Derivation of a fundamental diagram for urban traffic flow

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6 The European Physical Journal B into (26) [andwithf i = f 0 i , T cyc = T 0 cyc ,seeEq.(33)], we get L i ρ av i (u i , {fj 0 av })=ΔNi (u i , {fj 0 }) (1 − fi 0 = u i ̂Qi )2 Tcyc 0 ({f j 0}) , (39) 2(1 − u i ) which finally yields u i (ρ av i L i , {f 0 j }) = ( ̂Q 1+(1− fi 0 ) 2 i Tcyc 0 ({f j 0}) ) −1 2ρ av . i L i (40) Thiscanbeinsertedintoequation(26) togive Ti av (ρ av i L i , {fj 0 }) = u i (ρ av i ρ av i L i L i , {fj 0}) ̂Q i 3.1 Transition to congested traffic = ρav i L i +(1− f 0 T cyc({f 0 i ̂Q )2 j 0}) . (41) i 2 The utilizations u i increase proportionally to the arrival flows A i , i.e. they go up during the rush hour. Eventually, ∑ f j = ∑ (1 + δ j )u j → 1, (42) j j which means that the intersection capacity is reached. Sooner or later, there will be no excess capacities anymore, which implies δ i → 0andf i → u i .Inthiscase, we do not have any finite time periods ΔT i − T i , during which there are no delayed vehicles and where the departure flow O i (t) agrees with the arrival flow A i . Therefore, O i (t) =γ i (t) ̂Q i according to equation (12), and certain relationships simplify. For example, the utilization is given as integral of the permeability over one cycle time T cyc , divided by the cycle time itself: u i = 1 T cyc t i0+T ∫ cyc t i0 dt ′ γ i (t ′ ). (43) Moreover, in the case of constant in- and outflows, i.e. linear increase and decrease of the queue length, the average number ΔNi av of delayed vehicles is just given by half of the maximum number of delayed vehicles 1 ΔN av i = max ΔNi . (44) 2 1 Note that the formulas derived in this paper are exact under the assumption of continuous flows. The fact that vehicle flows consist of discrete vehicles implies deviations from our formulas of upto 1 vehicle, which slightly affects the average travel times as well. In Figure 2, for example, the number of vehicles in the queue is Ni max − 1 rather than Ni max . Therefore, the related maximum delay time is (1 − u i)T cyc(Ni max − 1)/Ni max , which reduces the average delay time Ti av by (1 − u i)T cyc/(2Ni max ). As a consequence, we have T i ({u j })= L i V 0 i + (1 − u i)T los 2 ( 1 − ∑ j u ). (45) j The average speed Vi av of traffic stream i is often determined by dividing the length L i of the road section reserved for it by the average travel time T i = Ti 0 + Ti av , which gives V av i ({u j })= ( L i T i ({u j }) = 1 V 0 i ) −1 + (1 − u i)T los 2L i (1 − ∑ j u , j) (46) and can be generalized with equation (28) tocaseswith an efficiencies ɛ i ≠0: V av i ({u j },ɛ i )= ( 1 V 0 i ) −1 (1 − u i )T los +(1− ɛ i ) 2L i (1 − ∑ j u . j) (47) Note, however, that the above formulas for the average speed are implicitly based on a harmonic rather than an arithmetic average. When correcting for this, equation (46), for example, becomes V av i ({u j })= ∣ L i u i ̂Qi ∣∣∣∣ ({u j }) ln 1+ ΔN max i ΔN max i ({u j }) u i ̂Qi T 0 i ∣ . (48) As is shown in Appendix A, this has a similar Taylor approximation as the harmonic average (47). The latter is therefore reasonable to use, and it is simpler to calculate. As expected from queuing theory, the average travel time (45) diverges, when the sum of utilizations reaches the intersection capacity, i.e. ∑ j u j → 1. In this practically relevant case, the traffic light would not switch anymore, which would frustrate drivers. For this reason, the cycle time is limited to a finite value T max cyc ({u0 j })= T los 1 − ∑ , (49) j u0 j where typically u 0 j ≤ u j. This implies that the sum of utilizations must fulfill ∑ u j ≤ ∑ u 0 j =1− T los T max . (50) j j cyc As soon as this condition is violated, we will have an increase of the average number of delayed vehicles in time, which characterizes the congested regime discussed in the next section. 4 Fundamental relationships for congested traffic conditions In the congested regime, the number of delayed vehicles does not reach zero anymore, and platoons cannot be served without delay. Vehicles will usually have to
D. Helbing: Derivation of a fundamental diagram for urban traffic flow 7 wait several cycle times until they can finally pass the traffic light. This increases the average delay time enormously. It also implies that there are no excess green times, which means δ i = 0. Consequently, we can also assume O i (t) = ̂Q i , as long as the outflow from road sections during the green phase is not (yet) obstructed (otherwise see Sect. 5). Formula (13) applies again and implies for time-independent arrival flows A i ΔN i (t i0 + kT max cyc )=ΔN i(t i0 )+ where u 0 i = 1 kT max cyc t i0+kT max cyc ∫ t i0 dt ′[ A i − γ i (t ′ ) ̂Q i ] = ΔN i (t i0 )+(A i − u 0 i ̂Q i )kT max cyc , (51) t i0+kT max cyc ∫ t i0 dt ′ γ i (t ′ ) 0). Assuming that t i0 is the time at which congestion sets in, equation (58) can be generalized to continuous time t, allowing us to estimate the number of additional stops of a vehicle arriving at time t: ⌊ n s (u i , {u 0 j},t)= ⌋ A i (t − t i0 ) ̂Q i u 0 i T cyc max({u0 j }) = ⌊ ⌋ u i (t − t i0 ) u 0 i T cyc max ({u 0 j }) . (59)
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Derivation of a fundamental diagram for urban traffic flow

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