how cooperative behavior emerges among interacting individuals is a long - standing problem that has attracted , starting from darwin , the attention of a large number of researches . in the context of game theory ,different mechanisms and models have been proposed to explain the observed cooperative behavior , two prominent examples being the ultimatum game and the prisoner s dilemma .theoretical studies , have shown that , selection at the individual level , may lead to altruistic behavior , in contrast to the general belief that only group selection can give raise to altruism .aiming at understanding the mechanisms leading to altruism as the core of cooperative behavior , the ultimatum game is one of the paradigmatic theoretical games used to understand their interrelation .the simplicity of this game has allowed to obtain a large set of experimental results that clearly show the presence of altruistic behavior , in the form of _ altruistic punishment_. altruistic punishment , meaning that individuals react to an unfavorable action by an opponent although the punishment is costly for them and yields no material gain , is a key ingredient for the explanation of cooperation as it emerges if altruistic punishment is possible , while breaks down if it is ruled out .the ultimatum game consists of two agents , who have to share a given amount of money .one of them , the _ proposer _ , makes an offer on how to share the money to the other agent , the _ responder_. the proposer can only make one offer .the responder decides whether he accepts or rejects the offer .if the responder accepts , the money is shared as proposed ; otherwise , none of them get anything . given that a narrowly rational responder would accept no matter what he has been offered something is better than nothing , a narrowly rational proposer would offer to the responder the minimum amount .however , real agents behave differently : offers are typically close to a 50 - 50 ratio and offers below 20% are typically rejected .recently , an extensive research performed among small societies around the world has shown that social structure is a key element in determining the degree of cooperation among its members .on the one hand , individual - based features seem not to be relevant in order to determine the degree of cooperation . on the other hand , in the ultimatum game , the proposals and rejection levels are different depending on the social structure .for example , societies based on cooperation and sharing of food , show higher offering levels . from the theoretical point of viewthe effect of structured populations in social dilemmas , e.g. , the prisoner s dilemma , public good games , or snowdrift games , has been analyzed mainly from the perspective of spatially extended populations . in the simplest case ,local interactions are considered in regular lattices where each individual interacts only within its local neighborhood in contrast to global random interactions considered in well mixed populations .however , recent progress in this area has shown that many social and biological interaction networks are not regular , as typically used in theoretical models , but display the small - world behavior and broad degree distributions .the question we address here is precisely how social structure affects the degree of altruistic behavior .this paper is organized as follows : in the next section we define the evolutionary ultimatum game model in a complex network . in section [ single - scale ] , following the tradition of spatial games , we first consider interactions given by a regular lattice ; later extending our analysis to random and small - world networks , which are characterized by single - scale degree distributions ; in section [ scale - free ] , we also consider scale - free networks .finally we discuss our results and draw the conclusions in section [ conclusion ] .a set of agents are arranged in the nodes of a network .we set 1 unit the amount to be shared in each interaction .each agent is characterized by a threshold  ] . in fig .[ fig : time ] we display the threshold of the agent with the lowest payoff in the population at each time step .agents with high and low thresholds are removed from the population . on the one hand ,agents with high thresholds make large offers that are likely to be accepted by their neighbors , contributing to the neighbors payoff .however quite unlikely they receive large enough offers to be accepted due to their high threshold .this behavior can be illustrated calculating the expected payoff per interaction in a completely mixed population , in the limiting case of agents with threshold , they obtain on average a payoff close to 0 .on the other hand , the opposite situation is observed for agents with low threshold : their offers are hardly accepted while they accept most of the offers they receive . in the limiting case of agents with threshold , they obtain on average a payoff close to 1/2 per neighbor .thus agents with low thresholds have in average a larger payoff than agents with high thresholds and are able to survive .although the previous argument has been obtained for a completely mixed population , we observe that it is still valid in the regular case ( fig .[ fig : threshold ] ) .thus , after a transient , the distribution of thresholds reaches a stationary distribution .thresholds above a critical value are removed from the population .for the one - dimensional lattice considered here , a critical threshold value is obtained . below this critical valuethe distribution of thresholds is not uniform : it increases approaching the critical value .the payoff distribution also shows a nonuniform distribution : only values above a critical value are found , displaying a maximum at a value around .the question we next address is whether the ultimatum model self - organizes in a critical state . in order to characterize the dynamics , we have measured the distribution of the distance between two consecutive selection events , , and the first return time distribution , , the time elapsed between two selection events affecting the same agent .the results are plotted in fig .[ fig : firstr ] . in both casesthe tails of the distributions are well fitted by power - laws .for the spatial correlation with , and for the first return time with .these results suggest that the system self - organizes in a critical state where the distribution of avalanches is also a power - law .the critical state would emerge despite the nonuniform distribution of thresholds ( and payoffs ) .an avalanche is typically defined as follows : it starts when the lowest payoff gets larger than a preset value ( close to the critical payoff ) , and stops when it drops below this value .the size , , of an avalanche is the number of time steps it lasts . in fig .[ fig : ava ] , we show the distribution of avalanche sizes when we use a payoff as the indication of an avalanche .the probability distribution displays a power - law decay with an exponent .it is worth noting that if we had used the threshold value , , as an indication of when an avalanche starts and stops , the distribution of avalanches would have decayed exponentially ( see inset of fig .[ fig : ava ] ) .this behavior reflects that the thresholds being removed are not always above a critical value ( as shown in fig .[ fig : time ] ) : agents with low thresholds are removed often from the population .thus , the fitness of an agent is given by its payoff .agents in a scale - free network in the asymptotic state after iterations .the distributions are averaged over 100 realizations . ]regular lattices are just a crude simplification of social and biological interaction. more realistic models of interaction networks include the small - world behavior : the average distance between agents in the network is similar to the one obtained in a random network , and the clustering , the fraction of neighbors of an agent that are also neighbors between them is large , as occurs in a regular lattice .we have performed simulations in small - world networks generated by rewiring the links of a one - dimensional lattice , using the algorithm introduced in ref . in order to keep all the agents with the same number of links : with a probability , two edges exchange their end nodes .the distribution of thresholds and payoff are similar to the one - dimensional lattice , as shown in fig .[ fig : threshold ] .it can be seen that for the limiting case of a random network , the distribution of thresholds is broader , and has a higher critical threshold . in this case, the system also self - organizes in a critical state where the avalanche size distribution displays a power - law scaling . similarly to the one - dimensional lattice , the avalanche size distribution ( when considering the threshold as the dynamical variable ) is not power - law , but exponential . for the small - world networks , the distribution of avalanche size is also power - law with an exponent that depends on the rewiring probability .for instance , for a system of agents we find an exponent , for a value ( fig .[ fig : ava ] ) .   in all the results presented so far , the network of interactions is such that the number of links of each node , its _ degree _ , is constant for all the agents . beyond the small - world behavior ,another important ingredient of interaction networks is that they often display a scale - free degree distribution .we now turn into the study of the results of this model when the topology of interactions is not a regular one , but a scale - free one . as the model for the generation of the network, we used the barabsi - albert algorithm , generated as follows : starting from a fully connected network of size , at time a node is added , and attached to existing nodes , where the probability to be attached to a node is proportional to its degree .this algorithm generates networks with a power - law degree distribution with an exponent .we have fixed the value .once the network is grown , it is kept fixed and the dynamics is played as indicated by rules ( i ) and ( ii ) .  