Heavy-tailed Distributions In Stochastic Dynamical Models

Hea vy-tai led Distribut ions In Sto chastic Dynami cal Mo dels Ph. Blanchard 1 , 2 , T. Kr ¨ uger 2 D. V olchenk o v 2 1 1 Dep a rtment of Physics, Universit¨ at Bielefel d, Postfach 10 01 31, 33501 Bielefel d, Germany 2 Center of Exc el lenc e Co gnitive Inter action T e chnolo gy, Universit¨ at Biele- feld, Postfach 10 01 31, 33501 Bielefeld, Germany 1 E-Mail: v olchenk@physik.uni-bielefeld.de Pr eprint submitte d to Co mmunic ations in Nonline ar Scienc e an d Numeric al SimulationOctob er 25, 2018 Hea vy-tai led Distribut ions In Sto chastic Dynami cal Mo dels Ph. Blanchard 1 , 2 , T. Kr ¨ uger 2 D. V olchenk ov 2 1 Abstract Heavy-tailed distributions are found throughout many naturally o ccur r ing phenomena. W e have reviewed the models of sto chastic dynamics that lea d to hea vy- tailed distributions (and pow er law distributions, in particular) in- cluding the multiplicativ e noise mo dels, the mo dels sub jected to the Degree- Mass-Action principle (the genera lized preferential attachmen t principle), the int e r mitten t b ehavior occurring in complex ph ysical systems near a bifurcation po int , queuing systems, and the models o f Self-or g anized criticality . Hea vy - tailed distributions app ear in them as the emer gent phenomena sens itive for coupling rules essential for the entire dynamics. Keywor ds: Heavy-tailed distributions, Prefer e ntial attachmen t, Int e r mittency , Queueing systems, Self-organize d criticality Con tents 1 In tro duction 3 2 Degree-Mass-Action pri nciple in random graphs formatio n 6 2.1 A random graph space of the preferential attachmen t model . . 6 2.2 The Cameo Principle. The or igins of scale- free graphs in so cial net works. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3 The statistics o f bursts in systems close to a threshold of insta- bility 11 3.1 Systems at a threshold of instabilit y . . . . . . . . . . . . . . . . 12 3.2 Distribution of residence times below the threshold . . . . . . . . 13 3.2.1 Some examples of decay in the correlated case η = 1 . . . 15 3.2.2 Upper and lower bo unds for P ( T ) for an y η . . . . . . . . 16 3.2.3 Behavior of P ( T ) for in ter mediate times . . . . . . . . . . 17 3.3 Distribution of quiescent times for the case of uniform densities . 17 1 E-Mail: v olchenk@physik.uni-bielefeld.de Pr eprint submitte d to Co mmunic ations in Nonline ar Scienc e an d Numeric al SimulationOctob er 25, 2018 4 F at tails in queuing systems 21 4.1 W aiting time distributions for queuing systems with fat tail ser- vice times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.2 Schedulin g based on time dependent priority indexes . . . . . . . 24 4.2.1 T asks p opulation dyna mics with time dependent prior it y indexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.2.2 Sto ch astic dynamics. Real-time queuing dynamics . . . . 27 4.2.3 ”First come ﬁrst served” (F CFS) schedulin g polic ies . . . 30 4.3 The impo rtance of adopting perfor mance scheduling p olicies . . . 33 5 P ow er l a w di stributions in Self–Organized Criti calit y 34 5.1 Coarse- graining of micro s copic evolution rules for SOC–mo dels . 36 5.2 Cov ar iance o f r andom forces . . . . . . . . . . . . . . . . . . . . . 39 5.3 An inﬁnite num b er of critical regimes in SOC– models . . . . . . 42 5.4 F at tails in SOC-mo dels . . . . . . . . . . . . . . . . . . . . . . . 43 6 Conclusions 45 7 Ac kno w ledgments 46 1. Introduction In 1906, a lecturer in economics a t the Univ e rsity of Lausanne in Switzerland V.F. Pareto had discov ered that the allo cation of wealth among individuals can be eﬃcien tly describ ed by a p ower law proba bilit y distribution, as b eing self- similar ov er a wide ra nge of wealth magnitudes. A great many suc h distr ibutions hav e b e en fo und in diverse ﬁelds o f sc ience b eing a long-known feature of many empirical distributions such as Pareto, Zipf, and L´ e v y distributio ns use d to mo del real-world pheno mena . In particular, P a ul L ´ evy w or ked on a class of probability distributions with ” heavy tails”, which he called stable distributions largely considered probabilistic curiosities at the time, as heavy-tailed distri- butions have prope r ties that are qualitatively diﬀerent to the many commonly used distributions such as exp onential, norma l or Poisson. Since then, there has b een a p ermanent surge of interest to heavy-tailed and power law distributions fro m scientists working in ﬁelds a s diverse as weather forecasting to sto ck mark et analysis. The growth rate of the interest to the heavy-tailed and p ow er law distr ibutions can be attested by the y ea rly in- crease of the total nu mber o f publications on the topic (see Fig . 1) in the arXiv ( http://arxiv. or g ), the ma jor forum for disseminating scientiﬁc res ults in Physics, Mathematics, Nonlinea r Sciences, Co mputer Science, a nd Quantitativ e Biology . The purp ose o f this review pap er is to explain the mo dels of sto chastic dynamics that lead to heavy-tailed and power law distributions. 3 Figure 1: The total n umber of publications dev oted to the hea vy-tailed and p ow er law distri- butions in the arXiv ( http://arxiv.or g ) gro ws fast each year. In this sur vey , w e ca ll a distribution he avy-taile d if it do es not have the exp onentially b ounded asymptotes . Namely , given a random v ariable X char- acterized by the probability distribution function F ( x ) = Pr [ x < X ] , we say that it has a he avy right tail if lim x →∞ F ( x ) · exp λx = ∞ , ∀ λ > 0 . (1) An imp ortant sub clas s o f heavy-tailed distr ibutions is the sub-exp onential distri- butions [2], for which a s um of n indep endent random v ariables X 1 , X 2 , . . . , X n , with common distribution F ( x ) is given, Pr [ x < X 1 + X 2 + . . . + X n ] ∼ x →∞ Pr [ x < max ( X 1 , X 2 , . . . , X n )] . (2) Among s ub-exp onential distributions, we shall b e es sentially interested in those distributions of a random v ar iable X which are characterized by a p ower law de c ay , Pr [ x < X ] ∼ x →∞ 1 x 1+ γ , (3) for some γ > 0 . There are s everal ph ys ical mechanisms known a s underly ing the p ow er law behaviors. Pow er law distributions often manifest a form of regular it y ar ising through growth processe s whic h is comp osed of a larg e n umber of commo n even ts a nd a small n umber of rar er even ts happ ened randomly . As the num b er of even ts grows, their distribution, under certain conditions, might co n verge to 4 a steady state (3). The mechanism of pr efer ential attachment in whic h some quantit y is dis tributed at random among a n umber of individuals a ccording to how m uch they a lready hav e ha d b een prop osed in [3] as a n algo rithm explaining power law degr ee distr ibutions in some s c ale-free net works. Gra phs in that grow by succes s ively adding a new v er tex say x at each time step that links to a n already existing one o f the degr ee k , with the pro bability ∝ k / N where N is the total num b er o f vertices present in the graph. The pr eferential attachmen t pro cess generates a heavy-tailed distribution following a p ow er law in its tail. Po wer laws are also found in the study of stochastic proces s es in volving m ultiplicative no ises [1]. A t ypica l equation of multip licative sto chastic pro cess is given b y a linear L angevin e qu ation with a r a ndomly changing coeﬃcient. The eﬀect of such a rando m co eﬃcient dr astically enhance the additive ra ndom force in the Langev in e quation and increase ﬂuctuations. F ollowing [4 ], let us consider tempora l ﬂuctuations for a simple discrete time version of the Langevin equation, x ( t + 1 ) = b ( t ) · x ( t ) + f ( t ) , (4) where f ( t ) represents a random a dditiv e noise, and b ( t ) is a non-nega tive sto chas- tic co eﬃcient in ter preted as dis sipation for b ( t ) < 1 and ampliﬁca tion for b ( t ) > 1 . If we assume for simplicity that b ( t ) a nd f ( t ) are indep endent white noises having stationary s tatistics, and f ( t ) is symmetric, we obta in, for the second order momen t,  x 2 ( t + 1 )  =  b 2   x 2 ( t )  +  f 2  , (5) in which the ang ular brack ets denote an av erage over realiza tions. F o r constant b 2 and f 2 , there is a statio nary solution,  x 2  =  f 2  1 − h b 2 i ,  b 2  < 1 , (6) but  x 2 ( t )  diverges a s t → ∞ , fo r  b 2  > 1 . The statistics of x ( t ) is estimated theoretically b y in tro ducing the characteristic function, F ( ξ , t ) = D e iξ x ( t ) E . (7) When  x 2 ( t )  diverges, F ( ξ , t ) has s ingularity at ξ = 0 in the limit of t → ∞ , so that T aylor expansio n cannot be applied for the steady solution. In such a cas e the following fractional p ow er term can b e as s umed for the low est order term bec ause the c ha racteristic function is generally a co nt inuous function F ( ξ , ∞ ) = 1 − c o nst · | ξ | β + . . . , 0 < β < 2 , (8) which is e q uiv alent to the assumption of p ow er la w tails for the cumulativ e probability distribution Pr [ | x | < X ] ∼ 1 x β . (9) The v a lidation o f sto chastic mechanisms g enerating the p ow er-law b ehavior re- mains an a ctive ﬁeld of resea rch in many a r eas of mo dern s cience. 5 In the forthcoming sections, we cons ide r in details other, mor e so phisticated mechanisms that might bring forth fat–tail statistics in dynamical systems. The plan of our r eview is following. In Sec. 2, we discuss the Degree-Mas s - Action pr inciple in r andom gr a phs formation - the preferential attac hments and their natural generalizations. In Sec. 3 , we consider the statistics of burs ts in sy stems clos e to a thr eshold of instabilit y . Then, in Sec. 4, we review the emergence of fat tails in queuing systems. Fina lly , in Sec. 5, we expla in the app earance of power law distr ibutions in mo dels of Self–Orga nized C r iticality . W e conclude in the last section. 2. Deg ree-Mass-Action principle i n random graphs formation Random gr aphs with scale–free pr obability deg ree distributions are ubiqui- tous while mo deling many real world net works such as the W orld Wide W eb, so cial, linguistic, citation and bio chemical netw o rks; an excellent survey is [5]. The preferential attachment principle, together with its v arious modiﬁca tions, could b e seen as a particular case of de gr e e-mass-action principle since the degree of a no de ac ts in that a s a p ositive aﬃnity par ameter quantifying the attr a c- tiveness of the no de for new vertices. Our ﬁrst aim is to co nstruct a family of static r a ndom gr aph mo dels, in which vertex degr ees are dis tr ibuted p ower-la w like, while edges still ha ve high degree o f indep endence. As usual in rando m graph theor y , we will entirely dea l with asymptotic pr op erties in the s e nse that the graph size g o es to inﬁnit y . 2.1. A r andom gr aph sp ac e of the pr efer ential attachment mo del W e consider gra phs with vertex set V = V n = { 1 , ..., n } where an edge betw een the vertices x and y (denoted by x ∼ y ) is in ter preted as a per sistent contact b etw een the t wo nodes . Giv en x ∈ V , its degree is denoted b y d ( x ). W e will think of edges as generated by a pair-for mation pro cess in whic h eac h vertex x - often denoted a s an individual - chooses a se t of partners a ccording to a sp eciﬁed x -dep endent rule. Therefore the s et of individuals which hav e contact with a given vertex x can be divided into t wo -pos sible non-disjoint sets: the set of nodes which are c ho s en b y x himself and the set of no des which hav e c ho sen x as one of their partners. W e ca ll the size o f the ﬁrst set the o ut-degree d out ( x ) of x a nd the size of the second one the in-deg ree d in ( x ) of x . Obviously , d ( x ) ≤ d out ( x ) + d in ( x ) , (10) and if the choices are suﬃciently indep endent one can expect eq uality to hold almost surely if n → ∞ . W e partition the set of vertices V n int o groups { C i ( n ) } i ≥ 1 where all members of a gro up C i ( n ) ch o ose exactly i pa rtners by themselves ( d out = i on C i ( n )). Let P 1 α ( n, j ) the probabilit y for x to choos e a ﬁxed pa r tner y ∈ C j ( n ) if n partners are av aila ble for the c hoice and just o ne choice will be ma de be P 1 α ( n, j ) = A α ( n ) j α n (11) 6 Here A α ( n ) is a nor malization constant such that A α ( n )   X i ≥ 1 | C i ( n ) | i α n   = 1 and α is a real parameter. Since w e wan t A α ( n ) → n →∞ A α we need P i | C i ( n ) | i α n to be bounded as a function o f n whic h will impo s e later on constraints on the consta nt α . The par ameter α acts as an aﬃnity pa rameter tuning the tendency to ch o ose a par tner with a high out-degree or low out- degree. If α = 0 , choices are made without any prefer ences and A α ( n ) ≡ 1. F o r α > 0 the ”highly a ctive” individuals are preferr ed whereas for α < 0 the ”low activity” individuals are fav ored. F r om this w e obtain the basic probability Pr[ x ∼ y ] ≃ A α ( n ) i · j α n , x ∈ C i , y ∈ C j . (12) Concerning the size of the sets C i ( n ) w e will mak e the follo wing as sumption: | C i ( n ) | n = p i ( n ) → n →∞ c 1 i γ . (13) With this choice we ha ve to imp ose the res triction α < γ − 1 to ensur e the conv erg ence of A α ( n ). W e require further more γ > 2 throughout the pap er since otherwise the exp ected in- degree for individuals from a ﬁxed gr oup would diverge. The bas ic pro ba bilities together with the ﬁxed out-de g ree distribution deﬁne a probability distr ibution on each gra ph with vertex-set V n , and therefo r e a random graph space G n ( α, γ ). First we wan t to compute the imp ortant pairing probabilities. W e start with the easier case α = 0. Pr ( x ∼ y | x ∈ C i ; y ∈ C k ) = i + k n − ik n 2 ∼ n →∞ i + k n (14) Likewise one can compute the cor resp onding probabilities for α 6 = 0. Dro pping the simple details w e just state the res ult: Pr ( x ∼ y | x ∈ C i ; y ∈ C k ) ≃ A α ( k i α + k α i ) n , n → ∞ . (15) It turns out,that the typical gra phs in this model still hav e for α < 2 a p ower- law distribution for the degr ee with a n exp onent which can b e diﬀerent fro m the exp o nent of the out-degree. F or α > 2 w e obtain a degree distribution which follows in mean a p ow er law but has ga ps. T o compare bo th domains w e will use the in tegrated tail distribution F k = P r ( d ( x ) > k ) . 