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## Fundamental relations of traffic flow

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**Fundamental relations of traffic flow**Transportation Systems Engineering**Introduction**• Fundamental relations • Time and space mean speed • Fundamental equation (q, k, v) • Fundamental diagrams (q, k, v) Fundamental relations of traffic flow**Mean speeds**• Time mean speed • average of all vehicles passing a point over a duration of time • It is the simple average of spot speed • Expression for vt • vi spot speed of ith vehicle • n number of observations Fundamental relations of traffic flow**Mean speeds**• Time mean speed • Speeds may be in the form of frequency table • then vt • qi number of vehicles having speed vi • n number of such speed categories Fundamental relations of traffic flow**Mean speeds**• Space mean speed • average speed in a stretch at an instant • It also averages the spot speed • But spatial weightage instead of temporal Fundamental relations of traffic flow**Mean speeds**• Space mean speed - derivation • Consider unit length of a road • let vi is the spot speed of ithvehicle • let ti is the time taken to complete unit distance • ti=1/ vi • If there are n such vehicles, then the average travel time ts is given by Fundamental relations of traffic flow**Mean speeds**• Space mean speed - derivation • If average travel time is ts then • average speed vs is 1/ts • the harmonic mean of the spot speed • If speeds are in a frequency table Fundamental relations of traffic flow**Mean speeds**• Space mean speed • If speeds are in a frequency table • then vs • qi number of vehicles having speed vi • n number of such speed categories Fundamental relations of traffic flow**Mean speeds**• Relation between Vt and Vs • If SMS is vs TMS is vt and the standard deviation of speed is σ • Then • vt > vs sine SD cannot be negative • If all the speed are same, then vs = vt • Derivation (assignment: Refer notes) Fundamental relations of traffic flow**Mean speeds: Illustration**Fundamental relations of traffic flow**Mean speeds**• Example 1 • If the spot speeds are 50, 40, 60,54 and 45, then find the TMS and SMS • Example 2 • The results of a speed study is given in the form of a frequency distribution table. Find the TMS and SMS Fundamental relations of traffic flow**Fundamental relations**• Relationship between q, k, v • Let there be a road with length v km • assume all vehicles are moving with v km/hr • Number of vehicles counted by an observer at A for one hour be n1 • By definition, number of vehicles counted in one hour is flow (q) Fundamental relations of traffic flow**Fundamental relations**• Relationship between q, k, v • n1=q • Density is the number of vehicles in unit distance • n2=kv • But, n1=n2 • since all vehicle have speed v and the distance is v • Therefore q=kv Fundamental relations of traffic flow**Fundamental diagrams**• Follows fundamental relations • Also phenomenological • Flow-density (qk) • Speed-density (kv) • Speed-flow (qv) Fundamental relations of traffic flow**Fundamental diagrams**• Flow-density (q k) curve Fundamental relations of traffic flow**Fundamental diagrams**• Flow-density (q k) curve • The relationship is normally represented by a parabolic curve • At jam density, flow will be zero because the vehicles are not moving. • There will be some density between zero density and jam density, when the flow is maximum. Fundamental relations of traffic flow**Fundamental diagrams**• Speed-density (v k) curve Fundamental relations of traffic flow**Fundamental diagrams**• Speed-density (v k) curve • Max. speed is free flow speed • Max. density is jam density • At zero density, speed is free flow speed • At jam density, speed becomes zero • Most simple assumption is a linear • Non-linear relationships also possible Fundamental relations of traffic flow**Fundamental diagrams**• Speed-flow (v q) curve Fundamental relations of traffic flow**Fundamental diagrams**• Speed-flow (v q) curve • Flow is zero either because there is no vehicles or there are too many vehicles so that they cannot move • At maximum flow, the speed will be in between zero and free flow speed Fundamental relations of traffic flow**Fundamental diagrams**• Combined Fundamental relations of traffic flow**Relation between Vt and Vs**• Derivation • Consider a stream of vehicles with a set of sub-stream flow q1, q2,… ... qi ,... qn having speed v1, v2 , ... vi, ... vn. • Fundamental relation between flow (q), density (k) and mean speed (vs) • Therefore for any sub-stream qi, the relationship is • Summation of all sub-stream flows is total flow q • Summation of all sub-stream density is total density k • Let f i denote the proportion of sub-stream density k i to the total density k (INCOMEPLETE) Fundamental relations of traffic flow**Relation between Vt and Vs**• Derivation • Consider a stream of vehicles with a set of sub-stream flow q1, q2,… ... qi ,... qn having speed v1, v2 , ... vi, ... vn. • Fundamental relation between flow (q), density (k) and mean speed (vs) • Therefore for any sub-stream qi, the relationship is • Summation of all sub-stream flows is total flow q • Summation of all sub-stream density is total density k • Let f i denote the proportion of sub-stream density k i to the total density k (INCOMEPLETE) Fundamental relations of traffic flow**Space mean speed averages the speed over space.**• If ki vehicles have vi speed, then space mean speed is • Time mean speed averages the speed over time, • Substituting in • Rewriting the above equation and substituting in , and then substituting in (INCOMEPLETE) Fundamental relations of traffic flow**By adding and subtracting vs and doing algebraic**manipulations, vt can be written as, • The third term of the equation will be zero because will be zero, since vs is the mean speed of vi . • The numerator of the second term gives the standard deviation of vi. • by definition is 1, Therefore (INCOMEPLETE) Fundamental relations of traffic flow**Thank You**tomvmathew@gmail.com