Fundamental relations of traffic flow Transportation Systems Engineering Transportation Systems Engineering Lecture Notes by Prof. Tom V Mathew

Fundamental relations of traffic flow Introduction Fundamental relations Time and space mean speed Fundamental equation (q, k, v) Fundamental diagrams (q, k, v) TVM_IITB_20080807 Fundamental relations of traffic flow

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 TVM_IITB_20080807 Fundamental relations of traffic flow

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 TVM_IITB_20080807 Fundamental relations of traffic flow

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 TVM_IITB_20080807 Fundamental relations of traffic flow

Fundamental relations of traffic flow Mean speeds Space mean speed - derivation Consider unit length of a road let vi is the spot speed of ith vehicle 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 TVM_IITB_20080807 Fundamental relations of traffic flow

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 TVM_IITB_20080807 Fundamental relations of traffic flow

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 TVM_IITB_20080807 Fundamental relations of traffic flow

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) TVM_IITB_20080807 Fundamental relations of traffic flow

Mean speeds: Illustration TVM_IITB_20080807 Fundamental relations of traffic flow

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 speed range frequency 2-5 1 6-9 4 10-13 14-17 7 TVM_IITB_20080807 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) TVM_IITB_20080807 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 TVM_IITB_20080807 Fundamental relations of traffic flow

Fundamental relations of traffic flow Fundamental diagrams Follows fundamental relations Also phenomenological Flow-density (qk) Speed-density (kv) Speed-flow (qv) TVM_IITB_20080807 Fundamental relations of traffic flow

Fundamental relations of traffic flow Fundamental diagrams Flow-density (q k) curve TVM_IITB_20080807 Fundamental relations of traffic flow

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. TVM_IITB_20080807 Fundamental relations of traffic flow

Fundamental relations of traffic flow Fundamental diagrams Speed-density (v k) curve TVM_IITB_20080807 Fundamental relations of traffic flow

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 TVM_IITB_20080807 Fundamental relations of traffic flow

Fundamental relations of traffic flow Fundamental diagrams Speed-flow (v q) curve TVM_IITB_20080807 Fundamental relations of traffic flow

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 TVM_IITB_20080807 Fundamental relations of traffic flow

Fundamental relations of traffic flow Fundamental diagrams Combined TVM_IITB_20080807 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) TVM_IITB_20080807 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) TVM_IITB_20080807 Fundamental relations of traffic flow

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) TVM_IITB_20080807 Fundamental relations of traffic flow

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) TVM_IITB_20080807 Fundamental relations of traffic flow

Thank You tomvmathew@gmail.com