41 research outputs found
Analysis of Push-type Epidemic Data Dissemination in Fully Connected Networks
Consider a fully connected network of nodes, some of which have a piece of
data to be disseminated to the whole network. We analyze the following
push-type epidemic algorithm: in each push round, every node that has the data,
i.e., every infected node, randomly chooses other nodes
in the network and transmits, i.e., pushes, the data to them. We write this
round as a random walk whose each step corresponds to a random selection of one
of the infected nodes; this gives recursive formulas for the distribution and
the moments of the number of newly infected nodes in a push round. We use the
formula for the distribution to compute the expected number of rounds so that a
given percentage of the network is infected and continue a numerical comparison
of the push algorithm and the pull algorithm (where the susceptible nodes
randomly choose peers) initiated in an earlier work. We then derive the fluid
and diffusion limits of the random walk as the network size goes to
and deduce a number of properties of the push algorithm: 1) the number of newly
infected nodes in a push round, and the number of random selections needed so
that a given percent of the network is infected, are both asymptotically normal
2) for large networks, starting with a nonzero proportion of infected nodes, a
pull round infects slightly more nodes on average 3) the number of rounds until
a given proportion of the network is infected converges to a constant
for almost all . Numerical examples for theoretical results
are provided.Comment: 28 pages, 5 figure
Stationary analysis of a single queue with remaining service time dependent arrivals
We study a generalization of the system (denoted by ) with
independent and identically distributed (iid) service times and with an arrival
process whose arrival rate depends on the remaining service
time of the current customer being served. We derive a natural stability
condition and provide a stationary analysis under it both at service completion
times (of the queue length process) and in continuous time (of the queue length
and the residual service time). In particular, we show that the stationary
measure of queue length at service completion times is equal to that of a
corresponding system. For we show that the continuous time
stationary measure of the system is linked to the system via a
time change. As opposed to the queue, the stationary measure of queue
length of the system at service completions differs from its marginal
distribution under the continuous time stationary measure. Thus, in general,
arrivals of the system do not see time averages. We derive formulas
for the average queue length, probability of an empty system and average
waiting time under the continuous time stationary measure. We provide examples
showing the effect of changing the reshaping function on the average waiting
time.Comment: 31 pages, 3 Figure
Asymptotically optimal importance sampling for Jackson networks with a tree topology
This note describes an importance sampling (IS) algorithm to estimate buffer overflows of stable Jackson networks with a tree topology. Three new measures of service capacity and traffic in Jackson networks are introduced and the algorithm is defined in their terms. These measures are effective service rate, effective utilization and effective service-to-arrival ratio of a node. They depend on the nonempty/empty states of the queues of the network. For a node with a nonempty queue, the effective service rate equals the node's nominal service rate. For a node i with an empty queue, it is either a weighted sum of the effective service rates of the nodes receiving traffic directly from node i, or the nominal service rate, whichever smaller. The effective utilization is the ratio of arrival rate to the effective service rate and the effective service-to-arrival ratio is its reciprocal. The rare overflow event of interest is the following: given that initially the network is empty, the system experiences a buffer overflow before returning to the empty state. Two types of buffer structures are considered: (1) a single system-wide buffer shared by all nodes, and (2) each node has its own fixed size buffer. The constructed IS algorithm is asymptotically optimal, i. e., the variance of the associated estimator decays exponentially in the buffer size at the maximum possible rate. This is proved using methods from (Dupuis et al. in Ann. Appl. Probab. 17(4): 1306-1346, 2007), which are based on a limit Hamilton-Jacobi-Bellman equation and its boundary conditions and their smooth subsolutions. Numerical examples involving networks with as many as eight nodes are provided
Dynamic importance sampling for queueing networks
Importance sampling is a technique that is commonly used to speed up Monte
Carlo simulation of rare events. However, little is known regarding the design
of efficient importance sampling algorithms in the context of queueing
networks. The standard approach, which simulates the system using an a priori
fixed change of measure suggested by large deviation analysis, has been shown
to fail in even the simplest network setting (e.g., a two-node tandem network).
Exploiting connections between importance sampling, differential games, and
classical subsolutions of the corresponding Isaacs equation, we show how to
design and analyze simple and efficient dynamic importance sampling schemes for
general classes of networks. The models used to illustrate the approach include
-node tandem Jackson networks and a two-node network with feedback, and the
rare events studied are those of large queueing backlogs, including total
population overflow and the overflow of individual buffers.Comment: Published in at http://dx.doi.org/10.1214/105051607000000122 the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Joint Hitting-Time Densities for Finite State Markov Processes
For a finite state Markov process and a finite collection of subsets of its state space, let be the first time the process
visits the set . We derive explicit/recursive formulas for the joint
density and tail probabilities of the stopping times .
The formulas are natural generalizations of those associated with the jump
times of a simple Poisson process. We give a numerical example and indicate the
relevance of our results to credit risk modeling.Comment: 25 pages, 3 figure