Nikita D. Vvedenskaya

Fast Jackson Networks with Dynamic Routing

A new class of models of queueing networks with load-balanced dynamic routing is considered. We propose a sufficient condition for positive recurrence of the arising Markov process and a limiting mean-field approximation where the process becomes deterministic and is described by a system of nonlinear ordinary differential equations.

Multi-access system with many users: Stability and metastability

A model of random multi-access system with an ALOHA-type protocol is analyzed when the number N of users is large and the system is overloaded. In the limit as N → ∞, the behavior of the system is described by a nonrandom dynamical system. We give a condition for the dynamical system to have an attractive fixed point and outline cases of several fixed points. The presence of several fixed points indicates that the finite system may exhibit a metastability phenomenon.

Improved Performance of Mobile Data Networks Using Stack Algorithms and Receiver Capture

We consider a slotted multi-user channel with a common packet receiver which may be captured by one user packet in the presence of other packet transmissions. The unsuccessful user terminals repeat their transmissions using a stack-algorithm protocol for conflict resolution. Three alternative algorithms are considered; these differ by the feed-back information available to the contending users. The capture probability is modelled by various analytical expressions, corresponding to advanced receivers in typical fading packet radio channels. The average packet delay and the channel throughput are calculated for different combinations of algorithms and capture models: considerably enhanced performance appears possible for certain of these combinations.

A non-linear model of limit order book dynamics

This paper introduces a prototype model in an attempt to capture some aspects of limit order book dynamics simulating market trading mechanisms. We start with a discrete time/space Markov process and then perform a re-scaling procedure leading to a deterministic dynamical system controlled by non-linear ODEs. This allows us to introduce approximants for the equilibrium distribution of the model represented by fixed points of deterministic dynamics.

Infinite parallel job allocation (extended abstract)

In recent years, the task of allocating jobs to servers has been studied with the “balls and bins” abstraction. Results in this area exploit the large decrease in maximum load that can be achieved by allowing each job (ball) a little freedom in choosing its destination server (bin). In this paper we examine an infinite and parallel allocation process (see [ABS98]) which is related to the “balls and bins” abstraction. The simple process can be used to model many problems arising in applications like load balancing, data accesses for parallel data servers, hashing, and PRAM simulations. Unfortunately, the parallel allocation process behaves in a highly non-uniform manner which makes its analysis challenging. Even the typically simple question of for which arrival rates the process is stable, is highly non-trivial. In order to cope with this non-uniform behavior we introduce a new sequential process and show (via simulations) that the sequential process models the behavior of the parallel one very accurately. We develop a system of ordinary differential equations in order to describe the behavior of our sequential process and present a thorough analysis of the performance this process. For example, we show that the queue length distribution decreases double-exponentially. Finally, we present simulation results indicating that the solutions to the differential equations very well predict the queue length distribution of our sequential process and the largest injection rate for which it is stable. Summarizing, we can conclude that in all the performance characteristics we have measured experimentally, the parallel and the sequential process are closely related. This indicates that the obtained solution of the differential equations and the results presented above are applicable to the parallel process, too.

Configuration of overloaded servers with dynamic routing

We consider overload of servers in a network with dynamic routing of messages. The system consists of k servers and ℓ independent Poisson input flows. Messages from each flow are directed to m servers, and each message is directed to a server that is the least loaded at the moment of its arrival. In such a system, configuration of overloaded servers depends on the intensity of input flows. A similar effect was considered in [1] for a system with another geometry.

Circle of interacting servers: Spontaneous collective behavior in the case of large fluctuations

We consider large fluctuations and overload of servers in a network with dynamic routing of messages. The servers form a circle. The number of input flows is equal to the number of servers; the messages of a flow are distributed between two neighboring servers; upon its arrival, a message is directed to the least loaded of these servers. Under the condition that at least two servers are overloaded, the number of overloaded servers in such a network depends on the rate of input flows. In particular, there exists a critical level of the input rate above which all servers are most probably overloaded.

Fast networks with dynamic routing

A class of queueing network models with dynamic routing based on the principle of balanced load is proposed. The principle of dynamic routing considered is to select a server with the shortest queue among a sample collection of servers chosen at random.

Single-Frequency Packet Network Using Stack Algorithm and Multiple Base Stations

This paper evaluates a new method to combine contiguous frequency reuse with random access in wireless packet-switched networks. We consider a radio network with two base stations receiving packets transmitted by a large population of mobile users. Both stations share the same channel and therefore transmissions to one base station in one cell interfere with transmissions to the other base station in the other cell. A simulation of a two-cell system is performed, to study the interaction of retransmission traffic in two cells. To consider the performance of such system analytically, we model a one-cell system with time-varying channel properties: if only one station is receiving messages from its cell, the channel is assumed to be in ”good state”; if terminals in both cells are busy the channels are assumed to be in ”bad state”. For conflict resolution, the stack-algorithm is used. The packet delay is calculated. Our results confirm that in wireless packet networks, it is advantageous to allow neighbouring cells to share the same channel.

A Local Large Deviation Principle for Inhomogeneous Birth–Death Processes

The paper considers a continuous-time birth–death process where the jump rate has an asymptotically polynomial dependence on the process position. We obtain a rough exponential asymptotic for the probability of trajectories of a re-scaled process contained within a neighborhood of a given continuous nonnegative function.