7 W e will show that in both cases we get the same integrated ta il distribution. Since we are interested in the dep endence of the epidemic thres ho ld from the power-law expo nent o f the total degree distribution we hav e to analyze how this exp onent v aries with the t wo par ameters α and γ Since the par tner choice is suﬃciently random and not to o strongly biased to ward hig h deg ree individuals (that’s the mea ning of the condition α ≤ γ − 1 ) it is ea sy to s e e tha t the in- degree dis tr ibution of a vertex from g roup C i conv erg es for n → ∞ to a Poisson distribution with mean const · i α . There are essen tially tw o regimes in the parameter space, o ne for which the exp ected in-deg ree is of smaller order than the o ut-degree ov er all gro ups and one where the in-degree is a symptotically of larger order. In the ﬁr st case it is clear that the in-degr ee is to o small to hav e an eﬀect on the degree distribution exponent. In other w o rds: the s et of individuals with deg ree k co nsists ma inly of individuals whose out-degree is of o r der k . An easy estimation using the formula for the pairing probabilities shows that the exp ected in-degree of individuals from group i is given by E ( d in ( x ) | x ∈ C i ) ≃ cons t · i α asymptotically . Therefore the in- de g ree is of smaller order than the out-degre e if α < 1. In the case γ − 1 ≥ α ≥ 1 the se t of individuals with deg ree k co nsists mainly of individuals from g roups with an index of o rder k 1 /α . 2.2. The Came o Principle. The origins of sc ale-fr e e gr aphs in so cial n etworks. In the present section, we descr ibe an edge formation principle re lated up o n a structure apriori imp ose d on the vertex s et. W e assume that s uc h a structure can be s p eciﬁed by a real p ositive random v ariable ω ∈ R + that quantiﬁes some so cial prop erty of an individual such as its wealth, po pularity , or b eauty b eing distributed ov er the po pulation with a g iven probability distr ibution ϕ ( ω ). F urthermor e, we ass ume that a link b etw een the tw o individuals, x and y , arises as a result of a directed c hoice made by either x or y (symbolized b y x → y or y → x r esp ectively); in ma ny real life netw or ks edg es ar e formed tha t wa y . Although the edge cr eation is certainly a directed pro cess, in the prese n t section we cons ider the resulting graph to be undir e c ted since for the ma jor it y of relev ant tra nsmission pro ces ses deﬁned on the net work the origina l o rientation of an edge is ir relev ant. W e s uppo s e that the pairing proba bility follows an inverse mass- action- principle : the pr o bability that x decides to connec t to y c har acterized by its aﬃnity v alue ω ( y ) re a ds as Pr { x → y | ω ( y ) } ∼ 1 N · 1 ϕ ( ω ( y )) α , α ∈ (0; 1) (16) where N b eing the total num b er of vertices. Let us note that it is not the actual v alue ω ( y ) which pla ys a decisive r ole while pairing, but rather its relative fre- quency of app ear ance ov er the p opula tio n. The prop os e d principle captures the essence o f antiquit y markets – the more r are a prop erty is, the higher is its v alue, and the more a ttractive it b ecomes fo r others. The par ing pro bability mo del 8 describ ed a bove ha d b een in tro duced in [34] a nd called the Came o-principle having in mind the attractiveness, r areness and beauty of the s mall medallion with a pro ﬁled head in relief called Ca meo. And it is e x actly their rareness a nd bea ut y whic h g ives them their high v a lue. In the Ec onomics of L o c ation the ory introduced by [6] and develop ed by [7], a city , or even more certa inly , a particular district in that may sp ecializ e in the pro duction of a go o d that can b e co nnected with natural r esources, education, po licy , or just low salary expenditur e s. City districts compete among themselves in a cit y market not necess arily connected with the quantit y of their inhabitants. The demand for these pro ducts comes into the city district from everywhere and can be co ns idered as exogeno us. Then, the loc a l a ttractiveness o f a site determining the cr eation o f new spaces o f mo tio n in that is speciﬁed by a real po sitive r a ndom v ariable ω > 0. Indeed, it is r ather diﬃcult if ever be p ossible to es timate exa ctly the actual v alue ω ( i ) for any site in the urban pattern, since such an es timation can b e referred to bo th the eco nomic an cultural factor s that may v ary ov er the diﬀerent historical ep o c hs and o ver the certain gro ups of p opula tio n. In the framework of a pr o babilistic appr o ach, it seems natural to consider the v alue ω as a r e al p os itiv e indep endent random v ariable distributed ov er the v er tex set of the graph representation of the site uniformly in accordance to a s mo o th monotone decreasing probability density function f ( ω ). While in vestigating the model o f Cameo graphs, w e assume that • The pa rameter ω is independent identically dis tr ibuted (i.i.d.) over the vertex set with a smo oth monotone decreasing density function ϕ ( ω ) • Edges are formed by a sequence of choi c es . By a c ho ice we mean that a vertex x c ho o ses another v ertex, say y , to form an edge betw een y and x. A vertex ca n make several c ho ices. All choices are a ssumed to b e independent of each other. • If x mak es a c ho ice the proba bilit y of c ho os ing y depends only on the relative density o f ω ( y ) and is of the form (16). • A pre-deﬁned out-degree distribution determines the n umber of choices made b y the vertices. The to ta l n umber of choices (and ther efore the nu m be r of edges) is assumed to b e ab o ut co nst · N . W e focus on the str iking obs e rv ation that under the a bove assumptions a scale-free degre e distribution emerges indep endent ly of the particular choice of the ω − distribution. F urther more, it can b e s hown that the exp onent in the degree distribution be c omes independent of ϕ ( ω ) if the tail o f ϕ decays fas ter then any p ow er law. Let V N = { 1 , ..., N } be the vertex set of a random graph space. W e are mainly in terested in the asymptotic prop erties for N be ing very lar ge. W e assign i.i.d. to each elemen t x from the s et V N a contin uo us p ositive rea l random v ariable (r.v.) ω ( x ) taken fro m a distributio n with dens ity function ϕ ( ω ). The v ariable ω can be interpreted as a par ametrization of V N . F or a set C ω 0 ,ω 1 = { x : ω ( x ) ∈ [ ω 0 , ω 1 ] } , (17) 9 we obtain E ( ♯C ω 0 ,ω 1 ) = N · ω 1 Z ω 0 ϕ ( ω ) dω (18) where ♯C ω 0 ,ω 1 denotes the ca rdinality of the set (17). Without lo ss of generality , we a lwa ys assume that ϕ > 0 on [0 , ∞ ) , and that the the tail of the distribution for ϕ is a monotone function, namely that ϕ ∈ C 2 ([0 , ∞ )) and the second deriv atives, D 2 ( ϕ µ ) , have no zeros for | µ | ∈ (0 , µ 0 ) and ω > ω 0 ( µ ) . Edges a re created by a directed pro cess in which the bas ic even ts are choices made by the vertices. All choices ar e assumed to b e i.i.d. The n umber of times a v ertex x makes a choice is itself a rando m v ariable whic h may dep end on x . W e call this r .v. d out ( x ). The num b er of times a vertex x was chosen in the edge for mation pro cess is called the in-degree d in ( x ) . Each c ho ice gener ates a directed edge. W e ar e mos tly interested in the cor resp onding undir ected graph. If we sp eak in the following a b o ut out-degr ee and in- degree we refer just to the original direction in the edge formation pro cess. Let p ω = P r { x → y | ω = ω ( y ) } be the basic pr obability that a vertex y w ith a ﬁxed v alue of ω is chosen by x if x is ab out to make a c hoice. F or a given rea liz ation ξ of the r.v. ω ov er V N we assume: p ω ( ξ , N ) = 1 N · A ( ξ , N ) [ ϕ ( ω )] α . (19) where α ∈ (0 , 1 ) a nd A ( ξ , N ) is a normalization constant. It is ea sy to see that the condition ∞ Z 0 [ ϕ ( ω )] 1 − α dω < ∞ is necessary and s uﬃcien t to get A ( ξ , N ) → A > 0 , for N → ∞ where conv erg ence is in the sense o f probability . Ther efore, we need α < 1 . One migh t ar gue that the choice pro ba bilities should dep end more ex plic itly on the actual realiz a tion ξ of the r.v. ω over V N -not only via the normaliz a tion constant. The r eason not to do so is tw ofold. First it is mathematica lly un- pleasant to work with the empirical distribution of ω induced by the realization ξ since one had to use a somehow ar tiﬁcial N − dep endent coar se gr a ining. Sec- ond the empirical distribution is not really ”observed” by the vertices (ha ving in mind for instance individuals in a so cia l net work). What seems to b e rele v a nt is more the common b elieve ab out the distribution o f ω . In this sense our setting is a na tural one. The emergence of a p ow er law distribution in the a bove settings is not a surprise as it might seem for the ﬁrs t glance . The situation is b est expla ined by 10 the following example. Let us take ϕ ( ω ) = C · e − ω and deﬁne a new v a riable ω ∗ = 1 [ ϕ ( ω )] α = e ω α C α . The new v ariable ω ∗ can b e seen as the eﬀective parameter to whic h the vertex choice pro cess applies. What is the induced distribution of ω ∗ ? With F ( z ) = Pr { ω ∗ < z } we obtain F ( z ) = 1 α ln C α · z Z 0 ϕ ( ω ) dω = − 1 z 1 /α − C, (20) and therefore the ω ∗ − distribution φ ( ω ∗ ) = 1 α · 1 ( ω ∗ ) 1+1 /α . This is a pow er law distribution with an exp onent depending only on α. The detailed re sults on the degr ee-degree correla tions, the clustering co eﬃ- cients, and the second moment of degree distributions are discussed in [34]. 3. The statistics of bursts i n systems close to a threshold of instabil it y Systems driven by r andom pro cesses at a threshold of sta bility may exhibit a r andom switc hing of a signa l betw een a quiescent (stable) and a bursting (unstable) s tate. Such an intermittent behavior is observed ov er a br o ad cla ss of diﬀerent s ystems in physics and nonlinear dyna mics . Dep ending on the origin, the in termittent b ehavior either meets the class iﬁcation pro p o sed b y Pomeau- Manneville [9 ] (the I-I I I types intermittency) or ﬁts the features of the c r isis- induced intermittency [10]. In b oth case s , the par ameters of the models ar e static. Another exa mple o f intermitten t behavior, called on-oﬀ intermittency has been introduced in [11] and then observed numerically and exp erimentally , [12, 13, 14, 15, 16, 17, 18, 19, 20, 2 1, 2 3]. The mechanism for this intermittency t yp e relies on a ra ndom forcing of a bifurcation para meter throug h a bifurcation po int . The erg o dic prope r ties of a system a t the threshold of sta bilit y can be par- tially c ha racterized b y the distribution of the q uiescent times (the duratio ns of laminar phases) P ( t ) where t ∈ N . Indeed, a co mplete characterization o f the statistical prop erties of the system will imply the knowledge of residence times distribution for all the r egions of the phase space and not only o f the laminar regions. Ho wev er , the for mer is the ﬁr st impo rtant sta tis tica l indicator of such 11 dynamics and this is a re a son why we fo cus a t the quiescent times distributions in the present study . Depending on the particular t yp e of in ter mittency ex hibited by the system, the statistics of this distribution can as y mptotically meet either exponential laws, or p ower laws of ex po nent γ . P a rticularly , the p ower-la w statistics for the quiescent times distribution is claimed to b e typical for the s ystems demon- strating the on-oﬀ type intermittency , and the v alue of expo nent γ depends in general from the nonlinearity characteristic of the dynamical system consider ed [22]. F or example, in the exp eriments with ion- acoustic instabilities in a lab o- ratory plasma [23], due to nonlinear eﬀects, the ex po nen t of p ower law dep ends on the v alue of a control parameter. In the pr esent section, we discuss the net eﬀect pr o duced on the sta tis tics of laminar phases by the stochastic ﬂuctuations o f a sy stem state v ar iable (a bifurcation para meter) near the ﬂuctuating thr eshold o f stability (a bifurcation po int ). W e do not refer to any deﬁnite physical system displaying an int ermittent behavior. F or the to y model whic h we introduce, the bifurc a tion pa r ameter and the bifurcatio n po int are considere d as random indep endent v ar iables. It is suppo sed that intermittency takes place in the system when the pro ces s cr osses the threshold v alue. The c o nt rol pa rameter of the system is the num b er η ∈ [0 , 1], which rep- resents a re lative frequency of ﬂuctuation of the threshold v alue: v arying the parameter η amounts to modifying the r elation betw een the characteristic time scales of the threshold v ariable a nd thos e of the state v ar iable. A t η = 0 (when the state and the thresho ld v ariables hav e the same time scale) the s tatistics of la minar phas e s is ex p one ntial, while at η = 1 (the limiting case o f quenc hed threshold) it ca n b e power law; for the intermediate v alues 0 < η < 1 , the statistics is mixed b ecoming exponential for suﬃcien tly large times. In general, the statistics of laminar phases dep ends on the statistics of the random system state v ar iable a nd threshold describ ed by the pro bability distri- butions F and G r esp ectively . F o r man y distributions F and G , the prop osed mo del can be so lved a nalytically . 