in fig .[ fig : u : pt](a ) we plot the stationary distribution of payoffs in the population .there is a well defined critical payoff , below which the agents are removed .this critical value is . for large payoffs ,the distribution decays as a power - law with an exponent of , reflecting the decay of the degree distribution of the network . for the distribution of thresholds in the population ( fig .[ fig : u : pt](b ) ) , there is not cutoff in this distribution .all thresholds in the range $ ] are present , with a maximum in the distribution around .the distribution is highly asymmetric : it approaches zero for thresholds close to 1 , while it reaches a finite fraction in the limit .thus , agents with low threshold values have more chances to survive .this is in agreement with the analytical argument in eq .[ eq1 ] . in scale - free networks ,the system also exhibits a power - law distribution of the size of the avalanches ( fig .[ fig : u : av ] ) . in this case , the payoff we set to define an avalanche is obtaining an exponent for the a power - law scaling .why can in the scale - free network agents with high thresholds survive better than in a regular or random network ? to address this questionwe have analyzed the average degree of the agents grouped according to their threshold , and the results are shown in fig .[ fig : u : tmed ] .it can be seen that agents with high threshold values are more likely to survive if they are located at the hubs of the network .the reason for this is that they can accumulate payoff via interaction with a large number of agents .it is worth noting that when the network is highly heterogeneous , for example when the degree distribution is scale - free , the dynamics depends on whether the payoff of each agent is normalized by its degree . for normalized payoffs in the scale - free networksdescribed previously the critical threshold depends on the number of agents . in the limit of large critical threshold tends to 1 , in contrast to the system size independent critical threshold reported previously , and the dynamics displays power - law distribution of avalanches when the threshold used to define an avalanche is chosen depending on . a deeper analysis of the normalized payoff case goes beyond the scope of this paper .how cooperation and altruism emerge among individuals is a withstanding question that has attracted much attention in the last years .also the relevance of the complex social organization , and the concomitant network of interactions in supporting altruistic behavior is an open question .we have proposed an evolutionary ultimatum game with local interactions that self - organizes in a critical state . to analyze how social structure influences the degree of cooperation we have considered different topologies for the network of interactions . assuming fairness as offer - acceptance around 50% ,the amount of altruistic behavior in the ultimatum game is reflected in the distribution of thresholds in the population . in regular and single - scale networks , high thresholds leading to low payoffs are removed , while intermediate thresholds are selected in the populations . in scale - free networks, the distribution of thresholds displays a maximum around a value of 50% , decaying for lower and larger threshold values . comparing with experimental results in small societies , both settings , the random and scale - free topologies ,capture the experimental findings where offers around 50% are the most common .however , in the scale - free networks the distribution of thresholds covers all the range of threshold values decaying slowing from the maximum around 50% , in contrast to the sharp cutoff obtained in the random case .thus , a hierarchical social structure may explain better the patterns of offer - acceptance found in some societies .  from a dynamical point of view , in all the different complex networkswe have analyzed , including regular lattices , small - world , random and scale - free networks , the distribution of avalanches displays a power - law scaling with an exponent that depends on network topology .this feature is typically a signature of self - organized criticality .many complex systems in nature are found to display this phenomenon .they characterize long - range correlations in a system , similar to the behavior near a critical point in a phase transition .the model introduced here can be compared with other evolutionary models .the exponents that characterize the dynamics is the same as in the simple model of evolution proposed by bak and sneppen ( bs model ) .for the one - dimensional lattice , the exponents characterizing the first return time , spatial correlation and avalanche distribution for the ultimatum model introduced here are the same as for the bs - model ; for random networks , the exponent of the avalanche size distributions corresponds to the mean - field exponent of the bs model .however , there is a crucial difference between the bs - model and the one introduced here : while in the bs - model the fitness is directly assigned to the agents randomly from a uniform distribution . in our model , the fitness is the outcome of the interactions and the distribution of thresholds is an emerging property of it .  the results presented here complement previous theoretical works on the emergence of cooperation . among the most used theoretical games to study the emergence of cooperation in social sciences ,the prisoner s dilemma is perhaps the most paradigmatic example . in this model, it has been shown that local interactions among agents can lead to a cooperative behavior in the so - called _ spatial games _ .in evolutionary models , it was also found that self - organized criticality can be present for this system .these results triggered the analysis of spatial games in different network topologies and strategies of the agents , aiming at uncovering the conditions under which cooperation can arise .in particular scale - free networks have been shown to sustain cooperation .together with our results , hierarchical social structures , represented for instance by scale - free networks of interaction , suggest that primitive societies displaying this kind of interactions favored the emergence of altruistic behavior . to elucidate whether the interaction patterns facilitated altruism or altruistic behavior led to complex interaction patterns we need to incorporate more realistic ingredients to the models as for instance the possibility of removal and establishment of social ties depending on the outcome of the interaction .another open question is the evolution of empathy itself . in this workwe have assumed that empathy has evolved before the thresholds are selected .however we expect that empathy will co - evolve simultaneously with the thresholds .other extensions of the model could consider , for example , several repeated interactions between agents before selection . in this case, it could be argued that a repeated interaction scheme could allow for an adaptation of the threshold by the least successful agents .the interplay between adaptation and selection is an open question to be addressed in future works .  in summary, experiments have shown that altruistic behavior is common in human societies .furthermore , it has been shown that social structure is a fundamental ingredient for understanding the distribution of sharing offers in the ultimatum game .we show that selection level together with local interactions can lead to a critical dynamics , where the precise degree of fair offer - acceptance depends on social structure .recent developments on complex networks has allowed us to consider some simple models capturing basic features of networks of interaction .our work emphasizes the importance of considering the social structure and calls for the development of more realistic networks models of social interaction to understand the interplay between individual behavior and with whom they interact .we acknowledge anxo snchez , and the editor and the referees for providing valuable comments and insight .we acknowledge financial support from mec ( spain ) through project fisicos ( fis2007 - 60327 ) , and the european commission through the nest project eden ( 043251 ) ; cjt acknowledges sbf ( switzerland ) through ser project c05.0039 .                         , saramki , j. , hyvnen , j. , szab , g. , lazer , d. , kaski , k. , kertsz , j. and barabsi , a .-, a.l . , structure and tie strengths in mobile communication networks , _ proc .usa _ * 104 * , 73327336 ( 2007 ) .

Abstract: experimental studies have shown the ubiquity of altruistic behavior in human societies . the social structure is a fundamental ingredient to understand the degree of altruism displayed by the members of a society , in contrast to individual - based features , like for example age or gender , which have been shown not to be relevant to determine the level of altruistic behavior . we explore an evolutionary model aiming to delve how altruistic behavior is affected by social structure . we investigate the dynamics of interacting individuals playing the ultimatum game with their neighbors given by a social network of interaction . we show that a population self - organizes in a critical state where the degree of altruism depends on the topology characterizing the social structure . in general , individuals offering large shares but in turn accepting large shares , are removed from the population . in heterogeneous social networks , individuals offering intermediate shares are strongly selected in contrast to random homogeneous networks where a broad range of offers , below a critical one , is similarly present in the population .


