A Transient Queuing Model for Analyzing and Optimizing Gate Congestion of Railway Container Terminals

Algorithm 1

The state transition probabilities transient calculation procedure.

Procedure: the state to state transition probabilities transient calculation

Initialization:

  Input the initial value of 𝑘,  𝑛,  𝑡_max, 𝑡_𝑎, 𝜆. Set 𝑡=0, 𝑁=0, 𝑠=0,  𝑃_0,0(0)=1, isend = 0

  Initialize the 𝑃_𝑠,𝑣(𝑡_𝑖) and 𝑃^′_𝑠,𝑣(𝑡_𝑖)

while  isend = 0 do

   for  𝑡_𝑖=𝑡 : 𝑡+𝑡_𝑎−1  do

     if  𝑁 < 𝑛  then

       calculate 𝑃^′_0,0(𝑡_𝑖) by (8) and 𝑃_0,0(𝑡_𝑖+1) by (7)

       for  𝑣=1 : 𝑁  do

       calculate 𝑃^′_{𝑣𝑘,𝑣}(𝑡_𝑖) by (9) and 𝑃_{𝑣𝑘,𝑣}(𝑡_𝑖+1) by (7)

         for  𝑎=1 : 𝑣(𝑘−1)  do

             calculate 𝑃^′_{𝑣𝑘−𝑎,𝑣}(𝑡_𝑖) by (10) and 𝑃_{𝑣𝑘−𝑎,𝑣}(𝑡_𝑖+1) by (7)

         end for

       end for

     else if 𝑁≥  𝑛  then

          calculate 𝑃^′_0,0(𝑡_𝑖) and 𝑃_0,0(𝑡_𝑖+1)

       for 𝑣=1 : 𝑛−1  do // calculate the part of probabilities that 𝑣 < 𝑛 //

       calculate 𝑃^′_{𝑣𝑘,𝑣}(𝑡_𝑖), 𝑃^’_{𝑣𝑘−𝑎,𝑣}(𝑡_𝑖) and 𝑃_{𝑣𝑘,𝑣}(𝑡_𝑖+1), 𝑃_{𝑣𝑘−𝑎,𝑣}(𝑡_𝑖+1)

       end for

       for  𝑣=𝑛 : 𝑁  do // calculate the part of probabilities that 𝑣 ≥ 𝑛 //

       calculate 𝑃^′_{𝑣𝑘,𝑣}(𝑡_𝑖) by (11) and 𝑃_{𝑣𝑘,𝑣}(𝑡_𝑖+1) by (7)

         for 𝑎=1 : (𝑛−1)(𝑘−1) do

            calculate 𝑃^′_{𝑣𝑘−𝑎,𝑣}(𝑡_𝑖) by (12) and 𝑃_{𝑣𝑘−𝑎,𝑣}(𝑡_𝑖+1) by (7)

         end for

         for 𝑏=(𝑛−1)(𝑘−1)+1 : 𝑛(𝑘−1)   do

            calculate 𝑃^′_{𝑣𝑘−𝑏,𝑣}(𝑡_𝑖) by (13) and 𝑃_{𝑣𝑘−𝑏,𝑣}(𝑡_𝑖+1) by (7)

         end for

       end for

     end if

   end for

   𝑡_𝑖=𝑡+𝑡_𝑎, 𝑣=𝑁+1

  calculate the probabilities when a new container truck arrives 𝑃_{𝑣𝑘−𝑎,𝑣}(𝑡_𝑖) by (14)

   𝑣=𝑣−1, 𝑡=𝑡_𝑖, 𝑁=𝑁+1

   if  𝑡 ≥ 𝑡_max then

  isend = 1

   end if

end while