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Paper text:
the formation of traffic jams is one of the fundamental problems of traffic flow theory .
traffic jams can form spontanously as well as due to hindrances , e.g. road works or slow cars .
although these hindrances often cover only a small part of the road , they can cause large jams , or , in a more physical language , one observes macroscopic effects due to local defects in the system .
this behaviour is one of the characteristic properties of nonequilibrium systems . in the context of driven lattice gases one can distinguish in principle two types of defects : 1 ) lattice defects , e. g. sites where the mobility of particles is reduced , and 2 ) particle defects , for example slow particles .    for the simplest model for a driven lattice gas in one dimension , the asymmetric exclusions process (
asep)@xcite , it has been shown that both types of defects can cause phase transitions .
slow particles , i.e. particles with a reduced hopping rate , determine completely the flow at small densities . moreover , if the distribution of the hopping rates fulfills certain conditions , one can show analytically that the variance of the distance distribution has a logarithmic divergence at the transition density @xcite .    also lattice defects in the asep
have been investigated extensively @xcite . implementing lattice defects one can observe three different phases : a high and a low density phase , where the average flow of the homogeneous system is recovered , and a segregated phase at intermediate densities , where the flow takes a constant value . despite extensive effort
, exact analytical results exist only for a special type of update @xcite , but approximative descriptions are in reasonable agreement with simulation results .    in this work
we generalize the results of the asep to the nagel - schreckenberg ( nasch ) model of traffic flow ( for a detailed explanation of the model see @xcite ) . compared with the asep ,
the particles ( cars ) can hop more than one site in a single update step and the update rules are applied in parallel to all cars .
therefore slow cars can be considered in two ways .
first , one can think of cars with a smaller maximum velocity and second , of cars with an enlarged braking probability .
this has been done very recently and analogous results to the @xmath0 case have been found @xcite .
in contrast to @xcite we discuss the case of two different maximum velocities in the third chapter . in the next chapter
we show simulation results for the nasch model with defect sites , implemented as sites where the braking probability of cars is higher compared to the rest of the lattice .
an alternative choice has been used in @xcite , where a speed limit in a part of the lattice has been considered .
as already mentioned above , in this chapter we show simulation results of the nasch model on a lattice with defect sites .    fig .
[ single_fund ] shows the fundamental diagram of a system with a _ single _ defect site .
obviously we can distinguish three different phases depending on the density .
in the high and low density phases the average flow of the defect systems takes the same value as in the homogeneous system . for intermediate densities
the flow is constant and limited by the capacity of the defect site .
this behaviour of the average flow can be explained looking at the density profile . in the high and low density phase only local deviations from a constant profile
can be observed , but at intermediate densities one observes a separation into macroscopic high and low density regions . changing the global density within the segregated phase ,
the bulk densities in the high ( @xmath1 ) and low density region ( @xmath2 ) remain constant , only the length of the high and low density region changes .
consequently the average flow is constant in the segregated phase , because the average density in the vicinity of the defect site does not depend on the global density .
near the average position of the shock the density profile decays exponentially from @xmath1 to @xmath2 . therefore one can introduce a length scale @xmath3 , which corresponds to the magnitude of the fluctuations of the shock position .
for @xmath4 we found numerically that @xmath5 for all densities in the segregated phase we took into account .
this scaling behaviour is already known for the asep at @xmath6 .
the modified scaling behaviour @xmath7 at @xmath8 was not found for @xmath9 .
this confirms the picture that the reduction of fluctuations is a consequence of the particle - hole symmetry @xcite .
a good estimate for the plateau value of the flow for the case of @xmath0 can be obtained using the assumption that the system is separated into two regions of constant density , where the results from the homogeneous system can be used . using an argumentation similar to @xcite we
obtain the plateau value of the average flow :    @xmath10    the bulk value of the density in the high ( low ) density region the density is given by @xmath11 ( @xmath12 ) with @xmath13 and @xmath14 .
the shock is located at @xmath15 if @xmath16 or at @xmath17 if @xmath18 .
this approximative treatment of the defect system is in a resonable agreement with the simulation results for large system sizes ( for a more detailed discussion see @xcite ) . in principle
one could treat higher velocities in the same way , but unfortunately no exact analytical description of the homogeneous system has been found by so far .
in order to study the effect of slow cars on the throughput of single lane traffic , we discuss a system which consists of fast cars with maximum velocity @xmath19 _ and _ one slow car with @xmath20 .    in fig .