3.1. S yst ems at a t hr eshold of instability Let us suppose that the state of a system can b e characterized b y a real nu m be r x ∈ [0 , 1]. Ano ther real n umber y ∈ [0 , 1] plays the role o f a threshold of stabilit y . The sys tem is stable as long as x < y and ex hibits a sudden transition to the irregular state otherwise ( x ≥ y ). W e co ns ider x as a random v ar iable distributed with resp ect to so me g iven probability distribution function P { x < u } = F ( u ) . In an analo gous w ay , the v alue of the threshold y is also a random v a riable distributed over the in terv al [0 , 1] with resp ect to some other probability distri- 12 bution function (pdf ) P { y < u } = G ( u ) . In gener al, F and G ar e tw o arbitr ary left-contin uous increa sing functions s at- isfying the normalization conditions F (0) = G (0) = 0 , F (1) = G (1) = 1 . Given a ﬁxed rea l num b er η ∈ [0 , 1], w e deﬁne a discrete time r andom pro cess in the following way . At time t = 0 , the v ar iable x is chosen with resp ect to p df F , and y is chosen with resp ect to pdf G . If x ≥ y , the pro cess contin ues and go es to time t = 1. At time t ≥ 1 , the following even ts happen: i ) with proba bilit y η , the random v aria ble x is chosen with p df F but the threshold y k e e ps the v alue it had at time t − 1 . Otherwise, ii ) with pro bability 1 − η , the random v ar iable x is chosen with p df F, a nd the threshold y is c ho sen with p df G . If x ≥ y , the pr o cess ends; if x < y , the pro cess contin ues and go es to time t + 1 . Even tually , at so me time step t, when the state v ar iable x ex ceeds the thresh- old v alue y , the pro cess stops, and the system destabilizes, so this in teg er v alue t = T a cquired in this random pro cess limits the dura tion of the episo de of laminar dynamics . In the la ps of time, the system reg a ins the comp osure state, when x < y , and the pro cess star ts again. While studying the ab ov e mo del, we are interested in the distribution o f the duration o f laminar phas es P η ( T ; F , G ) provided the probability distributions F and G are given a nd the co n trol pa rameter η is ﬁxed. Even if in our mo del the sta te v ar iable x is treated a s a r a ndom v aria ble, what is r e ally imp orta n t in what follows is the corresp onding p df F . It would in fact be p ossible to trea t x as a deterministic dynamica l v ar iable deﬁned b y the iterated images of a map of the interv al [0 , 1]. In this case we would a s sume the existence o f an inv aria nt ergo dic (Berno ulli) measure dF , fo r which x is a generic orbit. It is a lso to b e noticed that the mo del pr o po sed r e sembles clo sely the coherent - noise mo dels [25, 26] discussed in concern with a standar d sandpile mo del [27] in self-o rganized critica lit y , where the statistics of av alanche s iz es and durations take p ow er -law forms. No exact analy tical results conce r ning the coherent-noise mo dels hav e b een obtained so far . The prop ose d toy model has not b een dis- cussed in the literature b efore and, in principle, is muc h simpler than those discussed in [25, 2 6] since it do es not in volve an y spatial dynamics typical of such extended systems with quenc hed memo ry a s the original s andpile models. 3.2. D ist ribution of r esidenc e times b elow the t hr eshold W e are in tere s ted in the proba bilit y P η ( T ; F , G ) that the ra ndo m pr o cess int r o duced in the previous se ction ends pre c isely at time T with a cross ing of the thr eshold, provided the distributions F and G are given and η is ﬁxed. W e 13 shall denote P η ( T ; F , G ) simply as P ( T ). A stra ightforw ar d computation s hows directly from the deﬁnitions of Sec. 3 .1 that P (0) = Z 1 0 dG ( y ) (1 − F ( y )) . ( 21) F or T ≥ 1, the system can either stay b elow the thres hold in the laminar state ( a ”surviv al”) ( S ) or surmount the threshold to a burs t state (a ”death”) ( D ). Both event s can take place either in the cor r elated w ay (with probability η ; se e (i) in Sec. 3.1) (we denote them S c and D c ), or in the uncorr elated wa y (with probability 1 − η ; see (ii) in s ection Sec. 3.1) ( S u and D u ). F or T = 1, we hav e for example P (1) = P [ S D c ] + P [ S D u ] = η R 1 0 dG ( y ) F ( y ) (1 − F ( y )) + (1 − η ) R 1 0 dG ( y ) F ( y ) R 1 0 dG ( z ) (1 − F ( z )) = ηB (1) + (1 − η ) A (1) B (0) . (22) Similarly , P (2) = η 2 B (2) + η (1 − η ) ( A (1) B (1) + A (2) B (0)) +(1 − η ) 2 A (1) 2 B (0) , (23) where where w e hav e deﬁned, for n = 0 , 1 , 2 , . . . , A ( n ) = Z 1 0 dG ( y ) F ( y ) n (24) and B ( n ) = R 1 0 dG ( y ) F ( y ) n (1 − F ( y )) = A ( n ) − A ( n + 1) . (25) It is useful to in tro duce the gener ating function of P ( T ): ˆ P ( s ) = ∞ X T =0 s T P ( T ) . The generating prop erty of the function ˆ P ( s ) is suc h that P ( T ) = 1 T ! d T ˆ P ( s ) ds T      s =0 . (26) Deﬁning the following auxiliary functions p ( l ) = η l A ( l + 1) , for l ≥ 1 , p (0) = 0 , q ( l ) = (1 − η ) l A l − 1 (1) , fo r l ≥ 1 , q (0) = 0 , r ( l ) = η l [ η B ( l + 1) + (1 − η ) A ( l + 1) B (0)] , for l ≥ 1 , r (0) = 0 , ρ = η B (1) + (1 − η ) A (1) B (0) . (27) 14 we ﬁnd ˆ P ( s ) = B (0) + ρs + s [ ˆ r ( s ) + ρ ˆ p ( s ) ˆ q ( s ) + ρA (1) ˆ q ( s ) + A (1) ˆ q ( s ) ˆ r ( s )] 1 − ˆ p ( s ) ˆ q ( s ) , (28) where ˆ p ( s ) , ˆ q ( s ) , ˆ r ( s ) are the generating functions of p ( l ) , q ( l ) , r ( l ) , resp ectively . In the marginal cases of η = 0 and η = 1, the pro bability P ( T ) can b e readily calculated. F or η = 0 , equations (27) and (2 8 ) give ˆ P η =0 ( s ) = B (0) 1 − s A (1) . (29) Applying the in verse formula (26) to equation (29), w e g e t P η =0 ( T ) = A T (1) B (0) =  Z 1 0 dG ( y ) F ( y )  T Z 1 0 dG ( y ) (1 − F ( y )) . Therefore, in this case, for any choice of the pdf F ( u ) and G ( u ) , the proba bilit y P ( T ) decays exp onentially . F or η = 1, e q uations (27) and (28) yield ˆ P η =1 ( s ) = ˆ B ( s ) , (30) so that P η =1 ( T ) = B ( T ) = Z 1 0 dG ( y ) F ( y ) T (1 − F ( y )) . (31) 3.2.1. S ome examples of de c ay in the c orr elate d c ase η = 1 W e hav e just seen that the probability P ( T ) decays exp onentially , in the uncorrela ted case η = 0 , for any choice of the p df F a nd G . In the corr elated case η = 1, many diﬀerent types of b ehavior ar e po ssible, depe nding on the form of the p df F and G . W e will ex amine a n imp ortant class of F and G , for which P η =1 ( T ) ca n b e explicitly computed from eq uation (31 ). W e will take F and G as absolutely co nt inuous with resp ect to the Lebesg ue measure, with dF ( u ) = (1 + α ) u α du, α > − 1 , dG ( u ) = (1 + β )(1 − u ) β du, β > − 1 . (32) Here we recognize the family of inv ar iant measure s of a map of the interv al with a ﬁxed neutral point [28]. Equation (31) gives in this case: P η =1 ( T ) = Γ(2 + β ) Γ(1 + T (1 + α )) Γ(2 + β + T (1 + α )) − Γ(2 + β ) Γ (1 + ( T + 1)(1 + α )) Γ(2 + β + ( T + 1)(1 + α )) . Using Stirling’s approximation, we g et fo r T >> 1: P η =1 ( T ) = (1 + β ) Γ(2 + β ) (1 + α ) − 1 − β T 2+ β  1 + 0  1 T  . (33) 15 F or diﬀere nt v alues of β , the ex po nent of the threshold distribution, we g et all p ossible power law decays of P η =1 ( T ). Notice that the e x po nent ( − 2 − β ) characterizing the decay of P η =1 ( T ) is indep endent of the distribution F of the state v ariable. W e were not able to pr ov e that the asymptotic decay of P η =1 ( T ) is a lgebraic for any choice of the distributions F and G ; nev er theless, we hav e not found any counterexample contradicting this conjectur e . Let us consider in par ticular the case o f unifor m F (the results o f this section suggest in fact that wha t determines the decay of P ( T ) is mostly the thresho ld p df G ): P η =1 ( T ) is then a particular cas e of a Riemann-Liouville integral, and we did no t ﬁnd any case of non–algebraic deca y for large T in the tables [29]. 3.2.2. U pp er and lower b ou n ds for P ( T ) for any η W e will use the fact that A (1) n ≤ A ( n ) ≤ A (1) and 0 ≤ B ( n ) ≤ A (1) , n = 1 , 2 , . . . . (34) The upper bound for A ( n ) is trivial, since 0 ≤ F ( y ) ≤ 1 for any y ∈ [0 , 1]. The low er b ound is a c onsequence of Jensen’s inequa lit y , and of the fact that the function x → x n is con vex on the int e rv al ]0 , 1[ for an y integer n . W e now replace these b ounds for A ( n ) a nd B ( n ) in the general formula for P ( T ) a nd the r esulting ex pr essions in all the terms o f the sums, ex cept the one corres p o nding to the index n = 0 in P I ( T ). (This term, which c orresp onds to a sequence of corr elated surviv als, has to b e treated separ ately , in order not to lose information on the cas e η = 1 . ) Labelling b y the index k the num b er of uncorrela ted surviv als in the s equence of even ts consider ed in the sum for P ( T ), we get P η ( T ) ≤  η T B ( T ) + η T − 1 (1 − η ) A ( T ) B (0)  + [ η A (1) + (1 − η ) A (1) B (0)] P T − 1 k =1 γ T − 1 k [(1 − η ) A (1)] k η T − 1 − k (35) and P η ( T ) ≥  η T B ( T ) + η T − 1 (1 − η ) A ( T ) B (0)  + (1 − η ) A (1) B (0) P T − 1 k =1 γ T − 1 k [(1 − η ) A (1)] k η T − 1 − k (36) where γ T − 1 k represents the num b er of sequences of T − 1 ev ents c, u ( c = corr e - lated surviv al, u = unco rrelated s urviv al) c ontaining a num b er k of even ts u , s o that γ T − 1 k =  T − 1 k  . This implies the upper b ound P η ( T ) ≤ η T B ( T ) + (1 − η ) A (1) B (0) [ η + (1 − η ) A (1)] T − 1 + η A (1) n [ η + (1 − η ) A (1)] T − 1 − η T − 1 o (37) 16 and the lo wer b ound P η ( T ) ≥ η T B ( T ) + (1 − η ) A (1) T B (0) = η T P η =1 ( T ) + (1 − η ) P η =0 ( T ) . (38) W e th us see that, for any 0 ≤ η < 1, the decay o f distr ibutio n P ( T ) is bo unded by expo nent ials. F urthermore, the b ounds (37 ) and (38) are exact in the marginal cases η = 0 and η = 1. 3.2.3. Behavior of P ( T ) for interme diate times W e have seen in Sec. 3.2.1 that there exists a class of p dfs for which P η ( T ) decays like a p ow er law whe n η = 1. In s ection 3.2.2, we s how that for a ny η < 1 the asy mptotic decay of P η ( T ) is expo nent ia l. W e now make some remarks ab out the b ehavior of P η ( T ) for η close to 1. The ﬁrst thing to be noted is that, for T ﬁxed, P η ( T ) is a contin uous function o f η , since it is a ﬁnite sum of contin uo us functions (see Sec. 3.2.2). This results, of cour se, imply that the contin uity ca nno t be uniform in T . This mea ns that, for any ﬁxed in ter v al of times [ T − , T + ], with T − in the range of v a lidit y of the pow e r -law asy mpto tes (3 3) of P 1 ( T ), P η ( T ) will be a rbitrarily close to the same p ower la w for η suﬃciently close to 1. F or times T ≫ T + , the decay b eco mes exp onential. W e shall see in the next section that for the case of unifor m densities, it is po ssible to estimate the v alue o f the crossover time to the exp onential b ehavior a s a function of η . 