr ons, the o r igin of the mesoscopic phenomena can b e either quantum o r purely classical, see, e .g ., [29]). The aver age los s ra te and/o r transp or t delays w ere previously studied, e.g., in the theories of bulk queues [30, 3 1] or a jamming tr ansition in traffic flow [32]. What mak es pres ent co nsiderations in trinsica lly differ- ent from these theor ies is the very nature of the quan- tit y we consider: the lo sses (not existing in flow mo dels) make the descr iption of the traffic pro ce s s essentially non- Hermitian. Although fluctuations in netw or k dynamics were prev iously studied (see , e.g. [11, 33]), this was done through measurements or numerical simulations of da ta traffic. Due to the symmetry of the contin uum description o f a buffer with resp ect to its full (lossy) and empty (idle) states, w e also derive corr esp onding expressions for the statistics of idleness of the buffer ser ver (i.e. output link s from r outers). This quantit y is esse ntial in determining the way the statistics o f data traffic going in to a subse- quent buffer a long a data path are shap ed. This is self- evidently imp or tant when we ar e attempting to describ e the op eratio n of an en tire netw or k. II. THE MODEL W e conside r a single finite-size memory buffer fed with a random data -pack et stream. It stores the pack ets and then is servic e d by the data-link that sends this pack- ets further along the netw or k o n a firs t-in-first-out basis. This adequa tely mo de ls the output buffer a ttached to the switching device in the router. The s p eed of the input line of the buffer is muc h bigger than the sp eed o f the output line. The rea s on is that the input comes from the switching fabric o f a router which is des igned to op erate very fast indeed in order to feed a lar g e num b er of such buffers, but s equentially . The capa city of the o utput line is normally smalle r . Hence, we can mo del the packet arr iv al a s an ins tanta- neous re ne wal pro cess. The sto rage capa city of the buffer is L , measur ed in bits. The lengths of a rriving packets a re treated as rando m, all being m uch smaller than L . The service r ate (i.e. the rate at which pa ck ets depar t fro m the buffer) is considered to b e deterministic, as ra ndo m- ness in it is negligible as compared to that of the input traffic. W e nor malize the leng ths of pack ets p , the spe e d of the output link r out and the queue length ℓ by the size of the buffer L (which is hencefor th set to 1). The proc edure for the renewal cycle is describ ed as follows: at the moment of arriv al of a pack et o f size p , the state of the queue is ℓ , this is followed by the gap η (random inter-arriv al time) until the next arr iv al. W e int ro duce the time scale required to empt y a full buffer provided there are no new arriv als , η 0 ≡ 1 /r out . If ℓ + p ≤ 1 then the pa ck et joins the queue and the queue leng th prior the nex t a rriv al is ℓ ′ = ℓ + p − η /η 0 if ℓ ′ > 0 and ℓ ′ = 0 otherwis e. If ℓ + p > 1 then the pack et is disca r ded and the queue length prior the next arriv al is ℓ ′ = ℓ − η / η 0 if ℓ ′ > 0 and ℓ ′ = 0 other wise. Since the maxim um pack et size is mu ch less than 1 (the