In this paper we study the number of customers in infinite-server queues with a self-exciting (Hawkes) arrival process. Initially we assume that service requirements are exponentially distributed and that the Hawkes arrival process is of a Markovian nature. We obtain a system of differential equations that characterizes the joint distribution of the arrival intensity and the number of customers. Moreover, we provide a recursive procedure that explicitly identifies (transient and stationary) moments. Subsequently, we allow for non-Markovian Hawkes arrival processes and nonexponential service times. By viewing the Hawkes process as a branching process, we find that the probability generating function of the number of customers in the system can be expressed in terms of the solution of a fixed-point equation. We also include various asymptotic results: we derive the tail of the distribution of the number of customers for the case that the intensity jumps of the Hawkes process are heavy tailed, and we consider a heavy-traffic regime. We conclude by discussing how our results can be used computationally and by verifying the numerical results via simulations.

A complete understanding of the dynamics of Multipath TCP (MPTCP) over Cellular and WiFi networks is still lacking. This is a highly challenging issue, as a consequence of the complex interdependencies between the losses, packet reordering due to heterogeneous wireless channel features, errors and link layer retransmissions, as well as their (joint) influence on MPTCP's control mechanism. In this paper we develop a comprehensive approach that is capable of assessing the performance of long-lived MPTCP flows with joint WiFi and Cellular network access, taking into account the diverse characteristics of both types of networks. Relying on a parallel queueing model, we develop a framework that features the controllable network parameters, such as the retransmission limit and the buffer sizes, so as to capture their impact on the TCP-level performance. We include a variety of numerical and simulation results highlighting various non-trivial findings and insights for adaptive MPTCP design. In particular, we design a novel MPTCP algorithm to exploit the route heterogeneity, and demonstrate that the proposed algorithm is capable of learning the network's characteristics. It also identifies the improved MPTCP window increment parameters for any given set of channel errors, retransmission limits, and buffer sizes.

We analyze tail asymptotics of a two-node tandem queue with spectrally-positive Lévy input. A first focus lies in the tail probabilities of the type ℙ(Q ₁>α x,Q ₂>(1−α)x), for α∈(0,1) and x large, and Q _i denoting the steady-state workload in the ith queue. In case of light-tailed input, our analysis heavily uses the joint Laplace transform of the stationary buffer contents of the first and second queue; the logarithmic asymptotics can be expressed as the solution to a convex programming problem. In case of heavy-tailed input we rely on sample-path methods to derive the exact asymptotics. Then we specialize in the tail asymptotics of the downstream queue, again in case of both light-tailed and heavy-tailed Lévy inputs. It is also indicated how the results can be extended to tandem queues with more than two nodes.