perhaps the only thing which is more complicated and unpredictable than a human mind , is the collective human mind .this collective human mind is the driving force of financial market fluctuations .the intrinsic complexity of the market dynamics forms the basis for the exponential growth of the econophysics , since the publication of seminal papers of stanley et al . as in every new scientific discipline , the econophysics is developing rapidly in various directions .research in those various fields has contributed several stylized statistical properties of asset returns ( which , in principle , may cease to be valid at a certain moment of time , because the collective human mind is capable of learning ) , c.f .most advancements in econophysics are owing to the concepts of scale - invariance , intermittency and heteroscedasticity .so , viable approximations for the market movements have been either based on the continuous time random walk ( ctrw ) model , or derived from it , c.f . the model of fractional brownian motion ( fbm ) in multifractal trading time ( fbm - mtt ) .there is strong empirical evidence that fluctuations in financial markets possess multifractal statistics .multifractality has been found to be inherent both to the currency markets , and to the stock exchange markets .the time series analysis is usually based on the closing prices of two consecutive trading days , but higher frequency data have been also used .the focus of this paper is placed on the multi - scaling properties of currency and stock exchange time series .so , we refrain ourselves from disussing the advancements in understanding the other aspects of financial time series .it has been recently shown that in the case of intermittent time series , the scaling behaviour of low - variability periods can provide additional information , as compared to a multifractal analysis .preliminary study of the time series of stock indices and currency rates indicated that similarly to the heart rate variability data , the financial time series are also characterized by a multi - scaling behaviour of low - variability ( i.e. `` silent '' or `` calm '' ) periods .such a behaviour has been independently verified by kaizoji et al .  in this paper , analytic approachis used to derive the relationship between the multi - scaling of low - variability periods , and multi - affinity of the time series . besides , a detailed analysis of the statistics of low variability periods is performed for various daily stock indices and daily currency rates .the study of the high - frequency data of the usdeur exchange rate indicates that at the time - scale of one day , there is a cross - over between two scaling laws for the low - variability periods .most modern studies of the variability of intermittent time series are based on ( or derived from / related to ) the model of multi - affine fractional brownian motion ( mafbm ) , cf .this is not surprising , because multifractal behaviour is believed to be the most universal case of scale - invariance , c.f . . note that while ctrw and fbm - mtt models are somewhat more detailed than a data description by multifractal formalism ( by introducing the concept of waiting- or trading time ) , the time series generated by both these models can be also characterized by the multi - affine spectra and lead to the same scaling of low - variablity periods as the model of mafbm .one of the disadvantages of the multi - affine analysis is neglecting the long - term correlations in the dynamics of high - frequency variability .this claim is motivated as follows .consider a multi - affine time series , with a lower cut - off scale .then , each point of the time series is characterized by its own hurst exponent ( referred to as the lipschitz - h " older exponent ) ; this exponent describes the local scaling of fluctuations , latexmath:[\[\label{mf }  the distribution of points of certain values of is self - similar , and is described by a fractal dimension .so , according to the mafbm model , we study the scaling of increments at , without specific attention to the values of them at . in order to shed light into this phenomenon , the method of scaling analysis of low - variability periods has been devised .  on the other hand , a low - variability period of length defined as such a continuous time interval $ ] ( ) that _ ( a ) _ is a threshold parameter and angular braces denote sliding average over a window of width ( in principle , the window width can be arbitrarily large ; however , the highest time resolution of the metod and widest scaling range is achieved when it is as small as possible , i.e. just few cut - off scales ) ; _ ( b ) _ each period has maximal possible length , implying that decreasing or increasing would lead to violation of eq .[ delta ] .we speak about multiscaling behaviour , if the cumulative distribution function of the low - variability periods ( the number of periods with ) scales as where is a scaling exponent and is a constant .it should be noted that the scaling exponent can be considered as the extension of the concept of multi - affine spectrum to the highest possible time resolution .this is because of two circumstances .first , the effective time resolution of the multi - affine description by eq .[ mf ] is limited by the required scaling range ; meanwhile , the effective time resolution of the description based on low - variability periods is the cut - off scale .second , in the case of infinite time resolution , there is one - to - one correspondence between the spectra and , see below .the relationship between and spectra can be derived as follows .if a time moment is characterized by a lipschitz - h " older exponent , the mismatch between the local average and the instant value is of the same order of magnitude as the variability amplitude of over the time window of width : , we conclude that for a period , all the points are characterized by .similarly , the edges and of the period are characterized by .therefore , the the low - variability periods are the contiguous pieces which remain from the entire time - axis after the removal of the high - variability points with .these high - variability points form a sparse cantor - dust - like set , assuming that , where corresponds to the global maximum of the spectrum [ i.e. =1 ] .indeed , is the fractal dimension of the set of points described by the exponent .so , the fractal dimension of the set of points described by exponents is found as if an interval is divided into pieces by a cantor dust of dimension , the cumulative length - distribution exponent of the pieces is also ( because the number of pieces larger than is estimated as the number of boxes of size required to cover the cantor dust , according to the box - counting method ) .so , we have correspondence   note that in the case of simple fbm ( which is studied in more details in ref . ) , there is no scaling of low - variability periods with lengths . indeed , for simple fbm , , if [ otherwise , is not defined ] .so , the high - variability points either are absent ( i.e. there is a single low - variability period ) , or they are populated quasi - homogeneously with fractal dimension ( then , all the low - variability periods are shorter than or of the order of the window width ) .   the approach used for mafbm can be also applied to the fbm - mtt model with , where is a fractional brownian function of its argument(described by the hurst exponent ) and is a multi - affine function of time .indeed , in that case eq . [ h ]is substituted by is the increment of the trading time and is the local lipschitz - h " older exponent for the function .accordingly , eq . [ alpha ] is substituted by   so , both in the case of the mafbm and fbm - mtt models , the scaling described by multifractal spectrum with can be also described by the scaling of low - variability periods .the approach to describing the lipschitz - h " older exponent range is straightforward : instead of low - variability periods , high variability periods have to be used. then , eqns .[ alpha ] and [ malpha ] will remain valid , except that the inequalities will be opposite ; details of this aspect will be analysed elsewhere .now , we are left with two questions .first , the multifractal dimension can not exceed the topological dimension one . is it possible to have the scaling exponent of low - variability periods larger than one ?the answer is yes ( examples are provided below ) .more specifically , equality assumes that the most part of the aggregated time of the low - variability periods corresponds to shortest periods ( because ; here , the integral vanishes at large lengths ) .this assumes that the fractal dimension of the high variability periods is one , i.e.  . in that case , the low - variability moments form a cantor - dust like set of points . for real time series ,the time resolution is always finite ; then , in the regions of high `` dust density '' , the `` dust particles '' may overlap and form relatively long ( much longer than the time resolution , but much shorter than the total time interval ) low - variability periods .such a clustering of small--points is not described by the multifractal spectrum .second , the multifractal spectrum is a one - parameter - curve ; meanwhile , is a two - parameter - curve , which is effectively reduced to one degree of freedom by eq .