[ slcfund ] the fundamental diagram of this system is compared with the fundamental diagram of the analogous homogeneous systems . obviously for low densities
the flow is given by @xmath21 in a agreement with the homogeneous system with @xmath22 . compared to the homogeneous system one
observes the linear density dependence also for larger values of the average density .
the average flow depends linearly on the global density if @xmath23 holds , where @xmath24 denotes the stationary flow of the homogeneous system with fast cars .
deviations from this form that are observable near the intersection point of @xmath25 and @xmath24 , are due to the finite size effects .
the reason for the stability of the free flow regime in the mixed system is the gap in front of the slow car in the stationary state . in fig .
[ slcprof ] the density profile relative to the position of the slow car is shown .
obviously a large jam behind the slow car appears and the density in front of the slow car vanishes .
therefore the slow car can move with its free flow velocity @xmath26 .
this velocity determines the velocity of the whole jam and therefore the average velocity of the cars is simply given by @xmath25 .
this is one possible simple scenario for the occurance of moving jams that are well known from measurements .
in addition to the macroscopic form , two details of the density profile have to be discussed . behind the slow car one
observes an oscillating amplitude of the density profile .
these oscillations are well know from the spatial correlation function of the homogeneous system for small @xmath27 and densities near @xmath28 @xcite .
obviously the slow car synchronizes the motion of the faster cars .
furthermore we want to discuss the decay of the density profile at the end of the jam . for finite values of the braking probability
the jamlength is fluctuating .
these fluctuations are responsible for the finite size effects discussed above , because temporarily the gap in front of the slow car vanishes due to fluctuations of the jam length
. measurements of the jamlength show that the fluctuations scale according to @xmath29 . due to the subextensive scaling of the fluctations the segregated states are observable up to @xmath30 if @xmath31 , where @xmath32 denotes the bulk density of the jam .
the investigation of local defects in the nasch model has shown that these defects can change the macroscopic properties of the model . in the case of lattice defects
three phases occur analogous to the asep ( i.e. @xmath33 ) with continuous time update .
the segregated phase also exists for higher maximum velocities , but some details of the density profile are changed . in the low density regime one observes an oscillating profile as a consequence of the parallel update .
the anomalous scaling of the fluctuations of the shock position found for the asep at @xmath34 is absent for @xmath9 , for all parameter combinations that have been taken into account .
this result confirms the picture that the particle - hole symmetry is essential for the reduction of fluctuations .
particle defects also produce phase separated stationary states at low densities .
these states consist of a large jam behind the slowest vehicle and a large gap in front of the slowest car .
the velocity of the moving jam is completely determined by the free flow velocity of the slow car .
the formation of large clusters due to slow cars can also be observed in two - lane traffic , but due to lane changing the lifetime of the large clusters is finite ( at least for low densities ) .
nevertheless it has been shown , that already a small concentration of slow cars changes leads to a drastic reduction of the average flow at low densities @xcite .
our results show that the system poperties can change completely , if local disorder is taken into account . these results might also be important for an understanding of the behaviour of real traffic which are often determined by imperfections. such imperfections are not only the defects considered here , but also other deviations from an ideal system , e.g. ramps or a finite system size .
we investigate the effect of quenched disorder in the nagel - schreckenberg model of traffic flow . 
 spatial inhomogenities , i.e. lattice sites where the braking probability is enlarged , are considered as well as particle disorder , i.e. cars of a different maximum velocity . 
 both types of disorder lead to segregated states .