Two-node fluid network with a heavy-tailed random input: the strong stability case

We consider a two-node fluid network with batch arrivals of random size having a heavy-tailed distribution. We are interested in the tail asymptotics for the stationary distribution of a two-dimensional queue-length process. The tail asymptotics have been well studied for two-dimensional reflecting processes where jumps have either a bounded or an unbounded light-tailed distribution. However, presence of heavy tails totally changes the asymptotics. Here we focus on the case of strong stability where both nodes release fluid with sufficiently high speeds to minimise their mutual influence. We show that, like in the one-dimensional case, big jumps provide the main cause for queues to become large, but now they may have multidimensional features. We first find the weak tail asymptotics for a directional marginal of the stationary distribution in an arbitrary direction under Poisson arrival instants. In this case, decomposition formulas for the stationary distribution play a key role. Then we employ the sample-path arguments to find the exact tail asymptotics for the directional marginal under renewal arrival instants assuming one-dimensional batch arrivals.