3.3. D ist ribution of quiesc ent times for the c ase of uniform densities In this section, we co nsider the distribution o f quiescent times for the sp ecial case of uniform densities dF ( u ) = dG ( u ) = du , for all u ∈ [0 , 1] and for a n y η ∈ [0 , 1]. In this ca se, simpler and implicit expr essions can be given for ˆ P ( s ) and P ( T ). After some tedio us but tr iv ial computatio n, we get fro m eq ua tion (28): ˆ P ( s ) = 1 1 + (1 − η ) γ ( s )  1 + γ ( s ) s − ηγ ( s )  , (39) where γ ( s ) is deﬁned by γ ( s ) = ln(1 − ηs ) η s . (40) The asymptotic b ehavior o f P ( T ) is determined by the singularity of the gen- erating function ˆ P ( s ) that is closest to the origin. F or η = 0 , the generating function has a simple pole ˆ P ( s ) = (2 − s ) − 1 , and therefore P ( T ) decays e xpo nentially , whic h agrees with the result o f the previous sectio n. In Fig. 3.3, we hav e presented the distribution of quiescent times P ( T ) in log-linear s cale for η = 0 . F or the in termediate v alues 1 > η > 0, the generating function ˆ P ( s ) has t wo singularities. One p ole, s = s 0 , c o rresp onds to the v anishing denominator 1 + (1 − η ) γ ( s ) , where s 0 is the unique non tr ivial solution of the equation − ln(1 − η s ) = s η 1 − η . (41) 17 –12 –10 –8 –6 –4 –2 ln P(T) 2 4 6 8 10 12 14 16 T Figure 2: Distribution of quiescent times f or the unifor m ly distributed v ar i ables x and y . P η ( T ) deca ys exponentially for η = 0, consisten tly with the analytical r esult P ( T ) = 2 − ( T +1) (solid line). Another singularity , s = s 1 = η − 1 , corr e spo nds to the v a nis hing argument of the lo g arithm. It is easy to see tha t 1 < s 0 < s 1 , so that the dominan t singularity of ˆ P ( s ) is of p olar type, and the c o rresp onding decay of P ( T ) is exp onential, with rate ln ( s 0 ( η )), for times muc h larg er than the crossover time T c ( η ) ∼ ln ( s 0 ( η )) − 1 . The results of Sec. 3.2.2 ab out the upp er b ound for the distribution P ( T ) allow us to b e mor e precise ab out this decay rate. In particula r, since B ( T ) ≤ A (1), it follows from (37) that P ( T ) ≤ η A (1) + (1 − η ) A (1) B (0) ( η + (1 − η ) A (1)) T − 1 (42) which in the case of uniform densities gives P ( T ) ≤ 1 2  1 + η 2  T . (43) W e hav e then 1 s 1 = η < 1 s 0 ≤ 1 + η 2 (44) and w e see that the rate ln ( s 0 ( η )) v anishes like 1 − η as η tends to 1. When η tends to 1, the tw o singula rities s 0 and s 1 merge. More precisely , we hav e ˆ P η =1 ( s ) = s + (1 − s ) ln(1 − s ) s 2 . (45) 18 The corresp onding dominan t term in (45) is of or der O( T − 2 ) [30]. This obviously agrees with the exact re sult w e get from equatio n (3 1), with dF ( u ) = dG ( u ) = du : P η =1 ( T ) = 1 ( T + 1 )( T + 2 ) . (46) In Fig. 3.3, w e ha ve drawn the distribution o f q uiescent times P η =1 ( T ) that exhibits the power-law decay , with the s lop e γ = − 2 . –16 –14 –12 –10 –8 –6 ln P(T) 3 4 5 6 7 ln(T) Figure 3: Di stribution of quiescen t times for the uniformly di stributed v ar iables x and y . W e sho w the p ow er-l a w deca y of P η = 1 ( T ) plotted in the log-log scale. The solid line is given by (46). In the ca se of uniform densities, it is p ossible to g et a n expre s sion o f P ( T ) for a ll times, and for any v alue of η , by applying the inversion formu la (26) to (39): P ( T ) = η T ( T +1)( T +2) + P T k =1 η T ( T − k +1)( T − k +2) k P k m =1  1 − η η  m c ( m, k ) , (47) where c ( m, k ) is deﬁned b y c ( m, k ) = m ! X l 1 + l 2 + ··· + l m = k l i ≥ 1 l 1 l 2 · · · l m − 1 l m ( l 1 + 1) ( k − l 1 ) ( l 2 + 1) ( k − l 1 − l 2 ) · · · ( l m − 1 + 1) ( k − l 1 − l 2 − · · · − l m − 1 ) ( l m + 1) . 19 When η 6 = 0 , ther e is an a lternative wa y of writing the previous expression: P ( T ) = η T ( T +1)( T +2) + P T k =1 η T +1 ( T − k +1)( T − k +2) P ∞ l =1 (1 − η ) l b ( l , k ) , (48) where b ( l , k ) is deﬁned by b ( l , k ) = X i 1 + i 2 + ··· + i l = k i j ≥ 0 1 ( i 1 + 1) ( i 2 + 1) · · · ( i l − 1 + 1) ( i l + 1) . In Fig . 3.3, we have plotted the distribution o f quies cent times P η ( T ) for the int e r mediate v alues η = 0 . 5, η = 0 . 7, η = 0 . 9, together with the ana lytical result (47). Figure 4: Distribution of quiescen t times for the uniformly distributed v ar iables x, y at the int ermediate v alues η = 0 . 5, η = 0 . 7, η = 0 . 9 (circles). F or comparison, the dotted li ne 2 − T − 1 presen ts the exponent i al deca y for η = 0, and the dashed li ne corresp onds to 1 / ( T + 1)( T + 2) for η = 1, s ee (46). The sol id li ne is given by (47) for η = 0 . 5, η = 0 . 7, η = 0 . 9. Note that in Fig. 3.3 and Fig. 3.3 (where η 6 = 1 ), we only plot distributions P ( T ) up to relatively short quies cent times ( T = 1 6, T = 25), since these times are already greater than the cross ov er time T c ( η ) ∼ 1 / ln( s 0 ) to the exp onential decay exp ( − ln( s 0 ) T ) ( s 0 deﬁned by e quation (41 )). F or muc h longer times, very few surviv als ar e obser ved, and the statistics g ets bad. Of co urse, T c ( η ) grows as the par ameter η tends to 1, so that we have go o d statistics for longer and lo nger times (in Fig. 3.3, for η = 1 , the plot is for quiescent times 20 ≤ T ≤ 2000). 20 A natural q uestion arising in this con tex t is ab out the relationship b etw een the ergo dic inv ariants that qua n tify the dyna mics of deterministic s y stems, for example the Lyapunov exp onents, and the scaling laws. The corr e s po nding question for mo dels of self–o rganized cr iticality is certa inly also p ertinent since in that case a relation is known b etw een the Lyapunov s pec trum and the trans- po rt pro per ties [31 ]. In our case, ho wever, because of the dynamical character not only of the state v ariable but als o of the threshold, some extensio n o f the deﬁnition of the inv aria nt s would b e needed, which is beyond the scop e of o ur discussion. 4. F at tail s in queuing systems In a simpliﬁed model of human activity , [33, 47, 48], it is view ed a s a dec ision based queuing system (QS) where tasks to be exe c uted arr ive randomly and accumulate before a server S . Under the prio rity-based s chedulin g rules, in which eac h incoming task is endow ed with a priority index (PI) indica ting the urgency to pro cess the job, the timing of the tasks follows fat tails probability distribution, (i.e the activity of the ser ver exhibits burs ts sepa rated by long idle per io ds with the ubiquitous Poisson behavior). There are t wo t yp es of dynamics: i). Servic e p olicies b ase d on ﬁx e d priority indexes . This c a se which is consid- ered in [3 3, 47, 48] a ssumes that the v alue a of the PI is ﬁxed once for all. Accordingly , very low pr iority jobs are likely to nev er be served. T o cir- cum ven t this diﬃculty [33, 47, 48] in tro duce a n a d-ho c probability factor 0 ≤ p ≤ 1 in terms of which the limit p → 1 co rresp onds to a deterministic scheduling strictly based on the PI’s while in the other limit p → 0 the purely rando m scheduling is in use. In this setting, the waiting time distribution (WTD) of the tasks b efore service is shown to asymptotically exhibit a fat tail b ehavior. The main po int o f the Bara basi’s c o nt ribution is to show that P I-based scheduling rules can alone generate fat tails in the WTD o f unpro cessed jobs. ii). Servic e p olicies b ase d on time-dep endent priori t y indexes . Here the prior- it y index is time-dep endent. This typically mo dels situa tio ns where t he ur gency to pr o c ess a t ask incr e ases with t ime and a ( t ) will hence be r e p- resented by increa sing time functions. Clearly , scheduling rules based on such a time-depe ndent PI do oﬀer new sp eciﬁc dynamica l features. They a re directly r elev ant in several contexts such as : a). Flexibl e manufacturing systems with limite d r esour c e . Here a sin- gle server is conceived to pro cess diﬀerent types of jobs but only a single type can b e pro duce d at a given time t (i.e. this is the limited 21 resource constraint). Accordingly , the basic problem is to dynam- ically schedule the pro duction to o ptimally match random demand arriv als for ea ch types o f items. The dynamic scheduling can b e opti- mally ac hie ved by using time-dependent pr iority indexes ( the Gittins’ indexes ) which spec ify in r eal time, which type o f pro duction to en- gage [40]. Problems of this t yp e b elong to the wider class referred as the Multi-Arme d Bandit Pr oblems in o per ations research. b). T asks with de ad lines . This situation, can be idealized by a queu- ing system where each incoming item has a deadline b efore which it deﬁnitely must b e proces sed, [43, 44, 39]. In this case , to be later discussed in the present paper , we can ex plicitly derive the lead-time proﬁle o f the waiting jobs obtained under several scheduling r ules, including the (optimal) time-dep endent priority rule known as the e arliest-de ad line-ﬁrst policy . c). Wa iting time-dep endent fe e db ack qu euing systems . In queuing net works, prio rity indexe s based on the waiting times can b e use d to schedule the routing through the netw ork. F or netw o rks with lo ops, such scheduling polic ies are able to g e nerate generically stable oscillations of the p o pula tions cont ained in the waiting ro om of the queues, [41]. In the context of QS, the waiting time pr ob ability distribution (WTD), (i.e. the time the tasks sp end in the que ue b efore b eing pro cessed) is a central qua n- tit y to characterize the dynamics. It strongly dep ends on the a rriv al and service sto chastic pr o cesses - in particular to the distributions of the int er-arrival and servic e time in ter v als. The ﬁrst moments of these dis tributions, e nable to deﬁne the traﬃc load ρ = λ µ ≥ 0 , (i.e. the ra tio b et ween the mea n servic e time 1 /µ and the mea n a rriv al time 1 /λ ) and cle a rly the s tability of elementary QS is ensured when 0 ≤ ρ < 1. F o cusing on the WTD, [33, 47, 48] emphas ized that hea vy tails in the WTD can hav e se veral origins, three of whic h are listed below: 1). the he avy tr aﬃc lo ad of t he s erver whic h induces large ”bursty” ﬂuctu- ations in b oth the WTD a nd the busy p erio d (BP ) of the QS. F or QS with feedback control driving the dynamics to heavy traﬃc loads, this a llows to generate self-orga nized critical (SOC) dynamics, [34] and the re sulting fat tails distribution exhibit a decay following a − 3 / 2 exponent. 2). the presence o f fat tails in the servic e time distribution pro duce fat tails of the WTD a prop erty which is here indep endent of the scheduling rule [36]. F or the conv enience of the reader , we give here a short review of these results. 3). prior ity index scheduling rules as dis c us sed in [33, 47, 48]. 22 In this se c tio n we pay the essen tia l a tten tio n to the case iii) but c o ntrary to the discuss io n c a rried in [33 , 4 7, 48], we shall here consider the dyna mics in presence of age-dep endent priority indexes . As it could hav e b een expe c ted, these aging mechanisms generate new b ehaviors that will b e ex plicitly discussed for t wo classes o f mo dels . 4.1. Waiting time distributions for queuing systems with fat tail servic e times Let us repro duce here a r esult recently derived in [36]. Theorem 4.1 (Boxma). Assume that the (r andom) servic e time in a M / G/ 1 QS is dr awn fr om a PDF with a r e gularly varying tail a t inﬁnity with index ν ∈ ( − 1 , − 2) , (r e gularly varying with index ν ∈ ( − 1 , − 2) ⇒ fat tail with index ν ∈ ( − 1 , − 2) ). F or this r ange of asymptotic b ehaviors of t he PDF, the ﬁrst moment β of the servic e exists. Assume further that the servic e is deliver e d ac c or ding to a r andom or der discipli n e. Then the waiting t ime distribution W R OS exhibits a fat tail with index (1 − ν ) ∈ ( − 1 , 0) and mor e pr e cisely, we c an write Pr( W R OS > x ) ∝ C ρ 1 − ρ h ( ρ, ν ) β Γ(2 − ν ) x 1 − ν L ( x ) , (49) wher e ρ < 1 is the tr aﬃc intensity, β the aver age servic e time, L ( x ) a slow ly varying fun ction and h ( ρ, ν ) = Z 1 0 f ( u , ρ, ν ) du, with: f ( u , ρ, ν ) = ρ 1 − ρ  ρ u 1 − ρ  ν − 1 (1 − u ) 1 / (1 − ρ ) +  1 + ρ 1 − ρ  ν (1 − u ) 1 / (1 − ρ ) − 1 . The fat tail b ehavior given in (4 9 ) is therefore entirely inher ited fro m the fat tail b ehavior o f the ser v ice and is not aﬀected by any reduction o f the tr a ﬃc int e ns it y ρ . Note a lso that change of the scheduling rule canno t get rid of this fat ta il b ehavior. This p oint can b e explicitly obs e rved in [3 8, 45], who show that for the previous M /G/ 1 Q S with a random order service (ROS) servic e discipline, one obtains: Pr( W R OS > x ) ∝ x →∞ h ( ρ, ν ) · Pr( W FC FS > x ) , (50) from which we directly observe that the fat tail in t he asymptotic b eha vior in not alter e d by a change of the sche duling rule . Note ﬁnally that for the M / M / 1 QS, (i.e. exp onential s e rvice distributions and hence no fat tail), [42] shows that the r andom orde r ser vice scheduling rule yields: Pr( W R OS > x ) ∝ x →∞ C ρ x − 5 / 6 e − γ x − δx 1 / 3 , (51) 23 with C ( ρ ) = 2 2 / 3 3 − 1 / 2 π 5 / 6 ρ 17 / 12 1 + ρ 1 / 2  1 − ρ 1 / 2  3 exp  1 + ρ 1 / 2 1 − ρ 1 / 2  , and γ =  ρ − 1 / 2 − 1  2 and δ = 3 h π 2 i 2 / 3 ρ − 1 / 6 , which has to b e co mpared w ith the FCFS sc he duling discipline, which for the same M / M / 1 QS reads as, [38]: Pr( W FC FS > x ) = 1 β (1 − ρ ) e − (1 − ρ ) x/β . (52) While the detaile d b ehaviors given by (51) and (52)clearly diﬀer, they how ever bo th share, in accord with [3 3], an e x po nent ia l deca y . 4.2. S che duling b ase d on t ime dep endent priority indexes The most naive approach to discus s the dynamics of QS w ith sc heduling based on time-dep endent priority indexes is to think of a p opulatio n mo de l in which the mem be r s suﬀer aging mec ha nisms whic h ultimately will kill them. Naively , we may co nsider the p opulation of a cit y in which mem be rs are either b orn in the city or immig r ate int o it at a c e r tain age a nd ﬁnally die in the city . Assuming that the death pr o bability dep ends on e ach individual age , the study of the age str ucture of the p opulation exhibits some of the salient features of our original QS. First, we dis cuss this cla ss of mo dels a nd then r eturn to the or iginal mo del of L. Ba rab´ asi [3 3] to consider a simple QS whe r e e ach task waiting to b e pro c essed carries a deadline (playing the role of a PI) and as time ﬂows the these dea dlines steadily r educe - this implies a ( time dep endenc e of the PI ). A t a given time, the sc heduling of the tasks follows the ”earlies t- deadline-ﬁrst” (EDF) p olicy and given a queue length conﬁguration, w e shall discuss the lead-time (lead-time = deadline - current time) proﬁle of the tasks w a iting to b e ser ved. 4.2.1. T asks p opulation dynamics with t ime dep endent priority indexes Consider a p opulation of tasks waiting to b e proc e s sed by S with the fol- lowing characteristics: i). An inﬂow of new tasks steadily enters int o the queuing system. Ea ch tasks is endow ed with a pr iority index (PI) a which indica tes its degr ee of urg e ncy to be pro cessed. In general, the tasks are heterogenous as the PI are diﬀerent. In the time interv al [ t, t + ∆ t ], the num be r of incoming jobs exhibiting an initial PI in the in terv al [ a, a + ∆ a ] is characterize d by G ( a, t )∆ t ∆ a . 