Stabilization of Branching Queueing Networks

Queueing networks are gaining attraction for the performance analysis of parallel computer systems. A Jackson network is a set of interconnected servers, where the completion of a job at server i may result in the creation of a new job for server j. We propose to extend Jackson networks by "branching" and by "control" features. Both extensions are new and substantially expand the modelling power of Jackson networks. On the other hand, the extensions raise computational questions, particularly concerning the stability of the networks, i.e, the ergodicity of the underlying Markov chain. We show for our extended model that it is decidable in polynomial time if there exists a controller that achieves stability. Moreover, if such a controller exists, one can efficiently compute a static randomized controller which stabilizes the network in a very strong sense; in particular, all moments of the queue sizes are finite

Cooperative dynamics of loyal customers in queueing networks

We consider queueing networks (QN's) with feedback loops roamed by "intelligent” agents, able to select their routing on the basis of their measured waiting times at the QN nodes. This is an idealized model to discuss the dynamics of customers who stay loyal to a service supplier, provided their service time remains below a critical threshold. For these QN's, we show that the traffic flows may exhibit collective patterns typically encountered in multi-agent systems. In simple network topologies, the emergent cooperative behaviors manifest themselves via stable macroscopic temporal oscillations, synchronization of the queue contents and stabilization by noise phenomena. For a wide range of control parameters, the underlying presence of the law of large numbers enables us to use deterministic evolution laws to analytically characterize the cooperative evolution of our multi-agent systems. In particular, we study the case where the servers are sporadically subject to failures altering their ordinary behavio

Reformulation and decomposition of integer programs

In this survey we examine ways to reformulate integer and mixed integer programs. Typically, but not exclusively, one reformulates so as to obtain stronger linear programming relaxations, and hence better bounds for use in a branch-and-bound based algorithm. First we cover in detail reformulations based on decomposition, such as Lagrangean relaxation, Dantzig-Wolfe column generation and the resulting branch-and-price algorithms. This is followed by an examination of Benders’ type algorithms based on projection. Finally we discuss in detail extended formulations involving additional variables that are based on problem structure. These can often be used to provide strengthened a priori formulations. Reformulations obtained by adding cutting planes in the original variables are not treated here.Integer program, Lagrangean relaxation, column generation, branch-and-price, extended formulation, Benders' algorithm

Interference Queueing Networks on Grids

Consider a countably infinite collection of interacting queues, with a queue located at each point of the $d$-dimensional integer grid, having independent Poisson arrivals, but dependent service rates. The service discipline is of the processor sharing type,with the service rate in each queue slowed down, when the neighboring queues have a larger workload. The interactions are translation invariant in space and is neither of the Jackson Networks type, nor of the mean-field type. Coupling and percolation techniques are first used to show that this dynamics has well defined trajectories. Coupling from the past techniques are then proposed to build its minimal stationary regime. The rate conservation principle of Palm calculus is then used to identify the stability condition of this system, where the notion of stability is appropriately defined for an infinite dimensional process. We show that the identified condition is also necessary in certain special cases and conjecture it to be true in all cases. Remarkably, the rate conservation principle also provides a closed form expression for the mean queue size. When the stability condition holds, this minimal solution is the unique translation invariant stationary regime. In addition, there exists a range of small initial conditions for which the dynamics is attracted to the minimal regime. Nevertheless, there exists another range of larger though finite initial conditions for which the dynamics diverges, even though stability criterion holds.Comment: Minor Spell Change

Optimal Information Transmission in Organizations: Search and Congestion

We propose a stylized model of a problem-solving organization whose internal communication structure is given by a fixed network. Problems arrive randomly anywhere in this network and must find their way to their respective “specialized solvers” by relying on local information alone. The organization handles multiple problems simultaneously. For this reason, the process may be subject to congestion. We provide a characterization of the threshold of collapse of the network and of the stock of floating problems (or average delay) that prevails below that threshold. We build upon this characterization to address a design problem: the determination of what kind of network architecture optimizes performance for any given problem arrival rate. We conclude that, for low arrival rates, the optimal network is very polarized (i.e. star-like or “centralized”), whereas it is largely homogenous (or “decentralized”) for high arrival rates. We also show that, if an auxiliary assumption holds, the transition between these two opposite structures is sharp and they are the only ones to ever qualify as optimal. Keywords: Networks, information transmission, search, organization design.Networks, Information transmission, Search, Organization design

Stability Problems for Stochastic Models: Theory and Applications II

Most papers published in this Special Issue of Mathematics are written by the participants of the XXXVI International Seminar on Stability Problems for Stochastic Models, 21­25 June, 2021, Petrozavodsk, Russia. The scope of the seminar embraces the following topics: Limit theorems and stability problems; Asymptotic theory of stochastic processes; Stable distributions and processes; Asymptotic statistics; Discrete probability models; Characterization of probability distributions; Insurance and financial mathematics; Applied statistics; Queueing theory; and other fields. This Special Issue contains 12 papers by specialists who represent 6 countries: Belarus, France, Hungary, India, Italy, and Russia

Exploiting Parallelism in the Design of Peer-to-Peer Overlays

Many peer-to-peer overlay operations are inherently parallel and this parallelism can be exploited by using multi-destination multicast routing, resulting in significant message reduction in the underlying network. We propose criteria for assessing when multicast routing can effectively be used, and compare multi-destination multicast and host group multicast using these criteria. We show that the assumptions underlying the Chuang-Sirbu multicast scaling law are valid in large-scale peer-to-peer overlays, and thus Chuang-Sirbu is suitable for estimating the message reduction when replacing unicast overlay messages with multicast messages. Using simulation, we evaluate message savings in two overlay algorithms when multi-destination multicast routing is used in place of unicast messages. We further describe parallelism in a range of overlay algorithms including multi-hop, variable-hop, load-balancing, random walk, and measurement overlay

A system for the simulation of hardware to software allocation and performance evaluation

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