Algebraic modelling and performance evaluation of acyclic fork-join   queueing networks
  Simple lower and upper bounds on mean cycle time in stochastic acyclic fork-join queueing networks are derived using a (max,+)-algebra based representation of network dynamics. The behaviour of the bounds under various assumptions concerning the service times in the networks is discussed, and related numerical examples are presented. 
Nikolai K. Krivulin
math.OC cs.SY