[ alpha ] [ or [ malpha ] ] .is it possible to have a such a class of scale - invariant time series , which is more generic than the class of multi - affine functions , so that the exponent can be arbitrary function of two independent variables and .this question is left open for further studies .to conclude , the scaling analysis of low variability periods provides a simple and superior ( in the sense of time resolution and applicability to the parameter range with ) alternative to the traditional multi - affine analysis of the time series .in particular ,eq . [ hh ] provides a way to check the validity of the assumption of multi - affinity , which requires .if there is no need for such verification , the route is even simpler , e.g. the analysis can be performed at a fixed window length .all this does not mean that studying the scaling of low - variability periods is always better than the multi - fractal formalism .instead , the two methods should be considered as complementing each other , because the accuracy and applicability of both methods are sensitive with respect to the width of the scaling range and specifics of the scaling behaviour .now we apply the above developed theory to financial time series .the specifics of long - term stock - market time series is that prices can vary by orders of magnitude .therefore , it makes sense to work with a logarithmic scale , and substitute the definition of low variability ( eq .[ delta ] ) as follows : , it is assumed that average is taken not over a time window centered around the current moment of time , but instead , the large--edge of the window coincides with ( because for practical applications in real time , the future is not known ) : so , a low - variability period is the period between two consecutive price movements that are exceeding the given threshold .the length of a low variability period is measured in the same units as time  in the units of the sampling period of the market data . the local average in eqn [ delta2 ]gives us the second input parameter : that could be interpreted as a degree of `` locality '' . with these two input variableswe maintain the easy interpretation of the results : threshold value is simply the percentage change of asset at time compared against the average price of that asset in preceding period with length of time units .further we measure the lengths of low - variability periods and we count them .then we define a cumulative distribution function that represents the amounts of periods where low - variability lasted at least time units .finally , we fit the data against the scaling law given by eq .[ scal ] .following analysis will give answers to two essential questions :  1 .do the low variability periods obey power law and if so , what is the scaling exponent ? 2 .assuming that there is a power law , how does the scaling exponent depend on the parameters and ?we used the following data in our analysis :  [ algandmed ]    * remark : from january , 1st 1999 german mark and french franc are fixed against euro .  in our analysis ,the scaling exponent was found by using least - squared fit for the whole data series . for visualizations , the dependence is plotted in log - log scale .then , the scaling exponent is equal to slope of the fit - line .financial time series are single realizations of intermittently fluctuating nonstationary time series , which makes calculation of exact error estimation of the scaling exponent impossible .however , rough estimates of the uncertainties have been obtained as follows .the least - squares fitted trend - line was found as described above , except that the slope was not optimized , i.e. it was considered as a fixed parameter .further , the sum of squared residuals was calculated as a function of .the error estimate was found as , where is the least - squares fitted value of the slope , and satisfies the condition .the technique used for obtaining the scaling exponent is illustrated in fig [ alfa ] .for spx time series under the following conditions : =1.6% and =10 days , title="fig:",width=491 ] [ alfa ]  above - mentioned procedure was carried out for all of the time series and with different thresholds and window widths .the quality of the data fit was measured by the correlation coefficient .total 1004 measurements were carried out with all the daily time series . in fig [ histogr ] , the validity of power law hypothesis is demonstrated .histogram shows that in most cases , the power law provided a good data fit .calculated values of and error estimations for currencies and equity indices for are presented in tables [ curtable ] and [ eqtable ] respectively .  , title="fig:",width=491 ] [ histogr ]  it is important to outline two effects :  * there is reverse relationship between the correlation coefficient and the threshold parameter .the reason is that the amount of large movements is small and not representative for statistical analysis .* there is also reverse relationship between the correlation coefficient and the window width , due to similar reasons ( longer averaging smoothes time series and large movements become more frequent ) .  due to these effectsthere is no sense to measure the scaling exponents beyond certain parameter range , e.g.  for and .[ curtable ]   [ eqtable ]   two phenomena can be observed :  1 .exponent and threshold tend to be negatively related : larger values correspond to lower values .2 .  for equities ,the dependence of the exponent on the threshold is relatively strong , but relatively weak for currencies .  in order to illustrate this observation , the values of plotted in fig [ demspx ] against the threshold for dem and spx time series .values for dem and spx under thresholds = 0.10% ... 2.95% and =10 days.,title="fig:",width=491 ] [ demspx ]  next we study the dependence of on the window width using the high frequency data of eur - usd exchange rate . as mentioned above , low amount of statistics becomes an issue , when the threshold parameter is too high for the given window width .therefore only those values of are plotted in fig [ eurhifrq ] , where .   for eur time series with different window widths under thresholds= 0.10% ... 2.95%.,title="fig:",width=491 ] [ eurhifrq ]   according to the theoretical expectations , should be a decreasing function of , owing to eq .[ alpha ] and to the circumstance that for ( i.e. for ) , is a decreasing function of .this trend is expected to held for the range , as well , due to simple reasoning .increasing the threshold parameter leads to some fluctuations ceasing to be qualified as `` large '' , so that longer low - variability periods will emerge . increasing the number of long periods , in its turn, corresponds to decreasing the scaling exponent , in accordance with the results depicted in fig [ demspx ] .for eur time series with different values of and are plotted versus , providing a data collapse for , .,title="fig:",width=491 ] [ collaps ]  in order to study , how well the theoretically derived relationship eq .[ alpha ] ( or eq . [ malpha ] ) holds for the financial time series , the scaling exponent has been plotted versus in fig .[ collaps ] ( using the same data as in fig . [ eurhifrq ] ) .note that the data - points with  describe the scaling behaviour at the time scale above one day ; the data - points in the range are influenced significantly by the intra - day dynamics of the exchange rate .  for the range of validity of eq .[ alpha ] ( ) , the data - points with lay reasonably well into a single curve , confirming the validity of eq .[ alpha ] .the data of fig .[ collaps ] indicate also that at the time - scale around one day , there is a cross - over between different scaling regimes : the data with incorporate a mixed dynamics ( which can not be described by mafbm model ) and lay no longer on the same curve . at the range , there is no evident reason for the data to lay on a curve : all the corresponding data points are scattered in a form of a disperse cloud .in this paper a new method for time series analysis is derived and analytically motivated .it is shown that for multi - affine time series , the scaling properties of low - variability periods are described by scaling exponent as a function of threshold parameter and averaging window width .the scaling analysis of low variability periods offers a simple alternative to the multi - affine analysis of the time series , providing wider applicability range and somewhat higher time resolution .an open question is about the existence of a class of scale - invariant time series , more generic than multi - affine ones , violating eq .[ alpha ] .particular emphasize is paid to the financial time series .the analysis showed a good scaling behaviour of the currency exchange rate and stock index data and confirmed the theoretical expectations , particularly the relationship between the scaling exponent and the model parameters and ( c.f .. [ alpha ] ) .the analysis of high - frequency data of eur - usd exchange rate indicated that there is a cross - over between different scaling regimes at the time scale of one day .it was also observed that scaling exponent values for equity time series tend to be more sensitive with respect to the threshold parameter than currency time series .the support of estonian sf grant no .5036 is acknowledged .r. cont , quantitative finance , * 1 * ( 2001 ) 223 . t. lux , f. heitger , in : h. takayasu , ed , empirical science of financial fluctuations ( 2002 ) springer , tokyo , pp . 123 t. lux , m. ausloos , in : a.bunde , j. kropp , h .