buffer size) and assuming that the av e r age incoming traffic ra te r in (also normalize d to the buffer size) is close to the service r ate: | r in η 0 − 1 | ≪ 1 (1) we can treat p , η and ℓ as co nt in uous v ariables. Our aim is to c a lculate the statistics o f the amount of the dropp ed traffic and the service lost due to idleness of the output link dur ing time t ≫ ¯ η ( ¯ η is the average int er-arr iv al time) in the regime (1). In this reg ime and for observ ation times t ≫ ¯ η , the system ca n b e describ ed by the F o kker-Planck eq uation as follows (in terms o f the transitional probability densit y function w ( ℓ ′ , t ; ℓ )) ∂ t w ( ℓ ′ , t ; ℓ ) = − a∂ ℓ ′ w ( ℓ ′ , t ; ℓ ) + 1 2 σ 2 ∂ 2 ℓ ′ w ( ℓ ′ , t ; ℓ ) , (2) where a and σ 2 are av er age moments of the change of the queue size p er unit time a ≡ 1 ∆ t h ∆ ℓ i , σ 2 ≡ 1 ∆ t h ∆ ℓ 2 i , ∆ t → 0 (3) and the following b oundar y a nd initial conditio ns a re im- po sed J ( ℓ ′ , t ; ℓ ) | ℓ ′ =0 , 1 = 0 , (4) w ( ℓ ′ , t ; ℓ ) | t =0 = δ ( ℓ ′ − ℓ ) , (5) where J ( ℓ ′ , t ; ℓ ) ≡ aw ( ℓ ′ , t ; ℓ ) − 1 2 σ 2 ∂ ℓ ′ w ( ℓ ′ , t ; ℓ ) (6) is the pro ba bilit y cur rent. By ∆ t → 0 in eq. (3) we mean that ∆ t is muc h smaller than the observ ation time, but large enough so that the underlying sto chastic pro cesses can b e consider ed as contin uous: ¯ η ≪ ∆ t ≪ t (7) 3 The solution of (2,4 ,5) can b e expres sed as follows w ( ℓ ′ , t ; ℓ ) =2e v ( ℓ ′ − ℓ ) ∞ X k =1 exp  − (4 π 2 k 2 + v 2 ) τ  4 π 2 k 2 + v 2 × [2 π k cos(2 π k ℓ ′ ) + v s in(2 π k ℓ ′ )] × [2 π k cos(2 π k ℓ ) + v sin(2 π k ℓ )] (8) where v ≡ a σ 2 , τ ≡ σ 2 t 2 (9) Note that the solution (8) can b e expr essed in terms of θ - functions. F o r the Laplace tr ansform of w ( ℓ ′ , t ; ℓ ) we have W ( ℓ ′ , ǫ ; ℓ ) ≡ L τ w ( ℓ ′ , t ; ℓ ) = 1 2 e v ( ℓ ′ − ℓ ) κ sinh( κ ) × ( 2 v 2 ǫ cosh[ κ ( ℓ ′ + ℓ − 1 )] + 2 κv ǫ sinh[ κ ( ℓ ′ + ℓ − 1)] + cos h[ κ ( | ℓ ′ − ℓ | − 1)] + cosh[ κ ( ℓ ′ + ℓ − 1)] ) (10) where κ ≡ p ǫ + v 2 (11) F r om (10) we have for the probabilities of returning to the b oundaries W (0 , ǫ ; 0) = 1 ǫ [ κ cotanh( κ ) − v ] W (1 , ǫ ; 1) = 1 ǫ [ κ cotanh( κ ) + v ] (12) These will b e used in the next sectio n. II I. ST A TISTICS OF LOSSES In this section w e concentrate on the sta tistics of the losses due to the buffer ov er flowing. The c o rresp onding formulae for the sta tistics of the se rver idleness c a n b e obtained using tr ansformation ℓ → 1 − ℓ, v → − v . First, we estimate the size of fluctuations of the loss e s on a time s cale t ≪ 2 /σ 2 . In or der to do that w e con- sider the dynamics of the system near the bo undary ℓ = 1 which is gov erned by the following trans itio nal pr obabil- it y: w 0 ( ℓ ′ , t ; ℓ ) = 1 √ 2 π σ 2 t exp  − a ( ℓ ′ − ℓ ) σ 2 − a 2 t 2 σ 2  ×  exp  − ( ℓ ′ − ℓ ) 2 2 σ 2 t  + exp  − (2 − ℓ ′ − ℓ ) 2 2 σ 2 t  − a σ 2 exp  2 a (1 − ℓ ′ ) σ 2  erfc  2 − ℓ ′ − ℓ + at √ 2 σ 2 t  (13) which is the so lution of (2) when the b ounda ry ℓ = 0 is sent to −∞ . The change in the state of the s ystem during time t ca n then b e repr esented a s follows: ∆ ℓ ( t ) ≡ ℓ ′ − ℓ = ∆ ℓ 0 ( t ) + ∆ ℓ loss ( ℓ ′ , t ; ℓ ) (14) where ∆ ℓ 0 ( t ) is the change in the state of the system if there was no