The stability of join-the-shortest-queue models with general input and output processes

Consider a switched queueing network with general routing among its queues. The MaxWeight policy assigns available service by maximizing the objective function $\sum_j Q_j \sigma_j$ among the different feasible service options, where $Q_j$ denotes queue size and $\sigma_j$ denotes the amount of service to be executed at queue $j$. MaxWeight is a greedy policy that does not depend on knowledge of arrival rates and is straightforward to implement. These properties, as well as i...

We say that a random variable is $light$-$tailed$ if moments of order $2+\epsilon$ are finite for some $\epsilon>0$; otherwise, we say that it is $heavy$-$tailed$. We study queueing networks that operate under the Max-Weight scheduling policy, for the case where some queues receive heavy-tailed and some receive light-tailed traffic. Queues with light-tailed arrivals are often delay stable (that is, expected queue sizes are uniformly bounded over time) but can also become dela...

We introduce the \textit{generalized join the shortest queue model with retrials} and two infinite capacity orbit queues. Three independent Poisson streams of jobs, namely a \textit{smart}, and two \textit{dedicated} streams, flow into a single server system, which can hold at most one job. Arriving jobs that find the server occupied are routed to the orbits as follows: Blocked jobs from the \textit{smart} stream are routed to the shortest orbit queue, and in case of a tie, t...

This paper considers a parallel system of queues fed by independent arrival streams, where the service rate of each queue depends on the number of customers in all of the queues. Necessary and sufficient conditions for the stability of the system are derived, based on stochastic monotonicity and marginal drift properties of multiclass birth and death processes. These conditions yield a sharp characterization of stability for systems, where the service rate of each queue is de...

The Rate Control Protocol (RCP) is a congestion control protocol that relies on explicit feedback from routers. RCP estimates the flow rate using two forms of feedback: rate mismatch and queue size. However, it remains an open design question whether queue size feedback in RCP is useful, given the presence of rate mismatch. The model we consider has RCP flows operating over a single bottleneck, with heterogeneous time delays. We first derive a sufficient condition for global ...

The equilibrium properties of allocation algorithms for networks with a large number of nodes with finite capacity are investigated. Every node is receiving a flow of requests and when a request arrives at a saturated node, i.e. a node whose capacity is fully utilized, an allocation algorithm may attempt to re-allocate the request to a non-saturated node. For the algorithms considered, the re-allocation comes at a price: either an extra-capacity is required in the system or t...

The fluid model has proven to be one of the most effective tools for the analysis of stochastic queueing networks, specifically for the analysis of stability. It is known that stability of a fluid model implies positive (Harris) recurrence (stability) of a corresponding stochastic queueing network, and weak stability implies rate stability of a corresponding stochastic network. These results have been established both for cases of specific scheduling policies and for the clas...

Dynamical flow networks serve as macroscopic models for, e.g., transportation networks, queuing networks, and distribution networks. While the flow dynamics in such networks follow the conservation of mass on the links, the outflow from each link is often non-linear due to, e.g., flow capacity constraints and simultaneous service rate constraints. Such non-linear constraints imply a limit on the magnitude of exogenous inflow that is able to be accommodated by the network befo...

This work studies the mean-square stability and stabilization problem for networked feedback systems. Data transmission delays in the network channels of the systems are considered. It is assumed that these delays are i.i.d. processes with given probability mass functions (PMFs). A necessary and sufficient condition of mean-square (input-output) stability is studied for the networked feedback systems in terms of the input-output model and state-space model. Furthermore, accor...

In this note we analyse various stability properties of the max-min fair Rate Control Protocol (RCP) operating with small buffers. We first tackle the issue of stability for networks with arbitrary topologies. We prove that the max-min fair RCP fluid model is globally stable in the absence of propagation delays, and also derive a set of conditions for local stability when arbitrary heterogeneous propagation delays are present. The network delay stability result assumes that, ...