- j.schellnhuber , eds , the science of disaster : scaling laws governing weather , body , stock - market dynamics , ( 2002 ) springer verlag , berlin , pp.377 e.w .montroll and g.h .weiss , j. math .* 6 * ( 1965 ) 167 .e. scalas , r. gorenho , f. mainardi , physica a * 284 * ( 2000 ) 376 .f. mainardi , m. raberto , r. gorenho , e. scalas , physica a * 287 * ( 2000 ) 468 .m. raberto , e. scalas , r. goren , f. mainardi , arxiv : cond - mat/0012497 ( 2000 ) .raberto , e. scalas , f. mainardi , physica a * 314 * ( 2002 ) 749 .e. scalas , r. gorenho , f. mainardi , m. mantelli , m. raberto , arxiv : cond - mat/0310305 ( 2003 ) .mandelbrot , fractals and scaling in finance : discontinuity , concentration , risk .springer , berlin 1997 .mandelbrot , quantitative finance * 1 * ( 2001 ) 113.b.b .mandelbrot , quantitative finance * 1 * ( 2001 ) 124 .n. vandewalle , m. ausloos , eur .j. b * 4 * ( 1998 ) 257 k. ivanova , m. ausloos , the european physical journal b , * 8 * , ( 1999 ) 665 f. schmitt , d. schertzer , s. lovejoy , appl .stochastic models data anal . *15 * ( 1999 ) 29 f. schmitt , d. schertzer , s. levejoy , int .appl . fin .* 3 * ( 2000 ) 361 ausloos m. and ivanova k. , comp .. commun . * 147 * , ( 2002 ) , 582a.a tsonis , f.heller , h. takayasu , k. marumo , t shimizu , physica a. * 291 * ( 2001 ) 574 j. kalda , m. skki , m. vainu , m. laan , arxiv : physics/0110075 v1 26 oct 2001 , m.skki , j. kalda , m. vainu , m. laan , physica a , * 338 * , ( 2004 ) , 255 r. kitt , j. kalda , wseas transactions on business and economics , * 1 * january 2004 101.t .kaizoji , m. kaizoji , physica a * 336 * ( 2004 ) 563 .ivanov , p.ch . ,rosenblum , m.g . ,amaral , l.a.n ., struzik , z. , havlin , s. , goldberger , a.l . , and stanley , h.e . ,nature * 399 * ( 1999 ) 461 .amaral , l.a.n . ,goldberger , a.l . ,ivanov , p.ch . andstanley , h.e .* 81 * ( 1998 ) 2388 .lin , d. c.  and hughson , r. l. , phys .* 86 * ( 2001 ) 1650 .garrido , p. , s. lovejoy and schertzer d. , physica a * 225 * ( 1996 ) 294 .carbone , a. , castelli , g. , stanley , h.e , phys .e * 69 * 026105 ( 2004 ) .

Abstract: properties of low - variability periods in the time series are analysed . the theoretical approach is used to show the relationship between the multi - scaling of low - variability periods and multi - affinity of the time series . it is shown that this technically simple method is capable of reveling more details about time series than the traditional multi - affine analysis . we have applied this scaling analysis to financial time series : a number of daily currency and stock index time series . the results show a good scaling behaviour for different model parameters . the analysis of high - frequency usd - eur exchange rate data confirmed the theoretical expectations .  ,  econophysics , multiscaling , multifractality , time series analysis , scale - invariance 89.65.gh , 89.75.da , 05.45.-a , 05.45.tp


the complexity of distributed real - time systems is growing rapidly ; model - based development tools are used to accelerate the development process and increase the quality of the produced code .in addition , it is possible to integrate formal verification as analysis technique into these tools .currently , the standard verification process is achieved by first translating system models into verification models , followed by verifying relevant properties by verification engines using special algorithms . in the verification community , researchers focus on tighter theoretical complexity bounds or computationally faster algorithms to reduce the required time for verification .nevertheless , if it comes to verification of complex systems , an efficient model reflecting the properties of the system under consideration becomes essential . by _efficient model _ , we refer to a model containing `` just - enough '' information of the system behavior regarding these properties .in fact , an inefficient modeling with irrelevant details can simply render the verification intractable .  within this paper ,we introduce an approach for the construction of such an efficient model for the verification of timing assumptions and constraints .the approach is presented in , but not restricted to , the context of ftos  , a model - based development tool for the design of fault - tolerant systems .  in our presentation , we first introduce ftos , mention insights regarding differences in comparison to other development tools , and propose our two - phase verification methodology ( sec . [ sec.motivating.examples ] ) . then based on ftos and timed automata , we describe the model construction process , focusing on the aspects concerning expressiveness , modification , and efficiency .  * * ( expressiveness ) * we give observations regarding restrictions of timed automata to construct models of real - time systems ( sec .[ sec.network.finite.capacities ] , [ sec.task.element.finite.precision ] , [ sec.dispatcher ] ) ; these observations are valid not only in the context of ftos , but apply also for other systems . * * ( modification ) * we formulate mathematical representations how to perform model modification to derive verification models from system models ( sec .[ sec.job.processing.element ] ) . * * ( efficiency ) * with the understanding of ( 1 ) complexities of verification and ( 2 ) our problem structure , we propose some theoretical criteria regarding how to construct an efficient model , such that it is possible for existing model checkers to generate results within reasonable time ( sec .[ sec.invocation.faults.events ] ) .at last , we report our preliminary implementation ( sec .[ sec.implementation ] ) , mention related work ( sec .[ sec.related.work ] ) , and conclude this paper ( sec . [ sec.conclusion ] ) .ftos is a model - based development tool for the development of fault - tolerant real - time systems , that alleviates designers burden by offering code generation for non - functional aspects with high extensibility .the conceptual modeling in ftos uses multi - aspect techniques comprising four different perspectives :  * hardware model * : the hardware model specifies the hardware used , including specifications of _ electronic control units _ ( ecus ) and the interconnecting _ network_.  * software model * : the underlying model of computation in ftos shares large similarities with that of giotto , which is based on the concept of logical execution times .a designer should specify _ tasks _ , _ ports _ , _ inputs _ , _ outputs _ , and _jobs_.  * fault model * : the fault model specifies the _ fault hypothesis _ of the system , which includes the set of _ fault containment units _ ( fcus ) ( possible faults concerning locations , types , durations ) , and the set of _ fault configurations _ ( possible simultaneously activating fcus ) .examples for the fault hypothesis are :  1 .a network link can have message lost ( fault type : ` msgloss ` ) with minimum interval between consecutive occurrences equal to millisecondsa software task can produce errors ( fault type : ` wrongresult ` ) due to a fault within an associated sensor ; once happened , it will not be corrected unless explicitly done by the user or the fault - tolerant mechanism .the minimum interval for the correct operation between two consecutive faults of the sensor is expected to be milliseconds .* fault - tolerance model * : the fault - tolerance model specifies methods to detect errors and to repair and restore the system .  during code generation , ftos selects , adapts and combines pre - implemented code templates based on model features .a detailed description of ftos can be found in .the concepts presented in this paper do not only apply for ftos , but also a range of other related projects , such as giotto or event - driven tasks with fixed deadlines .figure [ fig : mapping ] shows the different models of execution .an aperiodic or sporadic function is event driven ; when such an event happens , a deadline is assigned to the task handling the event .giotto functions are functions that interact synchronously at macro step level ( logical level ) , while at micro step level the execution is asynchronous. for detailed description of giotto and the concept of logical execution time , see .ftos functions are extensions of giotto functions . intuitively they are equipped with fault - tolerance abilities such that the system can resist faults defined by the fault model . in fig .[ fig : mapping ] , three redundant copies ( , , ) are deployed on the three machines ( , , ) .the figure also shows the necessity of a mapping the behavioral model ( in ftos : software model ) to the architecture model ( in ftos : hardware model ) .note that in general a design space exploration is needed for finding such a mapping . for details, we refer readers to articles regarding platform - based design .since this mapping is specified in ftos by the developer , our analysis can start from a given selection of hardware and software settings .the main property of fault - tolerant systems that needs to be verified is the ability to withstand the assumed faults .the fault assumptions are summarized in the fault hypothesis ( in ftos : fault model ) that defines faults regarding its location , effect , and frequency .the verification of such systems is hindered by two aspects : deadline violations and non - determinism due to e.g. imperfect synchronization of redundant units .  1 .  