ON/OFF Sources in an Interconnection Network: Performance Analysis when Packets are Routed to the Shortest Queue of two Randomly Selected Nodes

The authors investigate a system with N servers and with N sources connected with the servers. A sources can be in state «on» or «off». In state «on» the source generates the Poisson flow of packets of rate \lb. The service time of a packet is distributed exponentially with mean one. Upon its arrival a packet is directed to the server with the shortest queues of the following two servers: the server where the packet has been generated and another randomly selected server. The queue length probability as N \to \infty$ is investigated

Nonlinear axisymmetric liquid currents in spherical annuli

A numerical analysis of non-linear axisymmetric viscous flows in spherical annuli of different gap sizes is presented. Only inner sphere was supposed to rotate at a constant angular velocity. The streamlines, lines of constant angular velocity, kinetic energy spectra, and spectra of velocity components are obtained. A total kinetic energy and torque needed to rotate the inner sphere are calculated as functions of Re for different gap sizes. In small-gap annulus nonuniqueness of steady solutions of Navier-Stokes equations is established and regions of different flow regime existences are found. Numerical solutions in a wide-gap annulus and experimental results are used in conclusions about flow stability in the considered range of Re. The comparison of experimental and numerical results shows close qualitative and quantitative agreement

Fast Jackson-Type Networks with Dynamic Routing

We propose a new class of models of queueing networks with load-balanced dynamic routing. The paper extends earlier works, including [FC], [FMcD], [VDK], where systems with no feedback were considered. The main results are: (a) a sufficient condition for positive recurrence of the arising Markov process and (b) a limiting mean-field picture where the process becomes deterministic and is described by a system of non-linear ODEs

Dynamic Routing in the Mean-Field Approximation

A queuing system is considered, with stations $1$, $2$, where station $i$ contains a collection $S_i$ of infinite-buffer FCFS single servers. The number of servers in each $S_i$ is $N$, and we assume that $N$ is large (formally, $N\to\infty$). The system is fed by a Poisson flow of rate 2\l N, with i.i.d. exponential service times of mean one. Upon arrival, a customer chooses two servers according to the following rule. He performs two independent trials, each time choosing station $1$ with probability $p_1$ and $2$ with probability $p_2=1-p_1$ and then choosing a server at random from the corresponding collection $S_i$. He then selects the server with a shortest queue from the two, breaking ties at random if necessary. We assume that $p_1\geq p_2\geq 0$; the main result is that (a) the inequalities $\lambda <1$, $2\lambda p_1^2 <1$, are necessary and sufficient for the existence (and uniqueness) of a (stable) stationary regime, and (b) under these inequalities, a stationary queue-size distribution has a super-exponentially decaying tail. These results are in a striking contrast with a `linear' model where a customer simply chooses a station with probabilities $p_1$ and $p_2$ and then selects a server at random from the corresponding $S_i$. Here, the necessary and sufficient condition is $2\lambda p_1<1$, and the queue-size distribution is geometric

Performance evaluation of a single queue under multi-user TCP/IP connections version 2

We study the performance of several TCP connections through the bottleneck of a slow network accessed via a single queue with high but finite capacity. Using mean-field asymptotic methology, we establish some asymptotical results about queue length distribution and window size distribution when the number of user increases proportionally to buffer capacity. We show that the difference between the actual queue length and its maximal capacity tends to be exponentially distributed. We give a precise determination of the window size asymptotic distribution and we prove that under our model the small window size has log-normal distribution

Performance Evaluation of a Single Queue under Multi-User TCP/IP Connections

We study the performance of several TCP connections through the bottleneck of a slow network accessed via a single queue with high capacity. Using mean-field approximation methology, we establish some asymptotical results about queue length distribution and windo size distribution when the number of user increases proportionally to buffer capacity. We also give an evaluatio- n of TCP fairness under these traffic conditions

Delay, memory, and messaging tradeoffs in distributed service systems

We consider the following distributed service model: jobs with unit mean, exponentially distributed, and independent processing times arrive as a Poisson process of rate $\lambda n$, with $0<\lambda<1$, and are immediately dispatched by a centralized dispatcher to one of $n$ First-In-First-Out queues associated with $n$ identical servers. The dispatcher is endowed with a finite memory, and with the ability to exchange messages with the servers. We propose and study a resource-constrained "pull-based" dispatching policy that involves two parameters: (i) the number of memory bits available at the dispatcher, and (ii) the average rate at which servers communicate with the dispatcher. We establish (using a fluid limit approach) that the asymptotic, as $n\to\infty$, expected queueing delay is zero when either (i) the number of memory bits grows logarithmically with $n$ and the message rate grows superlinearly with $n$, or (ii) the number of memory bits grows superlogarithmically with $n$ and the message rate is at least $\lambda n$. Furthermore, when the number of memory bits grows only logarithmically with $n$ and the message rate is proportional to $n$, we obtain a closed-form expression for the (now positive) asymptotic delay. Finally, we demonstrate an interesting phase transition in the resource-constrained regime where the asymptotic delay is non-zero. In particular, we show that for any given $\alpha>0$ (no matter how small), if our policy only uses a linear message rate $\alpha n$, the resulting asymptotic delay is upper bounded, uniformly over all $\lambda<1$; this is in sharp contrast to the delay obtained when no messages are used ($\alpha = 0$), which grows as $1/(1-\lambda)$ when $\lambda\uparrow 1$, or when the popular power-of-$d$-choices is used, in which the delay grows as $\log(1/(1-\lambda))$

Balanced Allocations and Double Hashing

Double hashing has recently found more common usage in schemes that use multiple hash functions. In double hashing, for an item $x$, one generates two hash values $f(x)$ and $g(x)$, and then uses combinations $(f(x) +k g(x)) \bmod n$ for $k=0,1,2,...$ to generate multiple hash values from the initial two. We first perform an empirical study showing that, surprisingly, the performance difference between double hashing and fully random hashing appears negligible in the standard balanced allocation paradigm, where each item is placed in the least loaded of $d$ choices, as well as several related variants. We then provide theoretical results that explain the behavior of double hashing in this context.Comment: Further updated, small improvements/typos fixe