Bounds on mean cycle time in acyclic fork-join queueing networks
  Simple lower and upper bounds on mean cycle time in stochastic acyclic fork-join networks are derived using the $(\max,+)$-algebra approach. 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