in ordinary systems ,correctness relies on the assumption that a scheduling never leads to deadline violations ( without loss of generality , we assume that deadlines specified in our model are hard ) . nevertheless , in fault - tolerant systems , the constraint can be loosened . due to replication , a deadline violation of one unit might be tolerated .in fact , the violation of the deadline can be categorized as an occurrence of a fault defined in the fault model .this brings dramatic differences between fault - tolerant systems and ordinary systems , i.e. , deadline violation is feasible or acceptable provided that there exists a fault - tolerance mechanism such that the effect of fault can be eliminated .2 .  on the other hand, replication also introduces further difficulties . in ordinary giotto systems ,internal determinism is guaranteed , meaning that two deployments having the same relative ordering in the micro step level will have the same behavior , irrelevent of the absolute timing .unfortunately , internal determinism will not be maintained if no constraints are added additionally on ftos functions .consider fig .[ fig : nondeterminism ] , where , and are three deployments .the send action will broadcast messages to other machines regarding its liveness . ideally , when no error happens , then each machine should conceive a consistent view of the system .however , when the scheduling of changes to that of , with zero time transmission , the result will be an inconsistent view at and .this brings semantic incompatibility between different deployments .  to solve these problems, we thus propose the concept called _ deterministic assumption _intuitively , the goal is to assume that the implementation of fault tolerance mechanisms will always provide a consistent view for all correct machines regardless of deadline violation and scheduling issues . in practice, this will place constraints regarding the earliest and latest arrival time between messages sent , which need to be verified .    for above purpose, we adapt a two - phase verification process in our tool ftos - verify :  * ( phase 1 : verification on the platform independent layer )* we first assume that the deterministic assumption holds in all deployments .based on this assumption , we construct a verification model . the model is an abstract machine ( closed model ) where injection of faults is regulated based on the fault model .the model offers precision by revealing detailed mechanisms of fault - tolerance .our theoretical foundation enables us to construct a concise model with huge benefits . for this phase, the mathematical formulation and the proof of theorems are stated in ; it will not be the focus of this paper .  *( phase 2 : validity checking of the behavior - architectural mappings ) * in this phase , we have to focus on two aspects .first , we have to check whether the deterministic assumption holds in the platform .second , we have to check if there exists possibilities where deadlines are violated , and the violation exceeds the constraint specified and regulated in the fault model . note that since the correctness of the data and mechanisms are checked in the first phase , in the latter phase only protocol checking ( timing ) is needed .this will be the focus and the main contribution of the paper .for the analysis of the temporal behavior , we transform the models in ftos to communicating timed automata ( cta ) . in the following sections, we will describe our observations , relevant parts of the construction process , and theoretical criteria for model efficiency . by using a generalized view ,the results are applicable not only in the context of ftos , but can be used for verifying temporal behavior for generic distributed real - time systems .we use an extended format of communicating timed automata ( cta )  using variables of finite domain to express the features of the behavioral model .it is important to mention that this extended format does not change the expressiveness of cta .a system of communicating timed automata is a tuple , where  is an automaton with the following constraints . is a finite set of _ modes _ ( _ locations _ ) . is the set of finite - domain integer variables . is the set of _ clock variables_.   is the set of _ synchronizers _ ; each synchronizer is of the format where elements in represent synchronizer symbols .conceptually , represents receiving , and represents sending . is the _ initial location _ of the automaton . is the jump from mode to mode .  1 .  is the conjunction of inequalities of the form or , where , , , and . is the set of assignments of the form or , where , , and . is the set of mode invariants mapping a mode to a subspace of indicating the possible clock values to maintain in the mode .  in the following ,we summarize required components of the verification model and outline our observations .     to model the network of the distributed system , an appropriate level of detail must be selected . in general , for a network with message delay and junction points , we have to model such a network with automata to handle point - to - point communication .[ fig : ta.p2p ] is the template ( defined in uppaal ) of a timed automaton which models the point - to - point transmission with storage capacity equal to , and one overflow location .the function is used to return the index of the channel .for modeling of network components , only finite capacity can be reached .furthermore , the number of controlled locations grows exponentially as the number of allowed storage increases , because the variable delay ( due to the fault model ) or the routing scheme may lead to an arbitrary ordering of arrived messages .   for giotto - like mocs , tasks are units which perform dedicated computations .modeling the task execution can vary based depending on whether the applied scheduling is preemptive .[ fig : ta.task ] shows a timed automaton representing the task execution with potential context switches .since context switch and preemption can occur , the task should keep the record for the remaining time ( portion ) to finish the task .the variable ` percentage ` represents the progress of execution and ` increment ` reflects the minimal advance related to the time accuracy used during verification .the constraints regarding time accuracy imply finite precision in the model .modeling of tasks can only be achieved with finite precision , since with context switch , we need to record the portion of executed tasks .this also brings issues between expressiveness and complexity ; a better accuracy regarding the timing behavior of the context switch ( with finer time unit ) leads to increasing complexity of the resulting model since it depends on the biggest integer used in the system .     the task of the job processing element is to manage the execution of tasks and to implement the inter - task communication .the construction in timed automata may vary due to the concrete application .however , for ftos , a fixed sequence of atomic actions is defined in the software model .each atomic action can be represented by a similar model as used for the task model described previously .the main difference is that the models of the atomic actions are linked together instead of having a closed loop in the automaton representing the job processing element .here we omit the detailed construction process for the original model , but focus on the transformation into the according verification model .we give two motivating examples .  * in order to model the effect of faults , we need to add additional edges on the original model to represent the occurrence of faults .* to observe deadline violation , additional clocks that reflect the time progress since event occurrence , locations that represent the deadlines , and jumps are required to annotate the original model .  for these purposes, we define this annotation as a sequence of edit - operations over a labeled graph ; this facilitates the mathematical formulation how we transform between models .define five atomic edit actions as follows .  1.  * clock add * : given a clock variable , is an operation that adds a clock to .formally speaking , given , the result of is a new timed automaton .2 .  * variable add * : given a variable , is an operation that adds a variable to . formally speaking , given  ,the result of is a new timed automaton .* location add * given a location and an invariant , where is the conjunction of inequalities of the form with clock , , and , is an operation that adds a location to with invariant condition . formally speaking ,let , the result of is a new timed automaton .* jump add * : given two locations with guard , assignment , and set of synchronizers , where 1 . is the conjunction of inequalities of the form or , where is a clock , is a variable , , and . is the set of assignments of the form or , where is a clock , is a variable , and .+ let , then the result of is a new timed automaton by adding an arc to . 5 .* jump edit * : given two locations with guards , assignments , and sets of synchronizers are conjunctions of inequalities of the form or , where is a clock , is a variable , , and .2 .  are sets of assignments of the form or , where is a clock , is a variable , and .+ let , then the result of is a new timed automaton by changing the arc to in .note that in our formulations , we assume due to simplification reasons that the added element is not identical to any elements in the original set , and every newly added location or jump is well defined ( e.g. , to add a new location with invariants using clock , should have been defined in ) .let an edit sequence be , where edit actions .define the result of on , in symbols inductively as follows .