Model II: Inter-site traffic with input (source) and output (sink) of vehicles
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We now introduce two physical modifications in the inter-site traffic model studied in the preceding sections: (a) there is a daily input of new vehicles from site $E$ to city $A$, that are integrated into the network and (b) there is also an output of old vehicles that are taken off the circuit from the same site $A$ and are stocked in the site $F$. These new features are drawn in the digraph with additional edges and loops, labeled as $v_{1}$ and $v_{2}$, $r_{5}$ and $r_{6},$ respectively, as shown in Fig. \[fig03\].

![[]{data-label="fig03"}](Trafficinputoutput2.jpg){height="2.44in" width="2.6in"}

In this digraph the letters $A$, $B$, $C$, $D$, $E$ and $F$ label the vertices, whereas the $r_{i}$’s are for the loops and $q_{i}$, $p_{i}$, $s_1$ and $v_1$ are for the edges. One expresses the network by the array in Table \[A5\],

           Vertices           $A$           $B$           $C$           $D$           $E$           $F$           Sum of the line entries
  --------------------------- ------------- ------------- ------------- ------------- ------------- ------------- -------------------------
              $A$             $R_{1}$       $P_{1}$       $P_{5}$       $Q_{4}$       $0$           $S_{1}$       
              $B$             $Q_{1}$       $R_{2}$       $P_{2}$       $P_{6}$       $0$           $0$           
              $C$             $Q_{5}$       $Q_{2}$       $R_{3}$       $P_{3}$       $0$           $0$           
              $D$             $P_{4}$       $Q_{6}$       $Q_{3}$       $R_{4}$       $0$           $0$           
              $E$             ${\ V}_{1}$   