Computable exponential bounds for intree networks with routing

In this paper, we refine the calculus proposed previously by Chang et al. (1994). The new calculus, including network operations for multiplexing, input-output relation, and routing, allows us to compute tighter exponential bounds for the tail distributions of queue lengths in intree networks with routing. In particular, if external arrival processes and routing processes are either Markov arrival processes or autoregressive processes, the stationary queue length at a local node is stochastically bounded above by the sum of a constant and an Erlang random variable. The decay rate of the Erlang random variable is not greater than (in some cases equal to) the decay rate of the tail distribution of the stationary queue length. The number of stages of the Erlang random variable is the number of external arrival processes and routing processes contributing to its queue length. For the single queue case, both the lower and upper-bounds are derived.