b ounda r y , its statistics is determined by h ∆ ℓ 0 ( t ) i = at , h [∆ ℓ 0 ( t )] 2 i = σ 2 t + o ( t ) , (15) and ∆ ℓ loss ( ℓ ′ , t ; ℓ ) is the amount of traffic lost due to buffer ov er flowing. The moments of (15) can b e defined as follows h [∆ ℓ ( t )] n i = Z d ℓ ′ d ℓ ( ℓ ′ − ℓ ) n w 0 ( ℓ ′ , t ; ℓ ) p ( ℓ ) (16) where p ( ℓ ) is the stationary dis tr ibution of buffer o ccu- pancy . F o r the first tw o moments (16) in the limit t → 0 w e hav e h ∆ ℓ ( t ) i = at + σ 2 t 2 p (1) , h [∆ ℓ ( t )] 2 i = σ 2 t (17) F r om (14,15,17) we c an conclude that h ∆ ℓ loss ( t ) i = σ 2 t 2 p (1) h [∆ ℓ loss ( t )] 2 i + 2 h ∆ ℓ 0 ( t )∆ ℓ loss ( t ) i = o ( t ) (18) The first o f the relations (1 8) means tha t ∆ ℓ loss ( ℓ ′ , t ; ℓ ) is non-zero o nly if ℓ ′ , ℓ ∼ 1 in the limit t → 0. The second relation means either h [∆ ℓ loss ( t )] 2 i , h ∆ ℓ 0 ( t )∆ ℓ loss ( t ) i = o ( t ) (19) or ∆ ℓ loss ( t ) = − 2∆ ℓ 0 ( t ) + o( √ t ) (20) The relation (20) do es no t make sense physically , so in what follows we accept option (19) and show that it is consistent with the later calculations . Next w e lift the res triction t ≪ 2 /σ 2 . It can b e shown that the co nditional moments (with the condition tha t the system w as in the state ℓ a t the beginning of the observ ation interv al) can b e expressed as follows: m ( k ) loss ( t ; ℓ ) = k ! r k loss k Y i =1 t i +1 Z 0 d t i k − 1 Y j =1 w (1 , t j +1 − t j ; 1) × w (1 , t 1 ; ℓ ) , t k +1 ≡ t (21) where w ( ℓ ′ , t ; ℓ ) is determined by (8) and r loss ≡ lim t → 0 1 t Z d ℓ ′ Z d ℓ ∆ ℓ loss ( ℓ ′ , t ; ℓ ) = lim t → 0 1 t 1 Z −∞ d ℓ ′ d ℓ ( ℓ ′ − ℓ − at ) w 0 ( ℓ ′ , t ; ℓ ) = σ 2 2 (22) 4 F o r unconditional mo ments in the stationar y regime we hav e m ( k ) loss ( t ) ≡ 1 Z 0 d ℓ m ( k ) loss ( t ; ℓ ) p ( ℓ ) = k ! k Y i =1 τ i +1 Z 0 d τ i k − 1 Y j =1 w (1 , t j +1 − t j ; 1) · p (1) (23) where τ is defined in (9) and p ( ℓ ) is the stationar y so lu- tion of (2): p ( ℓ ) = 2 v e 2 vℓ e 2 v − 1 (24) T o calcula te m ( k ) loss ( t ) w e consider its Laplace transform: M ( k ) loss ( ǫ ) ≡ L τ m ( k ) loss ( t ) = ∞ Z 0 d τ e − ǫτ m ( k ) loss ( t ) = k ! p (1) [ L τ w (1 , t ; 1)] k − 1 L τ τ = k ! p (1) [ W (1 , ǫ ; 1)] k − 1 1 ǫ 2 (25) where W (1 , ǫ ; 1) is is defined by (10). T a king now the in verse Laplace transform we have m ( k ) loss ( t ) ≡ L − 1 ǫ M ( k ) loss ( ǫ ) = 1 2 π i γ +i ∞ Z γ − i ∞ d ǫ e ǫτ M ( k ) loss ( ǫ ) (26) F r om (25) we obtain m (1) loss ( t ) = p (1) τ = p (1) σ 2 t 2 (27) F o r the moments (25) with k > 1 we can iden tify the following r e gimes: M ( k ) loss ( ǫ ) = ( k ! p (1) ǫ − ( k +3) / 2 ǫ ≫ 1 k ! p k (1) ǫ − ( k +1) ǫ ≪ 1 (28) Corresp o ndingly , for the moments in t -repre s ent ation we hav e m ( k ) loss ( t ) =    k ! p (1) τ ( k +1) / 2 Γ[( k + 3) / 2] τ ≪ 1 p k (1) τ k τ ≫ 1 (29) Now we calculate the P DF p loss ( x ; t ) of the amount of the lo s t traffic, x , dur ing time t . T o calculate it we con- sider its characteristic function in the ǫ - representation: ˜ P loss ( s ; ǫ ) ≡ L x P loss ( x ; ǫ ) , P loss ( x ; ǫ ) ≡ L τ p loss ( x ; t ) (30) F r om (30) we obtain ˜ P loss ( s ; ǫ ) = ∞ X k =0 ( − s ) k k ! ∞ Z 0 d x x k L τ p loss ( x ; t ) = P loss ( ǫ ) + ∞ X k =1 ( − s ) k k ! M ( k ) loss ( ǫ ) (31) where P loss ( ǫ ) = L τ p loss ( t ) , p loss ( t ) = ∞ Z 0 d x p loss ( x, t ) (32) with 1 − p loss ( t ) b eing the probability for the system not to drop a single packet ov e r the p erio d of time t . Substi- tuting (25) into (31 ) we have ˜ P loss ( s ; ǫ ) = P loss ( ǫ ) + p (1) ǫ 2 ∞ X k =1 ( − s ) k [ W (1 , ǫ ; 1)] k − 1 = P loss ( ǫ ) + p (1) ǫ 2 W (1 , ǫ ; 1)  − 1 + 1 1 + sW (1 , ǫ ; 1)  In order tha t P loss ( s ; ǫ ) did not have a n abnormal b e- haviour (in particular , it did no t contain terms like δ ( x )), we must a ssume that P loss ( ǫ ) = p (1) ǫ 2 W (1 , ǫ ; 1) (33) Hence, P loss ( x ; ǫ ) = p (1) ǫ 2 W 2 (1 , ǫ ; 1) exp  x W (1 , ǫ ; 1)  (34) Int egrating this rela tion ov er x , we recover (33), which shows that our a ssumption is indeed cor rect. In the reg imes o f short and long times we hav e p loss ( x ; t ) =        p (1)erfc  x √ 4 τ  τ ≪ 1 δ h x − τ p (1) i τ ≫ 1 (35) and p loss ( t ) =      p (1) r 4 τ π τ ≪ 1 1 τ ≫ 1 (36) The conditional PDF (with the condition that the system dropp ed a t least o ne pack et during the time t ) c a n b e defined as follows w loss ( x ; t ) ≡ p loss ( x ; t ) p loss ( t ) =        r π 4 τ erfc  x √ 4 τ  τ ≪ 1 δ h x − τ p (1) i τ ≫ 1 5 The central moments can be calculated in the same wa y as (23). Here we will co ns ider o nly the v ariance of the losses σ 2 loss ( t ) in the limit τ ≫ 1: σ 2 loss ( t ) = m (1) loss ( t )  1 | v | cotanh | v | − sinh − 2 | v |  =        2 3 m (1) loss ( t ) | v | ≪ 1 1 | v | m (1) loss ( t ) | v | ≫ 1 (37) This is essentially in agr eement with the result of cons id- erations in Ref. 23 where a simple discrete-time mo del fo r studying loss e s in a sing le buffer w a s int ro duced. In tha t mo del packets o f fixed size ar rive with proba bilit y p at the equidistant time ep o chs. The service was deter minis- tic, and half o f packet was s erved b etw een the s uccessive time ep o chs. In s pite of s uch oversimplification, the dis- crete mo del has delivered quantitativ ely the s ame results which indicates the universalit y of the approach. Finally , we calculate the correla tor of the fluctuations of losses mea sured during tw o time interv als of length t 1 and t 2 corres p o ndingly a nd separated by the time T : corr( t 1 , t 2 , T ) = 1 Z 0 d ℓ ρ ( t 1 , t 2 , T ) − m (1) loss ( t 1 ) m (1) loss ( t 2 ) where ρ ( t 1 , t 2 , T ) = r 2 loss t 1 Z 0 d t ′ 1 t 2 Z 0 d t ′ 2 w (1 , t ′ 1 + t 2 − t ′ 2 + T ; 1 ) p (1 ) with r loss defined in (22). In the regime T ≫ t 1 , t 2 and T ≫ 2 /σ 2 it can be shown that corr( t 1 , t 2 , T ) → T →∞ 0 , (38) as we would exp ect. In fact, the c orrela to r go es to zero exp onentially if v 6 = 0. In the opp osite reg ime 2 /σ 2 ≫ T ≫ t 1 , t 2 we hav e corr( t 1 , t 2 , T ) = m (1) loss ( t 1 ) m (1) loss ( t 2 ) 1 p (1) r 2 π σ 2 T , (39) which is a gain in agr eement with the res ults o f the discrete-time consider ations [23]. IV. DISCUSSION AND CONCLUSION As we would exp ect intuitiv ely , los s events sepa rated widely in time ar e uncor related a s shown by equa- tion (38). By widely separated in time, we mean that the time separ ation of the tw o observ ation interv a ls in which loss es occ ur is muc h longer than the time ov er which fluctuations of queue length b ecome compara ble or m uch bigger than the buffer size itself, i.e . 