Traffic assignment for capacitated transportation networks including queueing  with application to freeway corridor control  Page 47 
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where L 2 P (f) = ^ ^ [m a x (0 ,fj  U j) ] (25) j= l ^ Recall th a t S(f) is the network to ta l flow cost f o r the associated capacitated network T. I t may be recognized th a t P (f) is a standard penalty function. 28 Theorem 2, C o ro lla ry t Under assumption la , f o r t\ t ) there e x is ts a useroptim ized flow p a tte rn , denoted f * ( T ) , such th a t § '( f * ( T ) , T ) = min{S' ( f , T )  f f fe a s ib le } . For r o ta tio n a l convenience, we denote S '( f * ( T ) , T ) by S '(T ). Theorem 3, C o r o lla ry : Under assumption 2a f o r T '(T ) there e x is ts a unique useroptimized flow p a tte rn , f * ( T ) . Note th a t f o r T '(T ) there e x is ts a fe a s ib le flow assignment and flow p a tte rn . Now consider the sequence o f time in te r v a ls , { [ 0 ,T .j]} , i = l , 2 , . . . , such th a t T < T2 < . . . » which defines the sequence o f networks (T. ) } . From Theorem 2, c o r o lla r y , f o r each i there e x is ts a useroptimized flow pa tte rn f * ( T , ) and the corresponding queue vector X *(T^). Next we turn to the convergence p ro p e rtie s o f the sequences { )}and { x * ( T .j) } . From conservation o f flo w . Equation (2), the sequence { f* ( T ..) } is bounded; hence there e x is ts a convergent subsequence o f useroptimized flow p a tte rn s. 39
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Title  Traffic assignment for capacitated transportation networks including queueing  with application to freeway corridor control  Page 47 
Repository email  cisadmin@lib.usc.edu 
Full text  where L 2 P (f) = ^ ^ [m a x (0 ,fj  U j) ] (25) j= l ^ Recall th a t S(f) is the network to ta l flow cost f o r the associated capacitated network T. I t may be recognized th a t P (f) is a standard penalty function. 28 Theorem 2, C o ro lla ry t Under assumption la , f o r t\ t ) there e x is ts a useroptim ized flow p a tte rn , denoted f * ( T ) , such th a t § '( f * ( T ) , T ) = min{S' ( f , T )  f f fe a s ib le } . For r o ta tio n a l convenience, we denote S '( f * ( T ) , T ) by S '(T ). Theorem 3, C o r o lla ry : Under assumption 2a f o r T '(T ) there e x is ts a unique useroptimized flow p a tte rn , f * ( T ) . Note th a t f o r T '(T ) there e x is ts a fe a s ib le flow assignment and flow p a tte rn . Now consider the sequence o f time in te r v a ls , { [ 0 ,T .j]} , i = l , 2 , . . . , such th a t T < T2 < . . . » which defines the sequence o f networks (T. ) } . From Theorem 2, c o r o lla r y , f o r each i there e x is ts a useroptimized flow pa tte rn f * ( T , ) and the corresponding queue vector X *(T^). Next we turn to the convergence p ro p e rtie s o f the sequences { )}and { x * ( T .j) } . From conservation o f flo w . Equation (2), the sequence { f* ( T ..) } is bounded; hence there e x is ts a convergent subsequence o f useroptimized flow p a tte rn s. 39 