Traffic assignment for capacitated transportation networks including queueing  with application to freeway corridor control  Page 44 
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and tends to i n f i n i t y . In the standard approach, the same li n k cost fu n c tio n is used at each it e r a t io n . The advantage o f our method is th a t we can prove th a t the sequence o f assignments converges to an e q u ilib riu m assignment f o r the capacitated network and th a t the assignment at the f in a l it e r a t io n approximates the e q u ilib riu m assignment f o r the capacitated network (the amount o f e r ro r decreasing,! o f course, as more ite r a tio n s are taken). This method may be recognized as a penalty fu n ctio n o f mathematical programming; whereas, the standard method is an approximation o f a penalty fu n c tio n approach. Moreover, th is development provides fu r th e r in s ig h ts f o r e q u ilib riu m assignment as w ell as in s ig h t in to the time e vo lu tio n o f queues and the nature o f the physical flow process. 4.2 MODEL Given the network T and a fix e d time in te rv a l [ 0 , T ] , we define the re la te d uncapacitated network t '(T ) = ( G , d , y '( T ) ) , where y*(T) is a c o lle c tio n o f li n k u n it cost fu n c tio n s . The demand re la te d to od p a ir j is taken as djT. Demand djT is s a t is f ie d by the assignment o f commodities to routes kcRj; route assignments are nonnegative numbers p^(T), k = 1 , . . . , R , the vector o f route assignments being p(T). p(J) is a fe a s ib le assignment i f i t s a t is f ie s the conservation equation dT = Bp(T) (18) The amount q j(T ) associated w ith l i n k j is given by R q X T ) = E a .p. (T) (19) J k=l 36
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Title  Traffic assignment for capacitated transportation networks including queueing  with application to freeway corridor control  Page 44 
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Full text  and tends to i n f i n i t y . In the standard approach, the same li n k cost fu n c tio n is used at each it e r a t io n . The advantage o f our method is th a t we can prove th a t the sequence o f assignments converges to an e q u ilib riu m assignment f o r the capacitated network and th a t the assignment at the f in a l it e r a t io n approximates the e q u ilib riu m assignment f o r the capacitated network (the amount o f e r ro r decreasing,! o f course, as more ite r a tio n s are taken). This method may be recognized as a penalty fu n ctio n o f mathematical programming; whereas, the standard method is an approximation o f a penalty fu n c tio n approach. Moreover, th is development provides fu r th e r in s ig h ts f o r e q u ilib riu m assignment as w ell as in s ig h t in to the time e vo lu tio n o f queues and the nature o f the physical flow process. 4.2 MODEL Given the network T and a fix e d time in te rv a l [ 0 , T ] , we define the re la te d uncapacitated network t '(T ) = ( G , d , y '( T ) ) , where y*(T) is a c o lle c tio n o f li n k u n it cost fu n c tio n s . The demand re la te d to od p a ir j is taken as djT. Demand djT is s a t is f ie d by the assignment o f commodities to routes kcRj; route assignments are nonnegative numbers p^(T), k = 1 , . . . , R , the vector o f route assignments being p(T). p(J) is a fe a s ib le assignment i f i t s a t is f ie s the conservation equation dT = Bp(T) (18) The amount q j(T ) associated w ith l i n k j is given by R q X T ) = E a .p. (T) (19) J k=l 36 