2 / σ 2 . How ever, in the case when the s eparation time is muc h smaller than 2 /σ 2 , the co rrelatio ns of loss fluctuations are decaying very , very slowly , a s ca n b e seen fro m equa- tion (3 9). Such time interv als a re likely to b e compar able or even smaller tha n the round trip times for typical TC P connections. TCP is the pro to col that controls the rate at which data is se nt acr oss a netw o rk, b etw een a par - ticular s ource and destination. The e xact details of the congeestion control op er ation of TCP can be found in [34]. F o r our purp ose s we shall o nly focus on its salient congestion co nt rol fea tures a nd the implications of the result of equa tion (39) on it. TCP limits its sending r ate as a function of the p er- ceived ne tw ork conge stion. It op erates on a virtual con- trol lo op of sending pack ets, receiv ing ackno wledg ement s and estimating the round trip time. Once a pack et is lost, the se nder cuts its tra nsmission rate by half. If no other los s is detected it increa ses its sending ra te linearly by a s mall increment. But if a subsequent lo ss even t is detected it cuts its tr a nsmission rate in half aga in. If successive loss even ts o ccur, which accor ding to equa- tion (39 ) is likely on the r elev a nt time scale, the reduc- tion in tra nsmission r ate can b e dramatic and p o tent ially unnecessary . As there a re multip le TCP connections ex- per iencing lo sses at the sa me buffer this will lead to a cycle of rapid under- usage and slow co nv ergenc e to c o n- gestion, which is clearly undesirable and ineffective. Studying o f spa tial correlations o f loss fluctuations over a netw or k is w ork in pr ogres s . This will help us quantif y the second significant asp ect of TCP op eratio n which is its r eaction to time-o ut even ts, as this is co nnec ted to co r - related losses and delays around the sequence o f buffers forming each co ntrol lo op. T o conclude, we emphas ize that the stability of a net- work with resp ect to da ta loss w a s mo stly ana lyzed in the pa st from the viewpo int o f the loss of physical con- nectivity in the netw or k topo logy where a failure of a given node or link was treated as a (pro babilistic) input int o a netw o rk mo del. Here we have s tudied dynamic al fluctuations in data loss in a single no de (memory buffer) of the netw or k. W e have shown that the s trong fluctu- ations and long-time memory in losses inevitably follow from the discrete character of s ignal propagation in the pack et-switched netw ork s . T his single-no de fluctuations can po tentially trigger a casca de of failures in neighbor- ing no des and thus r esult in a tempor al failure of lar g e parts of the net work. In the nex t stage, w e intend to utilize these features of the lo cal data loss as dynamical inputs into the netw ork and thus study po ssible a brupt increase of data lo ss in the netw or k tr iggered by a lo cal ov erlo ad. 6 Ackno wledgme nt s This work is s uppo rted by EPSRC grant GR/T2372 5/01. [1] C. Zhou , L. Zemanov´ a, G. Zamora, C. C. Hilgetag, and J. Kurths, Phys. R ev. L ett. 97 , 238103 ( 2006). [2] M. A. Arbib, The Handb o ok of Br ain The ory and Neur al Networks , MIT Press, London (2003). [3] H. Jeong, B. T omb or, R . Alb ert, Z. N. Oltv ai, and A .