In the present paper, we study the evolution of an overloaded cyclic polling model that starts empty. Exploiting a connection with multitype branching processes, we derive fluid asymptotics for the joint queue length process. Under passage to the fluid dynamics, the server switches between the queues infinitely many times in any finite time interval causing frequent oscillatory behavior of the fluid limit in the neighborhood of zero. Moreover, the fluid limit is random. Additionally, we suggest a method that establishes finiteness of moments of the busy period in an M/G/1 queue.

model the dynamic interaction among an evolving population of elastic flows competing for several links. With policies based on optimization procedures, such models are of interest both from a queueing theory and operations research perspective. In the present paper, we focus on bandwidth-sharing networks with capacities and arrival rates of a large order of magnitude compared to transfer rates of individual flows. This regime is standard in practice. In particular, we extend previous work by Reed and Zwart [Reed J, Zwart B (2010) Limit theorems for bandwidth-sharing networks with rate constraints. Revised, preprint http://people.stern.nyu.edu/jreed/Papers/BARevised.pdf] on fluid approximations for such networks: we allow interarrival times, flow sizes, and patient times (i.e., abandonment times measured from the arrival epochs) to be generally distributed, rather than exponentially distributed. We also develop polynomial-time computable fixed-point approximations for stationary distributions of bandwidth-sharing networks, and suggest new techniques for deriving these types of results.

The synchronization process inherent to the Bitcoin network gives rise to an infiniteserver model with the unusual feature that customers interact. Among the closed-form characteristics that we derive for this model is the busy period distribution which, counterintuitively, does not depend on the arrival rate. We explain this by exploiting the equivalence between two specific service disciplines, which is also used to derive the model's stationary distribution. Next to these closed-form results, the second major contribution concerns an asymptotic result: a fluid limit in the presence of service delays. Since fluid limits arise under scalings of the law-of-large-numbers type, they are usually deterministic, but in the setting of the model discussed in this paper the fluid limit is random (more specifically, of growth-collapse type).

Random multiple-access protocols of type ALOHA are used to regulate networks with a star configuration where client nodes talk to the hub node at the same frequency (finding a wide range of applications among telecommunication systems, including mobile telephone networks and WiFi networks). Such protocols control who talks at what time sharing the common idea "try to send your data and, if your message collides with another transmission, try resending later".In the present paper, we consider a time-slotted ALOHA model where users are allowed to renege before transmission completion. We focus on the scenario that leads to overload in the absence of impatience. Under mild assumptions, we show that the fluid (or law-of-large-numbers) limit of the system workload coincides a.s. with the unique solution to a certain integral equation. We also demonstrate that the fluid limits for distinct initial conditions converge to the same value as time tends to infinity. Keywords

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