24 ii). Co n tr ary to the situations discuss e d in [33], an ”ag ing” pr o cess directly aﬀects the ur gency to pro ces s a given task. In other words the priority index a is not frozen in time but a = a ( t ) mono to nously incr eases with time t . F or an inﬁnitesima l time incre a se ∆ t , in the simplest case w e shall hav e a ( t + ∆ t ) = a ( t ) + ∆ t. Here we s lightly generalize this and allow inhomogeneous aging rates writ- ten as p ( a ) > 0 meaning that a ( t + ∆ t ) = a ( t ) + p ( a )∆ t. iii). The s cheduling p olicy dep ends on the PI o f the tasks in the queue and we will fo cus on the natural policy ” pr o c ess t he highest PI ﬁrst” . iv). at time t , a sca lar ﬁeld M ( a, t ) co un ts the num b er of w aiting tasks with priority index a . Hence M ( a, t )∆ a is the num b er with PI p ( a ) ∈ [ a + ∆ a ]. Accordingly , the total w o rkload facing the hum an server S w ill be given, at time t by: L ( t ) = Z ∞ 0 M ( a, t ) da. (53) v). In the time interv al [ t, t + ∆ t ], the se rver S pro cesses tasks with an a - dependent r ate µ ( a )∆ t . T ypically µ ( a ) could be a monotonous ly increasing function o f a . As the service rate µ ( a ) explicitly depend on the PI a , it therefore plays an eﬀective role of service discipline. The previous elemen tary h yp otheses imply an evolution in the for m: M ( a + p ( a )∆ t, t + ∆ t )∆ a ≈ M ( a, t ) ∆ a − µ ( a ) M ( a, t )∆ a ∆ t + G ( a, t ) ∆ t ∆ a. Dividing b y ∆ a ∆ t , we end, in the limits ∆ a → 0 and ∆ t → 0 , with the scalar linear ﬁeld equation: ∂ ∂ t M ( a, t ) + p ( a ) ∂ ∂ a M ( a, t ) + µ ( a ) M ( a, t ) = G ( a, t ) . (54) It is worth to re ma rk that the dynamics given b y (54) is closely related to the famous McKendrick’s age structured p opulation dynamics, [37]. Assuming stationarity for the inc o ming ﬂo w of task s (i.e. G ( a, t ) = G s ( a )), the linearity of (54) enables to explicitly write its stationary solution as: M ( a ) = π ( a )  C + Z a 0 G s ( z ) p ( z ) π ( z ) dz  , (55) 25 where π ( z ) = exp  − Z z µ ( y ) p ( y ) dy  , (56) with a n integration constant C < ∞ re ma ining yet to b e determined. Assume that the P I attac hed to the incoming jobs do not exceed a limiting v alue T , namely: G ( a, t ) = I ( a < T ) ˆ G ( a, t ) ⇒ G s ( a ) = I ( a < T ) ˆ G s ( a ) , (57) where I ( a < T ) is the indicator function. In o ther words (57) indica tes that the new coming jobs do no t exhibit arbitrarily high PI’s. This enables to deﬁne: Ψ( T ) = Z ∞ 0 G s ( z ) p ( z ) π ( z ) dz = Z T 0 ˆ G s ( z ) p ( z ) π ( z ) dz (58) and (55) r eads as: M ( a ) = ( π ( a ) h C + R a 0 ˆ G s ( z ) p ( z ) π ( z ) dz i if a ≤ T π ( a ) [ C + ψ ( T )] if a > T . (59) The asymptotic b ehavior of M ( a ) for a → ∞ is en tirely due to π ( a ), (the squar e brack et terms are bounded by co nstants) and therefore (56) and (59) imply: M ( a ) ≈ π ( a ) ≈ ( e − k/q a q when µ ( a ) p ( a ) ∝ k a q − 1 , q > 0 , a − k when µ ( a ) p ( a ) ∝ k /a , (60) In view of (60 ), the following alternatives o ccur: a). F or q < 0 , in (6 0 ), the integral R ∞ 0 M ( z ) dz does not ex ist. In this cas e an e ver growing populatio n of tas ks accumulates in front of the server and the queuing pro cess is explo ding. b). F or q > 0, a stationary regime exis ts and in this case the constant C < ∞ in (59) can b e determined by solv ing : Z ∞ 0 G s ( z ) dz = Z ∞ 0 M ( z ) µ ( z ) dz , (61) which ex pr esses a globa l bala nce betw een the statio na ry incoming and out going ﬂows of tasks. c). F or q = 0 whic h implies that µ ( a ) p ( a ) ∝ k a , 26 (60) pr o duc es an exp onent- k fat tail distribution for M ( a ) c ounting the numb er of waiting tasks with PI a in the system. F or T < ∞ and a → ∞ , the fat tail of M ( a ) is p opulated by lo ng w a iting tasks i.e. those having sp en t more tha n a − T waiting ins ide the s ystem b efore b eing served. In the limiting case, for whic h µ ( a ) = µ = const and p ( a ) = a (i.e. aging directly pr op ortional to time) which leads to q = 0 in (60), the densit y M ( a ) coincides directly with the WTD for a → ∞ . This p opulation mo del shar e s several fea tures with the Bar ab´ asi’s mo del [3 3], namely: a). When a stationa r y regime ex ists, the function ˆ G s ( a ) which here plays the ro le of the initial PI dis tr ibution in [33, 47, 48], do es not aﬀect the tail behavior given by (6 0). b). The s cheduling rule here is implicitly g ov erned by the s ervice rate µ ( a ) which itself de p end on time as the PI a = a ( t ) a r e time-dependent. Note that µ ( a ) directly inﬂuences the a symptotic b ehavior of (60). In particular for case c), the tail exp o nent explicitly dep ends on µ ( a ). Besides the similarities, we no w also p oint out the impo rtant diﬀerences b etw een the present p opulation mo del and the model dis c us sed in [33, 47, 48]: a) the servic e is not res tricted to a sing le task a t a giv en time (i.e. the service r esource is not limit ed). Indeed µ ( a ) describ es a n av era ge ﬂo w of service and hence s everal tasks can b e pro cessed simultaneously - (in the cit y p opulation mo del the s ervice corresp onds to death and several individual may die simultaneously). b) while the fat ta il in [33, 47, 48] is en tir ely due to the scheduling rule and therefore o ccur s ev e n fo r QS far from tra ﬃc saturation, this is not so in the p opulation model. Indeed in this last ca se, fat tails are due to heavy traﬃc lo a ds o ccurring when the ﬂow of inco ming tasks nea rly satura tes the server, (this is implied b y q = 0 in(60 )) - for lower lo ads o ccur ring when q > 0 the fat tail in (60) disa ppe ars. 4.2.2. S to chastic dynamics. R e al-time qu eu ing dynamics In this sectio n w e will use the results of the rea l-time queuing theory (R TQS), pioneered in [43], to explore situations where the inco ming jobs hav e a deadline - this problem is already suggested in [3 3]. Based on [43, 44, 32] and [39], ﬁrst r ecall the basic hypotheses and the relev ant results of R TQS’s. Consider a general single s e rver QS with arriv al and ser vice being descr ibed by indep endent renewal pro ce sses with av e rage 1 /λ resp ectively 1 /µ and ﬁnite v ariances for b oth 27 renewal pro ces s es. Ea ch incoming task ar rives with a random har d time relative deadline D drawn from a P DF G ( x ) with a density g ( x ): Pr { 0 ≤ D ≤ x } = G ( x ) , with av erag e hD i : hDi = Z ∞ 0 (1 − G ( x )) dx = Z ∞ 0 xg ( x ) dx. A t a given time t , we deﬁne the lead time L to b e given by: L = D − t, (62) Assume now that the lead time L pla ys the role of a prio rity index and the service is delivered by using the e arliest-de ad line-ﬁ rs t (EDF) rule with preemp- tion (i.e. the server alwa ys pro cesse s the job with the shor test le ad time L ). Preemption implies that whenev er an incoming job exhibits a s horter L than the one currently in service, this incoming job is proces sed b efor e, (i.e. pr e- empts ), the curre ntly engag ed task which service is po stpo ned. The EDF rule directly corresp onds to the deterministic p o licy (i.e. p = 0 , γ = 0 in the original Barab´ a s i’s contribution [33]. A t a given time, o ne can deﬁne a pro bability distribution cor r esp onding to the le ad time pr oﬁle (L TP), F ( x ) = P r {−∞ ≤ L ≤ x } , of the jobs waiting in the QS. The L TP sp eciﬁes the repartition o f tasks having a giv en L at time t . Knowing the queuing p opulation Q at a given time, it is shown in [39] that for hea v y traﬃc regimes , the L TP can, in a ﬁr st o rder approximation s cheme, express ed b y a simple analy tical form. Indeed, following [39], let us deﬁne the fron tier ˆ F ( Q ) > 0 a s the unique so lutio n o f the equation Q λ = Z ∞ ˆ F ( Q ) (1 − G ( x )) dx, x ∈ [ l , ∞ ) ⊂ R + . (63) Let us also deﬁne the fro n tier F ( Q ) as F ( Q ) = ( ˆ F ( Q ) when Q ≤ Q ∗ ,  hDi − Q λ  ≤ l when Q > Q ∗ , (64) with Q ∗ being deﬁned b y ˆ F ( Q ) = l where l deﬁned in (63). In [39], it is sho wn that tw o alternative reg imes can o ccur: a) Jobs serve d b efor e de ad line . When hD i > Q/λ and therefor e ˆ F ( Q ) = F ( Q ) , the L TP cumulativ e distribution F ( x ) takes the form (see Fig. 5) F ( x ) =    0 when x < F ( Q ) , 1 − λ Q R ∞ x [1 − G ( η )] dη  when 0 < F ( Q ) ≤ x. (65) 28 Figure 5: Density of the lead tim e proﬁle f ( x ) = dF ( x ) /dx when F ( Q ) > 0. b) Jobs serve d after de ad line . When F ( Q ) = ( hD i − Q/λ ) ≤ 0 , the L TP cum ulative distribution F ( x ) takes the for m (Fig. 6 ) F ( x ) =      0 , when x < hD i − Q λ < 0 , h 1 + λ Q ( x − hDi ) i Q − λ hDi Q − λ hDi + λl , when hD i − Q λ ≤ x < l , 1 − λ Q R ∞ x [1 − G ( η )] dη  , when l ≤ x. (66) Figure 6: Density of the lead tim e proﬁle f ( x ) = dF ( x ) /dx when F ( Q ) < 0 . Remark . The alternative regimes given by (65) and (66) c a n b e heur istically understo o d by invoking the Litt le law whic h connects the av erag e q ueue length h Q i with the av erag e waiting time h W i , [38], h Q i = λ h W i , (67) a result independent of the sch eduling policy . In v iew of (63) and (6 7), one obviously susp e cts that the LT P strongly depends on the sign of the diﬀerence hDi − h Q i λ = hD i − h W i . Int uitively , when h W i exceeds h D i , it is exp ected, in the av erag e, tha t pro c e ssed jobs will b e delivered to o late and conv er sely . While the ab ov e heuristic ar g u- men ts is strictly v alid o nly for the av erag es, [39], were able to sho w that in heavy traﬃc regimes, it also holds a lso for the L TP given in (65) and (66). Assuming that the arriving tasks hav e p ositive deadlines, the L TP as given by (65) and (66) imply: 29 a). If the left-hand supp or t o f the L TP is neg ative, then a job entering int o service is already late, (ca s e of (66)) se e Fig. 6 . b). If the left-hand suppo rt of the L TP is po sitive then a jobs enters int o service with a p os itive lea d time, (case o f (65)) see Fig. 5. Accordingly , it is likely that the tasks will be completed befor e the de a dline expired. c). The critical v alue Q ∗ = hD i / λ for whic h F ( Q ) = 0 , cor resp onds to a queue length for which custo mer s are likely to b ecome late. Choo sing Q exactly to Q ∗ , w e ca nno t ex pec t la teness to dis a ppe a r completely but for Q < Q ∗ lateness will b e strongly reduced a b ehavior clea rly co nﬁrmed by simulation exp eriments [32, 39]. d). F or dea dline distributions G ( x ) with fat tails, it follows immediately from (65) and (66) that the L TP do es po ssess a fat tail. 4.2.3. ” First c ome ﬁrst serve d” (FCFS) sche duling p olicies Cho osing the de a dline probability density as g ( x ) = δ ( x ), (i.e. zero dead- line), the EDF s chedulin g p o licy directly coincides with the FCFS rule. F or this case w e have Q ∗ = 0 , and (64) implies F ( Q ) =  ˆ F ( Q ) = 0 , when Q ≤ 0 , − Q λ , when Q > 0 . (68) Hence the L TP density is giv e n by (65) is mere ly the uniform probability densit y U [ − Q/λ, 0], ([ − Q/λ, 0] b eing its suppor t). This expres ses the fact that in the heavy traﬃc regime ρ = λ/µ ≈ 1 , the waiting time behav es a s Q × (1 /µ ) ≈ Q × (1 / λ ) leading to a LPT linearly growing with Q . F or gener al G ( x ), the L TP asso ciated with a FCFS scheduling rule will b e given by the c o nv olution of the deadline distribution G ( x ) with the unifor m distribution U [ − Q/ λ, 0]. Indeed, adding the task deadlines with the time spent in the queue, we recov er the tasks lead-time. Accordingly , in the heavy traﬃc regime and for a given queue length Q , one explicitly knows the L TP’s for both the EDF and the FCFS scheduling po licies th us enabling to explicitly appreciate their respective characteris tics. In particular, using (65) a nd (66), one can conclude that for a given queue length Q , with the FCFS scheduling rule a nd the a sso ciated L TP F ( x ) b eing the conv olution of G ( x ) with the U [ − Q/λ, 0], w e o btain F ( x ) =            0 when x < − Q /λ, λ Q R x − ( Q/λ ) [ G ( ξ + Q/λ )] dξ when − Q/λ ≤ x < 0 , κ + λ Q  R x 0 [ G ( ξ + Q/λ ) − G ( ξ )] dξ  when x ≥ 0 , (69) 30 where the constant κ reads as: κ = λ Q Z 0 − ( Q/λ )  G ( ξ + Q λ )  dξ . The latter equation allows to emphasize the following features: i). When the left-hand supp ort o f the dea dline distribution G ( x ) is larg er than Q/λ , the left b oundar y of the suppor t of F ( x ) is la rger than 0 and therefore the jobs exper ience no delay when ent e ring in to service. ii). If the left-hand s uppor t of G ( x ) is smaller than Q/λ , then it may happ en that the L TP exhibits a negative left-hand suppo rt under the F CFS p olic y and a p ositive left-hand s uppo rt under the EDF scheduling rule. Hence in this la st s ituation, the FCFS p olicy would deliver tasks with lateness while the EDF tas k s will b e pr o cessed in due time. This explicitly conﬁrms in tuition that EDF is indeed an eﬃcient p olicy . It ha s bee n s hown that the EDF scheduling rule is o ptimal fo r minimizing the nu m be r of jobs pro cessed after the dea dline [46]. iii). If G ( x ) exhibits a fat tail for x → ∞ so has the L TP and this whatever the s cheduling r ule in use. This ca n e directly v er iﬁed from (69) by studying the L TP density f ( x ) = dF ( x ) /dx for x → ∞ , w e hav e: f ( x ) = λ Q  G ( x + Q λ ) − G ( x )  , for x → ∞ , which when G ( x ) ∼ 1 − x − q and for Q/λ < const takes the for m f ( x ) ∼ x − ( q +1) , for x → ∞ . (70) Hence, the L TP inherits the fat tail prop er t y of G ( x ) and this even when using the optimal EDF scheduling rule. Below, we fo cus on a fully explicit illustra tion inv olving the Pareto proba - bilit y distribution G ( x ) =    0 when x < B , 1 −  B x  ( ω − 1) when x B ≥ 1 , ω > 1 , (71) which has no moment of order ω − 1 or higher . F or ω > 2, w e hav e hDi =  ω − 1 ω − 2  B . 