* where is the null sequence .* ,  . * , . * , . * , . * ,  .starting from the textual description of the fault model , we can construct the set of _ deadline requirements _ for the system model .intuitively this means that for all runs entering the location , it must subsequently enter within at most time units . based on above definitions ,we sketch the algorithm , is not reentered before entering because actions in the job processing element are chained . ] how to generate the verification model from the system model as follows :  ' '' ''  * algorithm * : ` genverificationmodelpart ( ) ` + \ {  + / * input : original system model * /  + / * output : verification model * /  + * * let * * .  + * * forall * * deadline requirements ,  + / * add new clock and new variable for testing * /  + .  + .+ / * is the location for deadline violation * /  + .  + * * forall * * incoming jumps of ,  + ,  + where .  + * * endfor * *  + * * forall * * incoming jumps of ,  + ,  + where .  + * * endfor * *  + * * forall * * reachable locations from ,  + .  + where is defined as .  + * * endfor * *  + * * endfor * *  + * * return * * ./ * apply changes in * / + }  ' '' ''  for the property of deterministic assumption mentioned in section [ subsec.verification.goals ] , similar algorithms can be applied to annotate clocks , locations , and jumps ; the problem for checking deterministic assumption in ftos turns to be a reachability problem in timed automata .with respect to the operating system , we have to model the dispatcher explicitly .the modeled dispatcher merely captures the scheme for the execution of threads ; deadline violation , fault - tolerance or error handling is modeled in the job processing element .therefore , it can be used in arbitrary settings and not only in ftos . due to different scheduling algorithms, the model of the dispatcher differs dramatically regarding actual verifiability . for our analysis, we use priority based dispatchers modeling either fifo or round - robin techniques .nevertheless , as context switch of tasks / threads occurs , we have the following observation .  using a round - robin dispatcher leads to exponential increase of possible behaviors compared to a fifo - based dispatcher with the number of parallel tasks , if no assumptions on the task behavior can be made .in summary , this section gave insight in the main components of the verification model and their construction .besides the job processing element , all components and related observations can be directly applied for arbitrary real - time systems . for the job processing element, we described a generic way to use annotations to construct a model to use for verifying the absence of deadline violations . in the next section ,we point out how aperiodic behavior introduced by faults or events can be considered .to perform verification , modeling the arrival of faults or aperiodic events is necessary to establish a closed model , and in this section we consider its effect . in ftos, the probability of faults is implicitly reflected by the concept called _ least time between faults _ ( ltbf ) . in our analysis , the invocation of aperiodic tasks can be done similarly - the least time between occurrences of events for aperiodic tasks is defined as _ least time between arrivals _ ( ltba ) . with ltba or ltbf, we can augment the original model with a timed automaton producing the event ( called _ event agent _ ) similar to fig .[ fig : ta.event.agent ] .however , since ltbf ( or ltba ) is an integer which might be relatively large , and the complexity of verification in timed systems is related to this integer to , intuitively the execution time can increase by the factor of , where . ] , the use of ltbf ( or ltba ) may hinder the practicability of model checking .thus we propose some methods to effectively reduce the value of ltbf ( or ltba ) with equivalence criterion . for simplicity reasons ,the following theorems are all discussed using event - triggered aperiodic functions with ltba without loss of generality .[ simple ] let system have one ftos function with periodic deadline and one event - triggered aperiodic function .  w.l.o.g ., let be the timed automaton of the event agent , where be the least time between two consecutive aperiodic events .let be the maximal time interval for the system to finish processing the event ( called deadline interval from now on ) be the time for the event arrival . if the system can not finish processing this event within time , then the system violates the deadline . ] . if , then consider another system , where , i.e. , the only difference is to change from to .then both systems are equivalent regarding their behavior concerning deadline violation .that is , for and , either they both satisfy the deadline , or they both miss the deadline .  an intuitive argument for the bound can be derived using fig .[ fig : proof.explanation ] . tasks and events influence the execution of each other .the execution of an arriving event is influenced by the currently running task . during the deadline interval of the event ,this and all preceding tasks are influenced as well .the chain of influence can only be stopped if the execution is decoupled . sincepreceding tasks are decoupled by definition , two events with a minimal bound of can not influence the execution of the same task . in fig .[ fig : proof.explanation ] , we call a time point _ decoupling point _ if two consecutive tasks immediately before and after are not mutually influenced due to the occurrence of an event .we consider four possible cases in :  1 .consider the case where in , it is proven that no deadline is violated . when the verification engine proofs that the deadline is never violated with in , the deadline of the ftos function in will never be violated because ; the verification engine has already considered all cases in .2 .  consider the case where in , the counter - example indicates that the -th aperiodic task violates the deadline .we further split the discussion in subjects whether it is the first time for to process the event .our goal is to construct a counter - example for deadline violation in from the counter - example in . + . ]1 .  if , i.e. , it is the first time for to execute the aperiodic task , then this deadline violation can also occur in , since no constraints are made for the first occurrence of events in or .2 .  if , consider the -th aperiodic execution which does not violate the deadline .let the time for the coming of event -th be , and let the interval between the -th and the -th event be .the system should finish the -th processing before time . since , from time to , ftos function should finish one of its execution and proceed a new one .let the time for the start of that cycle be .if we change the counter - example time trace such that no event has happened before , we still get a counter - example trace in .this new counter - example trace is also a counter - example trace in .3 .  consider the case where in , the counter - example indicates that the ftos function violates the deadline .let the time which violates the deadline be ( note that is the multiple of ) .let the occurrence of the nearest event be ( if there exists no such event , then both and can deadlock ) .if , then the event is processed before time , the starting of the period which violates the deadline . in this way , the system violates the deadline with only the existence of ftos function , thus in , the deadline will also be violated .if , we consider whether the event is the first one being processedif yes , then the counter - example in is also a counter - example in .2 .  if not , then consider the time where the previous event occurs , and let the time be .since , we can find a decoupling point , where , where at it starts a new period . in this way, we can perform the same technique stated in ( 2-b ) before .4 .  consider the case where in , the counter - example indicates that both the -th aperiodic task and the ftos function violate the deadline .let the time which violates the deadline be ( note that is the multiple of ) , then the event occurs in time . by an argumentation similar to point 3-b , a counter - example trace in can be established .remark : ( 1 ) proposition [ simple ] formulates the insight that previous events occurred long before can not influence the current processing and scheduling , and therefore , are not the root cause of deadline violation . in other words , we could also construct a counter example with a single event as root cause . ( 2 ) the introduction of faults can be viewed analogously . for faults , in ftos ( or similar fault - tolerant systems )the value of is much smaller then ( ltbf ) , and this brings significant advantages for construction of a model with smaller state space .proposition [ simple ] can only be used for very simple systems with only one task and one event . in the following, we will generalize the result to systems consisting of one periodic function with period and several aperiodic functions .[ more.events ] let be a system with aperiodic functions .each function with index , where , is associated with a pair describing the ltba and deadline interval .consider another system , where the only difference is to perform the following change : if for all , , then we change to .both systems and are equivalent regarding their behavior concerning deadline violation .we consider the following cases .  