Process Completing Sequences for Resource Allocation Systems with Synchronization

Algorithm 1

For every 𝑡∈𝑇_Synch

 //Check for violations of the first necessary condition

 Find 𝑟_𝑖 such that 𝑊(𝑟_𝑖,𝑡)+∑_{𝑝∈•𝑡∩𝑃𝑆}𝑢_𝑖(𝑝)>𝐶_𝑖

 If successful, return Not Quasi-live

 //Check for violations of the second necessary condition

 Else 𝑘=1

  While 𝑘<𝐾=|•𝑡∩𝑃_𝑆|

   subset_count =0

   For each 𝑆^𝑘⊆•𝑡∩𝑃_𝑆

    place_count =0

    For each 𝑝∈(𝑆^𝑘)^𝑐

     Find 𝑟_𝑖𝑠𝑡∑_{𝑝∈𝑆𝑘}𝑢_𝑖(𝑝)+𝑊(𝑟_𝑖,•𝑝)>𝐶_𝑖

     If successful, place_count++

    End For

    If place_count = |(𝑆^𝑘)^𝑐|, subset_counter++

   End For

   If subset_count = (^{|•𝑡∩𝑃𝑆|𝑘}), return Not Quasi-live

   𝑘++

  End While

 End For

 Return Unknown