31 Using (64) with l = B , which implies Q ∗ = λB ω − 2 , we obtain F ( Q ) =    B  B λ Q ( ω − 2)  1 / ( ω − 2) , when Q ≤ λB ω − 2 , h ω − 1 ω − 2 i B − Q λ , when Q > λB ω − 2 . (72) Using (65) and (66), we can ﬁnd that the L TP distribution reads as Q ≥ B λ ω − 2 ⇒ F ( x ) =              0 , when x ≤ F ( Q ) , 1 − λ Q  ω − 1 ω − 2 B − x  , when F ( Q ) ≤ x < B , 1 − B λ Q ( ω − 2)  B x  ω − 2 . when x ≥ B . (73) Q < λB ω − 2 ⇒ F ( x ) =    0 , when x ≤ F ( Q ) , 1 − B λ Q ( ω − 2)  B x  ω − 2 . when x > F ( Q ) . (74) The latter equations descr ibe a fat tail with the exp onent ω − 2 . It is worth to men tion that (74) implies that for ω > 2 and for Q λ < B ω − 2 , the EDF sc heduling p olic y part of the tas ks enter into the service b efore the due date expired. Finally no te a lso, that for ω ≤ 2 , no momen ts exists for the deadline distribution and hence the theor y [39] cannot be applied directly . W e conjecture that for these regimes no scheduling rule will b e able to deliver tasks in due time. The results o btained for the L TP , enable us to in vestigate the asymptotic prop erties of the waiting time distribution. Indeed, assume a heavy traﬃc regime with the EDF sc heduling p olicy . Let us also supp ose that for a giv en queue length, some jobs a re served to o late (i.e., the left bounda r y of the L TP is nega tive). As under the EDF rule, the mo re ur gent jobs are alwa y s served ﬁrst, the waiting times of the last jobs in the queue necessarily exceed their deadlines. Therefo re, when the deadline distribution exhibits a fat tail, so will the WTD dis tribution. Note that while the EDF p olicy decreases, compared with the FCFS rule, the num b er of jo bs ser ved after their deadline, it cannot get rid of the fat tail o f the WTD, which is due to the fa t tail of G ( x ). This result is fundament a lly diﬀeren t fr o m the s ituation that is v alid for the frozen in time P I mo dels discussed in [33, 47, 4 8], where the fat tail b ehavior do es no t 32 depe nd on G ( x ) itself. This can b e heuristically understo o d as, in [33, 47, 48], the fat tail is mainly due to the low prior it y jobs , which, a s no a ging mechanism o ccurs, ar e likely to never b e served. Note that in [33, 47, 48], stable queuing mo dels (i.e., tho s e for which the tra ﬃc) [33] and fat tails of the WTD disapp ear under a FCFS scheduling rule. Indeed witho ut prio rity scheduling, the WTD alwa ys follows a n exp onential asymptotic deca ying behavior. In the presence of time-dependent PI, all tasks do ﬁnally acquire a high priority and this aging mechanism precludes the formatio n o f a fat tail s olely due to the sc heduling rule. Accordingly , in the presence of aging PI, the app eara nce of WTD with fat tails is due to G ( x ). 4.3. The imp ortanc e of adopting p erformanc e sche duling p olicies The re s ults for the L TP deriv ed in the prec eding section ca n b e directly measured on the actua l Q S. Consider the queue conten t of a single-stag e QS. Assume that at a given time, Q is the observed queue c o nt ent, and at this instant take a snapshot of the lead time asso ciated with each waiting item and construct the asso ciated L TP (i.e., the his togram of the obse rved lead times). In heavy traﬃc regimes (i.e., typically 0 . 95 ≤ ρ < 1 leading to stationar y av er a ge queue lengths ρ/ (1 − ρ )) and under the EDF scheduling p olicy , the L TP will approximately b e g iven b y (65,66). Actual s im ula tion ex per iment s a re rep orted in [43, 4 4, 39] and [32], wher e an ex cellent a greement b etw een measured data and theory is o bserved. F rom the human activit y viewp oint, the explicit expressio ns of the L TP obtained b oth for the FIF O and E DS p olicies show clearly that or ganizing the work s cheduling is ex tremely impor tant. As an illustra tio n, cons ider a situation in which the deadline distribution G ( x ) follows an exp onential law: G ( x ) = 1 − e − αx ⇒ hDi = 1 α . (75) F or this situation, we compare tw o diﬀerent org anization p olicies: 1. EDF sche duling p olicy . Int ro ducing (75) in to (63), we obtain F ( Q ) = ( − 1 α log  αQ λ  when Q ≤ λ α , 1 α − Q λ when Q > λ α . (76) When F ( Q ) > 0 (i.e., the upper line in (76)), it follows from (65) that F ( x ) =      0 when x < F ( Q ) , 1 − λ (1 − αx ) αQ when F ( Q ) ≤ x < 0 , 1 − λ αQ e − αx when 0 ≤ x. (77) 2. FIFO sche duling p olicy. With G ( x ) given by (75), the result given in (69) rea ds as F ( x ) =          0 , x < − Q λ , 1 + λx Q − λ αQ  1 − e − α ( x + Q λ )  , − Q λ ≤ x < 0 , 1 − λ  1 − e − αQ λ  αQ e − αx , x ≥ 0 . (78) 33 Comparing (77) and (78), we conclude that in a heavy tr a ﬃc reg ime, for a given work loa d Q , the use of EDF ena bles us to pro cess tasks in due time with a high pr obability while the naive FIFO policy gener a tes lar ge delays. Spec iﬁc a lly , when Q < λ/α , the EDF p olicy guara nt ees that most jobs enter int o service b e fo re the deadline (see (77)) and will therefore be served b efore deadline, with a hig h probability . On the contrary , the FIFO policy res ult given in (78) (i.e., obta ined fo r x = 0 in the last line o f (78) shows that a pr op ortion of 1 −  λ  1 − e − αQ/λ  /αQ  jobs enter the service with delays and will ther efore be la te. As far as human res ources ar e conce r ned, this s imple mo del ena bles us to quantify the impo rtance of adopting p erformance scheduling polic ies to r esp ond to the burn out – generating c ha llenge: deliver mor e in less time with fewer r esour c es . Along the same lines, o ne of the key rules to av oid burnout is to le arn to say no to new incoming tasks if the q ue ue length ex ceeds a threshold. In our mo deling framework, the c ritical threshold do e s dep end closely on the level Q ∗ , ab ov e whic h lately ser ved tas ks (and hence c o mplaints) are unav oida ble. 5. Po w er law distributions in Sel f–Organized Criticality In the la st t wo deca des, a meticulous attention has b een drawn to the phe- nomenon o f self–or ganize d critic ality (SOC), a prop er ty of dynamica l s ystems which ha ve a critical p oint as an attractor , [27, 50, 4 9, 5 1]. A notable fea- ture of these mo dels submitted to a p ower law statistics is that they have no characteristic scales, simila rly to the s cale inv ar iant systems b e ing in a critical state. How ever, unlike s ystems tackled by the critical phenomena theory the critical state in the SOC mo dels seems to be an a ttractor of the dynamics and seems to be achiev e d without any tuning of control para meters. It is observed in slowly-driven no n-equilibrium systems with extended degrees of freedo m and a high lev e l of nonlinearity . The gener a l idea behind SOC mo dels is very app eal- ing. Consider for instance Z hangs sandpile mo del on Z 2 , where eac h site has an energ y v aria ble which e volves in discrete time-steps according to a simple ”toppling” rule: If a v aria ble exceeds a thresho ld v alue, the excess is distributed equally among the neighbors. The neighbor ing sites may th us turn sup ercr itical and the pro cess contin ues until the excess is ”thrown ov erb oa rd” at the system bo undary . What makes this dynamical rule intriguing is that if the toppling is initiated from a highly excited state, then the termina l s tate (i.e., the state where the toppling stops) is not the most s ta ble sta te, but one o f many least-stable , stable states. Moreov er, the latter sta te is critical in the sense that further insertion of a small excess typically leads to further larg e-scale events. Using the sa ndpile analogy , such even ts are referred to as av alanches. F rom the very b eginning, larg e theor etical eﬀorts ha ve b een made in order to understa nd a true relatio n b etw een criticality and se lf-o rganized criticality bo th exhibiting a p ow er law b ehavior [52, 53]. In par ticular, the use of v arious r enormalization gr oup te chniques (RG) which pr ov ed their exceptiona l eﬃciency 34 in justifying s caling prop er ties in the critical pheno mena theor y [54] has b een in the fo cus of ma n y studies devoted to the SOC pheno mena , [55, 56, 5 7, 58, 59, 60, 61]..This still deserves a thorough investigation a s a potential candidate for the ”SOC phenomena theory”. In the critical phenomena theory , the RG metho d usually helps to establish the lo ng time and large scale asy mptotic b ehavior in inﬁnite systems deﬁned by the sto chastic diﬀerential equations with the Gaussian distributed external random force [62] that models random boundar y conditio ns . R G is an eﬀective metho d of s tudying self-similar scaling b ehavior in such s ystems. On the con- trary , the ma jority o f mo dels ex hibiting SOC phenomena are deﬁned on a ﬁnite piece L of a discrete lattice Z d by disc r ete time dynamical rules [27, 49]. Mo re- ov er, in SOC models the energy is usually dis sipated at the op en boundar ies of the lattice piece, while, in the ma jority of cr itical phenomena, a quenched distribution of absorbing defects through the lattice is impose d. The primal goal for the SOC phenomena theor y is to inv estigate the sca ling prop erties of SOC mo dels , in par ticula rly , to justify the numerically o bserved r e- sults [63, 64] on the p ower law distribution o f av alanc he s izes in sa ndpile mo dels int r o duced b y Bak, T ang and Wiesenfeld, [2 7]. In more gener al fo rmulation, the ﬁnite s iz e scaling (FSS) hypothesis [65, 6 6] is us ually a s sumed in SOC sy stems, P ( x, L ) = x − τ x F  x L σ x  , x ≡ { s, t, a } , (79) where P ( x, L ) is the probability distribution of occur rence o f an av a la nche o f a given size s (the num b er of sites in volved in a relaxatio n pro cess ), area a, a nd time t , L is the size of the lattice piece. If the FSS Ansatz (7 9 ) is v alid, then the dynamical exp onents τ x and σ x determine the universalit y class of the mo del [27]-[51]. Within the fra mework o f numerous mo dels , the dynamical exp onents τ x and σ x are found from the diﬀerent phenomenolog ical appro a ches, which are lo osely r elated to underlying microscopic mo dels, and therefore some doubt remains abo ut the universality of represe ntations such as (79), [67, 68, 69]. T o iden tify the s c ale inv a riant dyna mics, the re a l-space RG method had been applied to the cellular automaton deﬁned on a 2-dimensional lattice[55, 56]. This approach ( Dynamic al ly D riven R enormalization Gr oup (DDR G)) dea ls w ith the critical prop erties of the system b y in tro ducing in the renorma lization equa tions a dyna mica l steady state condition which assumes non-equilibrium stationa ry statistical weigh ts to b e used in the calcula tion [58]. The ﬁxed p o ints of sca l- ing tra nsformations deﬁne the dynamical exp one nts whose v alue are in a go o d agreement with computer simu lation data. Nev ertheles s, it has b een shown [55] that the ﬁxed points related to these dynamical ex po nen ts pr escrib ed by the renormaliz a tion g roup are not accessible form the ph y sical doma in of para meter v alues. An a lternative approach referring to the standard critical phenomena the- ory is bas ed o n the coa r se-gra ining of micr oscopic evolution rules for the SOC mo dels. In [7 0, 71] a contin uous s to chastic pa rtial diﬀerential equation related to the randomly driven mo dels had b een pro p o s ed althoug h the threshold na - ture of the SOC pheno mena was not taken into account. A sto chastic pa rtial 35 diﬀerential equation sub jected to a threshold conditio n and driven by a Ga us- sian dis tributed externa l ra ndom force acting contin uo usly in time has been discussed in [57, 72, 7 3]. Therein the external r andom force intro duced in to the dynamical equation simult aneously mo dels: ﬁrst, the noise risen due to eliminatio n of microsco pic degrees of freedom; second, unknown (or undeﬁned) b oundary conditions; third, a mechanism injecting energy into the sy stem which is supp ose d to act contin u- ously in time, br eaking the time sca le sepa r ation, and co uld provok e av alanches to ov er la p. The threshold condition is taken into account by the Heaviside step function θ ( x ). This s tep function had bee n r egularized as a limit o f contin uous inﬁnitely diﬀerentiable functions and then expa nded int o p ow er ser ies giving rise to an inﬁnite series of nonlinearities in the sto chastic diﬀerential equation [57]. Solutions of this nonlinea r partial diﬀer en tial equation could b e found b y iterating in the nonlinearities follow ed b y av erag ing over the distribution of the random force . Then, the lo ng time large scale a symptotic behavior of the solutions could b e established b y means o f a dynamic R G pro c edure in the spirit of the dyna mic RG -approa ch [6 2]. Particularly , the v alues of dy namical exp onents could be found in the form of p ow e r series in ε = 4 − d . How ever, due to an inﬁnite num b er of nonlinearities risen in the sto chastic diﬀeren tial eq ua tion by the p ow er expansio n o f step function, the resulting theor y calls for an inﬁnite nu m be r of charges (coupling c onstants) and, therefor e , c a nnot be a nalyzed in the framework of the standar d dynamic RG sc heme. In [57], only the ﬁr st tw o nonlinear terms hav e been kept for the RG analys is of the appro priate sto chastic problem. All higher order terms had b een neglected without any es timation of their con tr ibutio ns to the long time large scale asymptotic behavior. Le t us note that the s ta ndard p ow er co unt ing analysis of suc h a mo del shows co nvincingly that a ll these terms are of equal imp ortanc e fo r the asymptotic b ehavior and all of them ha ve to b e taken in to considera tion on equal fo oting. It is imp ortant to emphasiz e tha t the cor resp ondence b etw een the mo dels of deterministic dyna mics and the ab ove sto chastic pr oblem is indee d question- able a nd usually lays beyond the s tudies. The obvious adv an tage of suc h a coarse gr aining approa ch is that it allows to use the mo dern critical phenom- ena techniques of analy sis ac hiev ing impr essive results on the self-similar scaling behavior. Here, w e present the results [59, 61] on the lo ng time large scale asymptotic behavior for the mo del based on the nonlinear s to chastic dynamics equation de- rived from the coarse-g raining pro ce dur e fr o m the discrete rules of deter ministic dynamics holding all no nlinear ter ms in c heck. This task is highly non tr ivial and of suﬃcient interest itself s tim ula ting further developments in mo dern critical phenomena theory [54]. 