1,800 research outputs found
On Stochastic Bounds for Monotonic Processor Sharing Networks
International audienceWe consider a network of processor sharing nodes with independent Poisson arrival processes. Nodes are coupled through their service capacity in that the speed of each node depends on the number of customers present at this and any other node. We assume the network is monotonic in the sense that removing a customer from any node increases the service rate of all customers. Under this assumption, we give stochastic bounds on the number of customers present at any node. We also identify limiting regimes that allow to test the tightness of these bounds. The bounds and the limiting regimes are insensitive to the service time distribution. We apply these results to a number of practically interesting systems, including the discriminatory processor sharing queue, the generalized processor sharing queue, and data networks whose resources are shared according to max–min fairness
Perfect Simulation of Queues
In this paper we describe a perfect simulation algorithm for the stable
queue. Sigman (2011: Exact Simulation of the Stationary Distribution of
the FIFO M/G/c Queue. Journal of Applied Probability, 48A, 209--213) showed how
to build a dominated CFTP algorithm for perfect simulation of the super-stable
queue operating under First Come First Served discipline, with
dominating process provided by the corresponding queue (using Wolff's
sample path monotonicity, which applies when service durations are coupled in
order of initiation of service), and exploiting the fact that the workload
process for the queue remains the same under different queueing
disciplines, in particular under the Processor Sharing discipline, for which a
dynamic reversibility property holds. We generalize Sigman's construction to
the stable case by comparing the queue to a copy run under Random
Assignment. This allows us to produce a naive perfect simulation algorithm
based on running the dominating process back to the time it first empties. We
also construct a more efficient algorithm that uses sandwiching by lower and
upper processes constructed as coupled queues started respectively from
the empty state and the state of the queue under Random Assignment. A
careful analysis shows that appropriate ordering relationships can still be
maintained, so long as service durations continue to be coupled in order of
initiation of service. We summarize statistical checks of simulation output,
and demonstrate that the mean run-time is finite so long as the second moment
of the service duration distribution is finite.Comment: 28 pages, 5 figure
Stability of Redundancy Systems with Processor Sharing
We investigate the stability condition for redundancy-d systems where each of
the servers follows a processor-sharing (PS) discipline. We allow for generally
distributed job sizes, with possible dependence among the d replica sizes being
governed by an arbitrary joint distribution. We establish that the stability
condition is characterized by the expectation of the minimum of d replica sizes
being less than the mean interarrival time per server. In the special case of
identical replicas, the stability condition is insensitive to the job size
distribution given its mean, and the stability condition is inversely
proportional to the number of replicas. In the special case of i.i.d. replicas,
the stability threshold decreases (increases) in the number of replicas for job
size distributions that are NBU (NWU). We also discuss extensions to scenarios
with heterogeneous servers.Comment: To appear in proceedings of ValueTools 202
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