-L. Barab´ asi, Natur e 407 , 651 (2000). [4] B. J. Kim, J. Liu, J. Um, and S.-I . Lee, Phys. R ev. E 72 , 041906 (2005). [5] J. E. Cohen, F. Briand, and C. M. N ewman, Comm u- nity F o o d Webs: data and the ory , Springer V erlag, Berlin (1990). [6] W. Li, X. Zh ang, and G. H u, Phys. R ev. E 76 , 045102 (2007). [7] P . G. Lind , L. R. da Silv a, J. S. Andrade Jr., and H. J. Herrmann, Phys. R ev. E 76 , 036117 (2007). [8] L. K. Gallos, F. Liljeros, P . Argyrak is, A. Bunde, and S. Havlin, Phys. R ev. E 75 , 045104 (2007). [9] F. Liljeros, C. R. Edling, L. A. N . A m aral, H. E. Stanley , and Y. Ab erg, Natur e 411 , 907 (2001). [10] M. Kurant, P . Thiran, and P . Hagmann, Phys. Re v. E 76 , 026103 (2007). [11] J. Duch, and A. Arenas, Phys. R ev. L ett. 96 , 218702 (2006). [12] R. Pastor-Satorras and A. V espignani, Phys. R ev. L ett. 86 , 3200 ( 2001). [13] A.-L. Barab´ asi and R. Alb ert, Scienc e 280 , 98 (1999). [14] D. J. W att s and S. H. Strogatz, Natur e 393 , 440 (1998). [15] D. J. W atts, Smal l worlds: the dynamics of networks b e- twe en or der and r andomness , Princeton UP (1999). [16] R. Alb ert and A.-L. Barab´ asi, R ev. Mo d. Phys. 74 , 47 (2002). [17] R. Alb ert, H. Jeong, and A.-L. Barab´ asi, Natur e 406 , 378 (2000). [18] R. Cohen, K . Erez, D. b en -Avraham, and S. Havlin, Phys. R ev. L ett. 85 , 4626 (2000); ibid 86 , 3682 (2001). [19] L. A. Braun stein, S. V. Buldy rev, R. Cohen, S. Havli n, and H . E. Stanley , Phys. R ev. L ett. 91 , 168701 (2003). [20] S. N. Dorogo vt sev and J. F. F. Mendes, Eur ophys. L ett. 50 , 1 (2000). [21] D. J. Ashton, T. C. Jarrett, and N. F. Johnson, Phys. R ev. L ett. 94 , 058701 (2005). [22] R. Guimer` a, A. D ´ ıaz-Guilera, F. V ega-Redondo, A. Cabrales, and A. Arenas, Phys. R ev. L ett. 89 , 248701 (2002). [23] I. Y u rkevic h, I. Lerner, A. Step an en ko and C. Constanti- nou, Phys. R ev. E 74 , 046120 (2006). [24] A. S. Step anenko, C. C. Constantinou, T. N . A rvanitis, Pr o c. R oy. So c. A 461 , 933–955 (2005). [25] M. A. de Menezes, and A.-L. Barab´ asi, Phys. R ev. L ett. 92 , 028701 (2004); i bid 93 , 068701 (2004). [26] A. Stepanenko, C. C. Constantinou, T. N. A rv anitis, and K. Baughan, LNCS 2345 , 769–777 (Sp ringer, Berlin, 2002). [27] Y. Moreno, R. Pa stor-Satorras, A . V´ azquez, and A. V espignani, Eur ophys. L ett. 62 , 292 (2003). [28] T. O hira and R. Saw atari, Phys. R ev. E 58 , 193 (1998); S. G` abor and I. Csabai, Physic a A 307 , 516 (2002). [29] I. V . Lerner, Nucl. Phys. A 560 , 274 (1993). [30] J. W. Cohen, Single Server Queue , North-H olland, A m- sterdam (1969). [31] M. Sch wartz, T ele c ommunic ation N etworks, Pr oto c ols, Mo deling and Analysis , Add ison-W esley (1987). [32] O. J. O’Loan, M. R. Ev ans, and M. E. Cates, Phys. R ev. E 58 , 1404 (1998); T. Nagatani, ibid 58 , 4271 (1998). [33] M. A. d e Menezes and A.-L. Barab´ asi, Phys. R ev. L ett. 92 , 028701 (2004). [34] M. Allmanm, V . Paxso n and W. S tevens, I nternet RFC 2581, I ETF (1999).