Explanation Formula used: Write the expression for math program of system for system optimal function. S ( x ) UE = ∑ n ∫ 0 x 2 t n ( w ) d w …… (I) Here, S ( x ) UE is system, optimal function, n is specific route, w is flow of x n , t n ( w ) is the performance function corresponding to route n . Write the expression for first route travel time. t 1 = 5 + 3.5 x 1 …… (II) Here, t 1 is first route travel time, x 1 is traffic flow in first route. Write the expression for second route travel time. t 2 = 1 + 0.5 x 2 2 …… (III) Here, t 2 is first route travel time, x 2 is traffic flow in second route Write the expression for total traffic flow. q = x 1 + x 2 …… (IV) Here, q is traffic flow, x 1 is traffic flow in first route and x 2 is traffic flow in second route. Write the expression for travel time for equilibrium condition. t = q × t 60 × 1000 …… (V) Here, q is total traffic flow, t is travel time. The expression for total traffic is expressed as, x 2 = x 1 − q …… (VI) Here, x 1 is traffic flow in first route and x 2 is traffic flow in second route. Write the expression for system optimal function, S ( x ) so = w 1 + w 2 …… (VII) Here, S ( x ) so is system optimal function, w 1 is flow for x 1 and w 2 is flow for x 2 . Write the expression for total travel time for system optimal function, t SO = 1000 ( x S1 × t S1 + x S2 × t S2 ) 60 …… (VIII) Here, t SO is travel time for system optimal function, x S 1 is traffic flow in first route and x S 2 is traffic flow in second route, t S1 is time travel for first rout and t S2 is time travel for second rout. Write the expression for difference equilibrium condition and system optimal function is, t diffrence = t − t SO Here, t diffrence is difference between equilibrium condition and system optimal function, t is travel time for equilibrium condition and t SO system optimal function. Calculation: Substitute, values from equation (II) and (III) in equation number (I). S ( x ) UE = ∫ 0 x 2 t 1 ( w ) d w + ∫ 0 x 2 t 2 ( w ) d w = ∫ 0 q − x 2 ( 5 + 3.5 w ) d w + ∫ 0 x 2 ( 1 + 0.5 w 2 ) d w = [ 5 w + 3.5 w 2 2 ] 0 q − x 2 + [ w + 0.5 w 3 3 ] 0 x 2 = [ [ ( 5 ( q − x 2 ) + 3.5 ( q − x 2 ) 2 2 ) − ( 5 ( 0 ) + 3.5 ( 0 ) 2 2 ) ] + [ ( ( x 2 ) + 0.5 ( x 2 ) 3 3 ) − ( ( 0 ) + 0.5 ( 0 ) 3 3 ) ] ] Solve further S ( x ) UE = [ ( 5 ( q − x 2 ) + 3.5 ( q − x 2 ) 2 2 ) ] + [ ( ( x 2 ) + 0.5 ( x 2 ) 3 3 ) ] = [ ( ( 5 q − 5 x 2 ) + ( 3.5 q 2 + 3.5 x 2 2 − 2 ( 3.5 ) q x 2 ) 2 2 ) + ( x 2 ) + 0.1667 ( x 2 ) 3 ] = 5 q − 4 x 2 + 1.75 q 2 + 1.75 x 2 2 − 3.5 q x 2 + 0.16667 x 2 3 …… (X) Differentiate equation number (X) with respect to x 2 . d d x 2 S ( x ) UE = d d x 2 ( 5 q − 4 x 2 + 1.75 q 2 + 1.75 x 2 2 − 3.5 q x 2 + 0.16667 x 2 3 ) = − 4 + 3.5 x 2 − 3

The difference in total vehicle travel time between user-equilibrium and system-optimal functions.

Principles of Highway Engineering and Traffic Analysi (NEW!!)

6th Edition

ISBN: 9781119305026

Author: Fred L. Mannering, Scott S. Washburn

Publisher: WILEY

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Intersection Design

An intersection is a location where two or more roads or railways meet or join. The traffic movement is controlled at the intersection that depends on its design and type. It governs the operation, efficiency, speed, capacity, and safety of the highways,…

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Chapter 8, Problem 33P

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The difference in total vehicle travel time between user-equilibrium and system-optimal functions.

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