5.1. Co arse-gr aining of micr osc opic evolution rules for SO C–mo dels Recently , tw o randomly driv en SOC mo dels prop osed b y Zhang [4 9] and by Bak et al. [2 7] (BTW) hav e b een connected to sto chastic diﬀerential equa tions [57]. F or the c o nv enience of the reader , we brieﬂy desc rib e the micr oscopic rules 36 of these SOC mo dels. Both models are deﬁned on a ﬁnite piece of d -dimensional lattice L ⊂ Z d in which a ny site i ∈ L can store some c o nt inuously dis tributed v ariable E i usually called energy [81]. F or Zhang’s mo del, the system is p ertur bed by a r andom amount of energy δ E > 0 at a r andomly chosen site i ∈ L . Once the v alue E i exceeds a given threshold v alue E c , this site b ecomes activ e, and transfers all energy to the nearest neighbo rs. As a result, the neigh b o r ing sites can b e als o activ ated a nd transfer ener gy to the next neighbors, etc. until it is absor b e d at the op en bo undary ∂ L . The av alanche ends when all sites reac hed a v alue of energy smaller than E c . The next energy input in to the system o ccurs only when the av ala nche has stopp ed. F or the BTW mo del, the a mount of energy p ertur bing the system is ﬁx ed δ E = E c /q where q is a coo rdination n umber, and the amount of energy trans- ferred to neigh b or s fro m an activ e site is a lso ﬁxed at E c . F or b oth mo dels, energy is pumped in to the system at the small-scale of lattice spacing a a nd then is transfer red to a large scale compara ble to the size of L a nd actively dissipated at the op en b o undaries. F or ea ch i ∈ L , the micr oscopic evolution rule s can b e wr itten in the form of a st o chastic c ouple d map lattic e (SCML), E i ( t + 1 ) − E i ( t ) (80) = 1 q X mm [( χE mm ( t ) + E c ) θ ( E mm ( t )) − ( χE i ( t ) + E c ) θ ( E i ( t ))] + ζ i ( E , t ) , in whic h E i ( t ) is the exceed of ener gy o ver the c r itical v alue E c , χ = 1 for Zhang’s model a nd χ = 0 for BTW. The external noise ζ i ( E , t ) = ( δ E ) · δ i, n ( t ) Y j ∈L [1 − θ ( E j ( t ))] (81) acts at a slow time scale, and is pres ent when there are no active sites in the lattice. Here n ( t ) is a ra ndom vector po inting the site of the lattice piece L that is pertur b ed with ener gy ( δ E ) > 0 . The dynamics of av alanches gov erned by the SCML (80) evolves inﬁnitely fast in co mparison with the dyna mics of energy feeding. It has been p ointed out [57] that the SCML (80) is inv a riant under spatial trans lations, rotations, and reﬂections. F urthermore, if χ = 0 , the equation is inv ariant under a parity transformatio n of the order parameter, E → − E . The SCML (80-81) is supplied with the absorbing bo unda ry condition E ∂ L ( t ) = 0 . The SCML (8 0) can b e coar se-gra ined in order to obtain a contin uum sto chas- tic diﬀerential equation [57] for the eﬀective contin uum scalar ﬁeld E ( r , t ), ∂ E ( r , t ) ∂ t = α 0 ∆ [( χE ( r , t ) + E c ) θ ( E ( r , t ))] + f ( r , t ) (82 ) where α 0 > 0 is the only dimensional par ameter in the mo del, [ α ] = L 2 T − 1 , and dep ends on the lattice s pacing a , the unit time step, and the co ordina tion 37 nu m be r q . ∆ is the Lapla ce op era tor. The noise f ( r , t ) is a sum of the mul- tiplicative external noise dep ending on the who le lattice state and the internal noise that a ppe a rs due to the eliminatio n of micro scopic degre e s o f freedom. In the contin uous model, energy is thought to disapp e a r in thos e reg ions of lattice where f ( r , t ) < 0 and to a rrive at the p oints for which f ( r , t ) > 0 . In a stationary state, these pro cesses are obviously balanced, therefore, h f ( r , t ) i = 0 . In [5 7], the imp orta nt eﬀect o f dissipation at the op en b ounda r ies ha s not bee n tak en into a ccount and a quenched distribution of energ y absor bing defects has been assumed. Time scale separation of dynamics has be e n also neglected, and the noise f ( x ) , x ≡ r , t, has b een understo o d just a s a quenched Gaussian pro cess uncorrelated in space and time with a c ov ariance h f ( x ) f ( x ′ ) i = 2Γ δ d ( r − r ′ ) δ ( t − t ′ ) , (83) which is t ypic a l for random w alks. In the pr esent section, w e use a diﬀere nt Ansatz fo r the c ov ariance h f ( x ) f ( x ′ ) i whic h takes the slow-time scale dynamics of the stochastic force f ( x ) into ac count (see the nex t section). The contin uum stochastic partial diﬀer en tial equa tion (82) require s a regu- larization of the s tep function at zero. F ollowing [57], we use θ ( E ) = lim Ω →∞ 1 + er f (Ω E ) 2 , (84) as a re g ularizing function wher e e r f ( x ) = π − 1 / 2 R x −∞ exp[ − y 2 ] dy is the error function a nd Ω is the r egulariza tio n parameter. The reas on for this choice of regular iz ation pro c e dur e is that it allows a power e xpansion with an inﬁnite radius of con vergence. Developing (84) in p ow er s of E and substituting the se r ies ex pa nsion into (82), one obtains the s trongly nonlinear sto chastic partia l diﬀerential equa tion ∂ E ( r , t ) ∂ t = α 0 ∞ X n ≥ 1 λ n 0 n ! ∆ E n ( r , t ) + f ( r , t ) , (85) where the eﬀective coupling constants take diﬀerent v alues depending on the mo del: λ n 0 = lim ǫ →∞ ǫ n h E c θ ( n ) (0) + nχ ǫ θ ( n − 1) (0) i , n ∈ N (86) in which θ ( n ) (0) is the n − th order de r iv ative of the reg ularizing function (84) at zero. The c o eﬃcien t λ n 0 bec omes formally inﬁnite as ǫ → ∞ , ho wever, the series in (85) conv er ges. In the eq uation (85), we have supplied the par ameter α 0 and the coupling co nstants λ n 0 with index ”0” to distinguish their bare v alues fro m the renormalize d analogs which we sha ll deno te in forthcoming sections simply as α a nd λ n consequently . It has b een noted [57] tha t since θ (2 n +2) (0) = 0 , all even coupling constants v anis h for the BTW mo del, wher eas they do not fo r the Zhang’s one. The set o f coupling constants λ n 0 for b oth mo dels are identical in the limit ǫ → ∞ . 38 The eq uation (85) describ es the diﬀusion of energ y in Z d issued from a source deﬁned by f ( x ) , x ≡ r , t . This eq ua tion (up to a minor change of notations ) has b een conside r ed in the w ork [57] in the whole spac e and the imp ortant eﬀect of diss ipation at the op en b oundar ies has no t b een taken into a ccount. Alternatively , a small proba bilit y of dissipa ting an amount of energ y E c /q has bee n assigned for each site when it topples, instea d of transferring it to a certain neighbor. This proc e dur e expr esses the assumption of r andom b oundaries a nd corres p o nds to a mo del deﬁned on an inﬁnite lattice with a diss ipation for each toppling site. W e dis cuss the possible changes to the c r itical b ehavior due to the presence of regular absorbing bounda r y in the sectio n 1 1. W e study the long time large scale asymptotic behavior in the system gov- erned by the sto chastic diﬀer e n tia l equation (85) in the whole s pace supp osing that the r andom for c e f ( x ) is Gaussian distributed and characteriz e d by the co- v ariance (see the next section) which go es b eyond the ”white noise” approxima- tion studied in[5 7]. The introduction of the random force f ( x ) in (85) express es the bo undary conditions at the ra n dom b oundaries [57]. 5.2. Covarianc e of r andom for c es The intro duction of the random forc e into the equation (8 5) phenomenolog i- cally mo dels a consequence of the elimina tio n of micr oscopic degre e s of freedom and, at the same time, the injection o f energy in to the system. W e take the time scale separa tion in to acc o unt supp os ing that the dynamics of the slow- time s c ale and fas t-time sc ale co mpone nts of the r andom for ce ar e esse n tia lly diﬀerent. Namely , we supp ose that in the s low-time scale (i.e., the time s cale of energy injection) this dynamics can b e taken as the white noise like in [57], h F ( x ) i = 0 , with the cov a riance D F ≡ h F ( x ) F ( x ′ ) i = 2Γ δ d ( r − r ′ ) δ ( t − t ′ ) , (87) where Γ is the Onsager co eﬃcie nt. How ever, in the equation (85) de ﬁning the dynamics of relaxatio n pro cesse s evolving in the fast-time scale [57], the rando m force f ( x )in tro duced into r.h.s. has to b e also o f fast-time sca le . W e take it as the generalized random walks gov er ned by the linear Lang evin equa tio n ∂ f ( x ) ∂ t + R f ( x ) = F ( x ) , x ≡ r , t, (88) driven by the slow-time sca le ”white noise” F ( x ) , where the kernel of the pseudo- diﬀerential op erato r R has the form R ( k ) = ρ 0 α 0 k 2 − 2 η (89) in the F o urier space. Here, the coupling constant ρ 0 > 0 and the exp onent 2 η > 0 are related to the recipro cal cor relation time at wa ve num b er k , t c ( k ) = k 2 η − 2 /ρ 0 α 0 . Dimensio nal considerations show that the co upling consta n t ρ 0 is related to the characteristic ult r a-violet (UV) ultra-violet momentum scale in SOC momentum scale Λ ≃ 1 /a by ρ 0 ≃ Λ 2 η and cor resp onds to microscopic 39 degrees of freedom exp elled from the main equation (85) as a result of the coarse- graining of deterministic dynamical rules [57]. The expo nen t η corresp onds clear ly to the anomalous diﬀusion co eﬃcient [51, 82] z = 2 (1 − η ). Let us note that the sca ling form of recipro cal corre la tion time t c ( k ) has interesting connections with the spectrum of Lyapunov exponent, for the Zhang mo del. It ha s b e en indeed shown [82] that the Lyapunov exp o- nent s and the cor resp onding mo des r elate to the e ner gy transp or t in the lattice. How ever, the transp ort in SOC mo del is a nomalous and the tra nsp ort mo des corres p o nd to diﬀusion mo de s in a non-ﬂa t metric given by the probability that a site is active. It is r emark able that the Lyapuno v sp ectrum ob eys a simple ﬁnite sc aling form, with an univ e r sal exp onent τ < 1, whic h is directly rela ted to η b y the relation η = d 2(1 − τ ) . The parameter ρ 0 corres p o nds to the energy injection rate. It is be lieved in the literature that the SOC regime corres p o nds to the case when the injection rate go es to zero, the dis sipation rate go es to z e r o, such that the r a tio injection/dissipation go e s to ze r o esta blis hing the time scale separation [55, 56]. In the framework of critical phenomena theory approach to the pr oblems o f sto chastic dyna mics (see [77, 8 3], for example), the mo del for the random force cov ar iance D F in (87) is chosen under the following reaso ns: i) for the use o f the standard q uantu m-ﬁeld RG tec hnique, it is imp ortant that the function D F hav e a power-la w a symptote at large k ; ii) s inc e the cov ariance in (87) is sta tic ( ∝ δ ( t − t ′ )), the req uired physic al dimension [ h F F i ] = L d T − 3 is put on b y a s uitable combination of the only dimensional parameters in the logarithmic theory ( α 0 and k ); iii) the ”white noise” a ssumption (87) means that D F ∝ Const in the F ourier space. T o meet this requirement, o ne introduces a reg ula rization para meter ε > 0 qua n tifying the deviation form the lo garithmic b ehavior that is similar to the w ell-know ε = 4 − d expa nsion parameter in the critical phenomena theory [54]. All ab ove requir ement s ar e satisﬁed b y the mo del h F ( k , ω ) F ( − k , ω ′ ) i ≡ D F ( k ) ∝ α 3 0 k 6 − d − 2 ε . (90) In this mo de l, 2 ε is co mpletely unrelated to the space dimension d (in contrast to the standa rd critical phenomena approach [54, 77], w her e usually ε = 4 − d ). The logarithmic theory corresp o nds to the v alue ε = 0 . Finally , the mo del for the cov a riance D F ( k ) has to b e consistent with the form of the linea r op erator (89 ). Namely , fr om the equation (88), it follows that the