1 .if does not violate the deadline , then so does .2 .  consider the case where in , the counter - example indicates that the -th aperiodic task of type violates the deadline. 1 .  if , let the time for the -th and -th arrival of type- events be and .our goal is to find the decoupling point such that we can overlook all previously happened events .+ since , then within  ] , there are potential decoupling points , where , . + due to the sparsity of events , each type of event arrives at most once within $ ] . for each type ,the according event with deadline interval will overlap in worst case at most of these potential decoupling points . thus the total number of overlapped points is at most , which is less than the number of points among .therefore , there exists at least one point such that it is not overlapped by any deadline interval .thus we can set the decoupling point as . as a result, we can construct an equivalent counter - example where no event has happened before .this new counter - example trace is also a counter - example trace in .2 .  if , let the time for the -th arrival of type- events be . 1 .if for all type of events , the according events occurred at most once before , then the counter - example is also a counter - example in .if there exists some type of events occurred more than once : let for type be the total number of events occurred in the counter - example and be the latest event arrival time .choose such that and , .then we can find the decoupling point between the -th and -th arrival of event with type , similar to the argument in case 2-a .3 .  consider cases where the deadlock happens in the ftos function . 1 .if in the counter - example no event has occurred , then both and can deadlock .2 .  otherwise, first we try to pick an event based on arguments in 2-b - ii . if possible, then the decoupling point can be found , and the counter - example for can be established . if selection based on 2-b - ii is not possible , it follows the statement of 2-b - i that the counter example for is also a counter - example in .lastly , we discuss the most general case .[ more.tasks ] let system have periodic ftos / giotto functions with periodic deadline , where , and aperiodic functions .each function with index , where , is associated with a pair describing the ltba and deadline interval .consider another system , where the only difference is to perform the following change : if for all , , where is the least common multiple of , then we change to .both systems and are equivalent regarding their behavior concerning deadline violation .the main difference to the previous case is that periodic functions with different tasks might influence each other .potential decoupling points occur only at points in time , where all tasks start together .the proof idea is to view multiple periodic functions as a whole by taking the least common multiple .here we omit the detailed proof .for implementation , we extend the functionality of ftos - verify to test the applicability .the verification model is constructed in a format acceptable by uppaal  .note that templates in uppaal are not completely suitable for our usage , since they only represent a fixed behavior with configurable parameters .therefore , algorithms to automatically generate timed automata based on ftos models are needed .we have implemented our automated m2 m transformation tool using openarchitecturewareunder the eclipse modeling framework .  as use case, we apply the verification in the context of our balanced - rod example , where the control functions are replicated on three redundant machines to guarantee fault - tolerance .all components mentioned previously are generated by our automatic conversion technique ; the resulting uppaal system has communicating timed automata .as timing information for the different components , we use currently user - specified assumptions .an integration of wcet - analyzers is foreseen .one desired property specified is the guarantee for the absence of deadline violation , which turns to be the reachability property in uppaal .the overall execution time varies from to minutes depending on the accuracy of the verification model on a intel 2.33 ghz machine using a fifo - based priority - driven scheduler .the memory consumption can reach up to mb .the verification of using a round - robin based scheduler showed to be too memory comsuming .we mention related work , but constrain ourselves in works regarding the analysis of giotto - like systems ; for techniques applying formal verification in real - time analysis , we refer readers to the survey paper by wang . in giotto , the giotto - compiler will perform hardware mapping and apply analysis techniques to check schedulability .many design tools with giotto - like mocs apply similar approaches , for example , tdl  or htl  , but analysis techniques are not explicitly mentioned .one interesting work comes from comdes - ii project  , which is also based on the concept of logical execution time ; here , researchers apply model transformation from system models to verification models .nevertheless , as we focus on fault - tolerant systems , our work differs from the above works with the following facts .first , we encounter a harder problem ; by applying software fault - tolerance , modeling the communication between multiple deployed units is required , and this is not required by other giotto - like mocs . for those mocs , scheduling analysis developed in real - time community could be enough without the use of model checking .furthermore , by proposing the similarity between aperiodic events and fault occurrences , our theoretical criteria is powerful to reduce dramatically the complexity of the model ( not the verification algorithm ) .this is based on our understanding regarding constituents for the complexity of timed verification .in this paper , we discussed the issue of constructing a model to verify timing assumptions in the context of ftos using timed automata .however , due to our general approach , the results can be applied to arbitrary distributed real - time systems .   1 .we give observations concerning modeling of general distributed real - time systems using timed automata and formulate our verification model construction process .2 .  with the context of systems consisting of periodic and aperiodic tasks , we give theoretical criteria how to reduce the size of the verification model , which is particularly useful for our approach .the change of the maximum integer used in the system decreases the required time for verification with exponential scale .a prototype software for the conversion process is constructed with preliminary experiments .our work is currently based on user - specified assumptions regarding the timing of involved components .the next step will be the integration of wcet analyzing tools to have a faithful verification result .gerd behrmann , alexandre david , and kim g. larsen . a tutorial on uppaal . in marco bernardo and flavio corradini , editors , _ formal methods for the design of real - time systems : 4th international school on formal methods for the design of computer , communication , and software systems ( sfm - rt04 ) _ , number 3185 in lncs , pages 200236 .verlag , september 2004 .a. ghosal , a. sangiovanni - vincentelli , c.m .kirsch , t.a .henzinger , and d. iercan . .in _ proceedings of the 6th acm & ieee international conference on embedded software ( emsoft06 ) _ , pages 132141 .acm new york , ny , usa , 2006 .thomas a. henzinger , benjamin horowitz , and christoph m. kirsch .giotto : a time - triggered language for embedded programming . in _ proceedings of the first international workshop on embedded software ( emsoft01 ) _ , pages 166184 .springer - verlag , 2001 .x. ke , p. pettersson , k. sierszecki , and c. angelov . .in _ proceedings of the 14th ieee international conference on embedded and real - time computing systems and applications ( rtcsa08 ) _ , pages 153160 .ieee computer society washington , dc , usa , 2008 .f. wang and c.h .in _ proceedings of the 28th ifip wg 6.1 international conference on formal techniques for networked and distributed systems ( forte08 ) _ , volume 5048 of _ lncs _ , pages 185200 . springer verlag , 2008 .

Abstract: the increasing use of model - based tools enables further use of formal verification techniques in the context of distributed real - time systems . to avoid state explosion , it is necessary to construct verification models that focus on the aspects under consideration .  in this paper , we discuss how we construct a verification model for timing analysis in distributed real - time systems . we ( 1 ) give observations concerning restrictions of timed automata to model these systems , ( 2 ) formulate mathematical representations on how to perform model - to - model transformation to derive verification models from system models , and ( 3 ) propose some theoretical criteria how to reduce the model size . the latter is in particular important , as for the verification of complex systems , an efficient model reflecting the properties of the system under consideration is equally important to the verification algorithm itself . finally , we present an extension of the model - based development tool ftos , designed to develop fault - tolerant systems , to demonstrate our approach .