•   diagonalization of vector with components for i=1 to k.
  
  
• a(t):  k X 1 surplus aggregate level state vector (surplus or shortfall of production).
  
  
• n(t):  k X 1 state vector for number of operational machines.
  
  
• P:  k X 1 maximum production capacity vector.
  
  
• c(t):  k X 1 production level vector (utilization of each machine for stage j at time t).
  
  
• u(t):  k X 1 control vector.
  
  
• d(t):  k X 1 demand rate vector.
  
  
• g(t):  k X 1 Gaussian Noise coefficient vector.
  
  
• dW(t):  k X 1 Guassian noise vector.
  
  
•   k X 1 Poisson noise vector for broken machines repair process.
  
  
•   k X 1 Poisson noise vector for operational machines failure process.