cov ariance D f ( ω , k ) for the pseudo-r andom force f in tro duced in the r.h.s. of the main equation (85), h f ( x ) f ( x ′ ) i = Z dω 2 π Z d k (2 π ) d D f ( ω , k ) exp [ − iω ( t − t ′ ) + i k ( r − r ′ )] , k ≡ | k | , (91) 40 is related to D F ( k ) as D f ( ω , k ) = D F ( k ) ω 2 + [ ρ 0 α 0 k 2 − 2 η ] 2 . ( 92) Then, it is natural that the spectra l densit y of energy injection, f W ( k ) = 1 2 Z dω 2 π D f ( ω , k ) , (93) is indep endent of the corr elation time at given wav e num b er, t c ( k ). This is true if o ne takes D F ( k ) ∝ ρ 0 k − 2 η . E ven tually , collecting the latter result with the previous Ansatz (90), one arrives at the model D F ( k ) = ρ 0 α 3 0 k 6 − d − 2 ε − 2 η . (94) Both exp onents 2 η and 2 ε in (94) ar e the par a meters of the double expans io n in the η − ε plane around the origin η = ε = 0 , with the additiona l conv ention that ε = O ( η ) . The po sitive amplitude factor ρ 0 k − 2 η is consider ed a s a dimensionless coupling constant (i.e., a forma l s mall parameter of the ordinar y p erturba tio n theory). F o r the case of random for ce uncorrela ted in space, D F ( k ) ∝ Co ns t, the ” real” v alues of ε and η a re ta ken suc h that 6 − d = 2( η + ε ) . The simila r power-law Ansatz for the corr elator of rando m force has been used to mo del the energy pump in to the inertia l r ange of fully dev elo p e d turbulence [84, 83]. The mo del (92) wher e the function D F ( k ) is deﬁned by (94) is then more realistic and more reach in be havior than the simple ”white noise” assumption (83) discussed in the literatur e befor e (for example in the work [57]) since it takes int o account the ﬁnite cor relation time of ener g y ﬁeld set by int e ractions at a level of microsco pic degrees of freedom. It has a formal resemblance with the mo dels of random walks in random environment with long -range correla tions. W e no te that the similar correlator for r andom force has b een discus s ed for the ﬁr st time in studies devoted to the anoma lous scaling of a passive sca lar advected by the synthetic compressible ﬂow [85]. The Ansatz (92) that we use contains the previously discus s ed [57] mo del (83) as a s p ecia l ca se. Indeed, for the rapid-change limit ρ 0 → ∞ , the cov ariance (92) has the form D f ( ω , k ) → α 0 ρ 0 k 2 − d − 2 ε +2 η , (95) and, for ε − η = 1 − d/ 2 , one arrives at the mo del (83) uncor related in s pace and time with Γ = α 0 / 2 ρ 0 . In the opp osite limit of ”fro zen” conﬁguration of the sto chastic fo rce, ρ 0 → 0, the cov ar iance is sta tic (i.e., indep endent of time argumen t ( t − t ′ ) in t - representation), D f ( ω , k ) → π ρ 0 α 2 0 k 4 − d − ε δ ( ω ) . (96 ) The latter case obviously corresp onds to an external random forc e acting con- tin uo usly in time. F o r ε = 4 − d , this random force is unco rrelated in space ( ∝ δ ( r − r ′ )). 41 5.3. An inﬁnit e numb er of critic al r e gimes in SOC– mo dels Quantum ﬁeld theory formulation for SOC mode ls has be e n intro duced and studied in [6 1] following the general approa ch developed in [7 8, 79, 76, 84, 83]. All correlatio n functions h E ( x 1 ) . . . E ( x k ) i , x ≡ r , t, and resp onse functions h δ m [ E ( x 1 ) . . . E ( x k )] /δ f ( x ′ 1 ) . . . δ f ( x ′ m ) i expressing the system resp onse for an externa l p er tur bation w er e renormalized by subtracting all ultra–violet superﬁcial divergences from Green’s functions. An inﬁnite num b er of reno r malization c onstants has b een calculated in the one– lo op approximation in [61]. Possible sc a ling regimes of a r enormalizable mo del ar e asso ciated with the infra–red (IR) stable ﬁxed po int s of the corr esp onding diﬀerential RG equation. The co o rdinates of ﬁxed p oints { ρ ∗ , λ n ∗ } in the inﬁnitely dimensional space of coupling c o nstants λ n of the s to chastic mo del (85) ar e the solutions of the equations β ρ ( ρ ∗ , λ n ∗ ) = β n ( ρ ∗ , λ n ∗ ) = 0 , n = 1 , 2 , . . . , ∞ (97) where β ρ,n − functions, the co eﬃcient s in the diﬀeren tia l RG equation, are some rational functions of the par a meters ρ and λ n . Any s olution o f (97 ) is a n IR- attractive (IR-stable) ﬁxed p oint of the RG equation if the corr esp onding Jaco- bian matrix J ik = ∂ β i ∂ λ k is p ositively deﬁned (i.e., the real pa rts of a ll eigenv alues of the matrix J ik are po sitive) for small η > 0 , ε > η , 0 < ρ < 1, where λ 0 ≡ ρ and β i denotes the complete set of the β − functions of the RG–equation. It follo ws fr o m the explicit expressions for a n inﬁnite set o f β − functions found in [6 1] that t wo coo rdinates of ﬁxed p oints, λ 1 ∗ and λ 2 ∗ , can be chosen arbitrar y , then all other co or dinates ρ ∗ and λ k ∗ , k > 2 ca n be found directly from the equations (9 7). Therefore, the R G diﬀeren tial equa tion for SOC mo dels has a t wo-dimensional surface of ﬁxed p oints spanned with λ 1 ∗ and λ 2 ∗ in the inﬁnite dimensional space of coupling constants { ρ, λ n } . The complete IR–s tabilit y analys is for this manifold of ﬁxed p oints is a formidable task. In the limiting case of ”white noise” mo del (83), ta king for - mally ρ ∗ → ∞ , η = 0 , it is possible to demonstr ate that J ik ≈ − ( n − 1) εδ ik < 0 , where δ ik is the Kroneck er symbol, so that there are no IR– stable ﬁxed p o ints in SOC for zer o corre la tion time at all wa ve num b ers. The time s cale separa tion is mandatory for the existence of a critical regime in SOC. In the opp osite limit of ”frozen” conﬁguration of the sto chastic force, the ﬁxed points of the RG equation hav e b een shown a lso IR unstable, since J ik = − ( n − 1) εδ ik < 0 . 42 In a genera l s etting, the matrix J ik in such a case has a blo ck triangular form, and therefore its eigen v alues coincide with the diagonal elements whic h can be calculated for an y β n . The s tabilit y doma ins are deﬁned by the ro ots o f po lynomials in ρ ∗ . F or insta nc e , the p os itivity o f the ﬁr st eigenv alues requir es that ρ 3 ∗ + 3 ρ 2 ∗ − 2 − 4 ρ ∗ (1 + ρ ∗ )(1 + ρ 2 ∗ ) < 0 , and 2 ρ 2 ∗ − 3 − 5 ρ ∗ (1 + ρ ∗ )(1 + ρ 2 ∗ ) < 0 that is true for | ρ | < 1 . As n grows up, p olyno mia ls of a ny large order can app ear splitting the stability domain into a num b er of stable and unstable strips. It is imp ortant to no te that in a multi-c harg e theory , even if the IR-stable ﬁxed p oints of the RG equation exist, the ac tua l tra jectory of the sys tem in the m ulti-dimensional (phas e ) space of coupling c o nstants (in our case, an inﬁnitely dimensional s pace { ρ, λ k } ) starting fr om the given initial v alues ρ 0 , λ k 0 may not achieve any of them. The tra jecto ry can lea ve the stabilit y domain (in the critical pheno mena theory , it is us ua lly interpreted as the ﬁr st order phas e transition [54, 77]) breaking the scaling a symptote. 5.4. F at t ails in SOC-mo dels In the IR– stable critical r egimes, the Gr een functions and the res po nse func- tions exhibit scaling behavior characterized b y the follo wing ”critical dimen- sions”: ∆[ t ] = − ∆[ ω ] = − 2 + γ α ∗ = 2 η − 2 , ∆[ E ] = 2 η − ε, ∆[ E ′ ] = d + ε − 2 η . (98) W e hav e p o int ed out be fo re that for the random force uncorrela ted in s pace ( D F ( k ) ∝ Co nst), the ”real” v alue ε r is tak en suc h that 3 − d/ 2 = η + ε r . In t wo a lternative limiting cases , we have ε r = 1 − d/ 2 + η (the ”white noise” assumption; the system lacks of IR-attractive ﬁxed p oints) and ε r = 4 − d (the ”frozen” conﬁguration of the random force). Substituting these v alues in to (98), we obtain diﬀeren t critical dimensions for the energy ﬁeld ∆[ E ] and the auxiliary ﬁled ∆[ E ′ ] listed in the T ab. 1 . F or instance, for the critical dimension of the simplest Green function < E E > ( ω , k ) , we obta in ∆ [ < E E > ] = 2∆[ E ] − d + ∆[ t ] , and ∆[ < E E > st ] = 2∆[ E ] − d, 43 T able 1: Critical dimensions of the ﬁelds E and E ′ depending on the v ar i ous models f or the co v ari ance D F ε ∆[ E ] ∆[ E ′ ] 3 − η − d/ 2 d/ 2 + 3( η − 1) d/ 2 + 3(1 − η ) 1 − d/ 2 + η d/ 2 + η − 1 d/ 2 + 1 − η 4 − d 2 η − 4 + d 4 − 2 η for its static analog, < E E > st ( k ) = (2 π ) − 1 Z dω < E E > R ( ω , k ) . The simplest re spo nse function < E ′ E > ev aluates the av erage size of the relax- ation pro cess arisen in the sy s tem as a reaction for a p oint-wise per turbation; its F our ier transfor m is the distribution of av alanche size P ( s ) obser ved in nu- merical exp eriments. F or < E ′ E >, in the IR–stable critical regime w e obtain ∆ [ h E ′ E i ] = ∆[ E ′ ] +∆[ E ] − d + ∆[ t ] = − 2 + 2 η . (99) The squared eﬀectiv e ra dius R 2 = Z d x x 2 h E ( x , t ) E ′ ( 0 , 0) i (100) of the r elaxation pro cess a t a momen t of time t > 0 started at t ′ = 0 a t the origin x = 0 , one can ﬁnd that it scales as dR 2 dt ∝ R 2 η . (101) Indeed, since ∆[ R ] = − 1 (by conv ention), ∆  dR 2 dt  = − 2 − ∆[ t ] , from where we hav e the result (101). The obtained relatio n is analogo us to the well-kno wn Ric ha rdson’s phenomenological law for the diﬀusion of passive admixtures in the ambien t turbulen t ﬂows [80]. W e hav e studied the long time large scale asy mptotic b ehavior for the strongly nonlinear sto chastic problem which rela tes b oth to the Zhang and BTW mo dels exhibiting the self-organized critical b ehavior. The pr op osed mo del is int e r esting as itself since it is co nnected to the problem of nonlinea r diﬀusion of the c hemica lly a ctive scalar a dvection in the turbulen t ﬂows [86]. The sto chastic problem ha s b een considered in the bulk, far from a regular b ounda r y . The en- ergy dissipation at the open bo undaries has not be en taken into a ccount, instead a quenc hed distribution of energy absorbing defects has been ass umed. 44 W e now make several imp ortant comments on the further in vestigations in the framework of RG approach to the sto chastic diﬀerential equations related to the SOC models . The ﬁrst co mment is on the p ossible changes for the c r itical b ehavior close to the regula r open boundar y exhibiting the absor bing proper ty . The equation (85) can b e cons idered in a half-space z > 0 where the op en b oundary coincides with the z = 0 plane. Then the eﬀect o f b ounda ry w o uld be due to the semi-inﬁnite geometry of the system: The abse nc e of sites from one- ha lf s pa ce ( z < 0) c hang es the energy tra nsfer along the surface. Using the critical phenomena a nalogy [88], one can say that the surface does not beco me critical simult aneously with the bulk, but tends to decouple fr o m the r e st of the system. F urthermo r e, the pseudo-rando m force a cting a t the bo unda ry is to be alwa ys negative to e ns ure the complete dissipation of energy , h ⊥ ≡ f | z =0 ( r , t ) < 0 . (102) This is equiv ale nt to in tro ducing the new ﬁeld h 0 ⊥ on the surface that provok es a perturbatio n which ca n spread inside the system. These tw o eﬀects ar e in comp etition: If the co or dination n umber q is la rge, the toppled amount of energy E c /q dissipated at the op e n b oundar y is muc h smaller than that one tra nsferred to neigh b or s. In this case, the p erturbation risen due to h ⊥ close to the boundar y cannot propag a te into bulk. Otherwise, if energ y is rather intensiv ely dissipated at the b oundary than tra nsferred to the neighboring sites, a critical slo pe can app ear. Another comment is on p ossible steps b eyond the Gaussia n a pproximation. Let us note that the study of comp osite op era tors o f the t y p e ( E ′ ) n ( x ) would manage the corrections for the non-Gaussia n distributions of rando m forc e. W e also remember that comp os ite ope r ators ar e imp ortant for the deﬁnition o f ﬁnite size scaling correctio ns to the leading R G predicted asymptotes . Scaling, renormaliz a tion and sta tistical conserv ation laws in the K raichnan mo del of turbulent advection in the context of the renormaliza tion group im- prov ed per tur bation theory hav e b een inv estigated in [89]. 6. Co nclusions Po wer laws (heavy-tailed) distributions are found throughout man y naturally o ccurring phenomena in physics, and eﬀorts to obser ve and v alidate them are an active a rea of scie ntiﬁc research. W e hav e considere d a num b er of stochastic dynamical mo de ls that migh t genera te power law asymptotic distr ibutions. In particular, we hav e r eviewed the sto chastic pr o cesses in volving multiplicativ e noise, Degree- Mass-Action principle (generalized preferential attachm e n t pr in- ciple), the in ter mitten t b ehavior o ccurr ing in more complex physical sys tems near a bifurcatio n p oint, so me case s of queuing systems, and the models of Self-orga nize d criticality . These mo dels mig h t b e a ground for many natural complex systems. Heavy- tailed distributions app ear in them as the emer gent phenomena sens itive for 45 coupling rules es s ent ial for the ent ire dynamics . Relationships b etw een the rules and the p ow er law sta tistics are strikingly non-linear, as even a small per turbation may caus e a larg e eﬀect, a prop or tional eﬀect, or even no eﬀect at all. 7. Ackno wledgments W e w ould lik e to thank Bruno Cessa c, Elena Floriani, Max Hongler, and Ricardo Lima fo r numerous discussions . References [1] M. Levy , S. Solomo n, In t. J. Mo d. Phys. C 7 (4), 595- 601 (1996 ). 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Heavy-tailed Distributions In Stochastic Dynamical Models

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