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arXiv 1707.01548cs.NImath.PR
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离散时间分散介质访问算法的稳定性条件

Stability conditions for a discrete-time decentralised medium access algorithm

Seva Shneer, Alexander Stolyar

摘要:

我们考虑一个随机的分布式系统建模的无线网络的行为与节点采用离散时间版本的标准分散介质访问算法。系统是{ \em不饱和} --每个节点以每个时隙$λ$分组的速率接收外部分组流。每个数据包需要一个时隙来传输,但相邻节点不能同时传输。我们研究的算法是{ \em standard}的,因为:具有空队列的节点{ \em not}竞争介质访问;节点的访问过程{ \em not}依赖于其队列长度,只要它不为零。两个系统的拓扑结构被认为是,与节点排列在一个圆圈和一条线。我们证明了,对于任一拓扑,系统在条件$λ< 2/5$下是随机稳定的。该结果对于环形拓扑是直观的,因为每个节点在饱和系统(具有无限队列)中接收的吞吐量等于所谓的{ \em停放常数},其大于$2/5$。(The然而,后一个事实无助于证明我们的结果。)对于线路拓扑,结果根本不直观,因为在饱和系统中,一些节点接收的吞吐量低于$2/5$。

英文摘要:

We consider a stochastic queueing system modelling the behaviour of a wireless network with nodes employing a discrete-time version of the standard decentralised medium access algorithm. The system is {\em unsaturated} -- each node receives an exogenous flow of packets at the rate $λ$ packets per time slot. Each packet takes one slot to transmit, but neighboring nodes cannot transmit simultaneously. The algorithm we study is {\em standard} in that: a node with empty queue does {\em not} compete for medium access; the access procedure by a node does {\em not} depend on its queue length, as long as it is non-zero. Two system topologies are considered, with nodes arranged in a circle and in a line. We prove that, for either topology, the system is stochastically stable under condition $λ< 2/5$. This result is intuitive for the circle topology as the throughput each node receives in a saturated system (with infinite queues) is equal to the so called {\em parking constant}, which is larger than $2/5$. (The latter fact, however, does not help to prove our result.) The result is not intuitive at all for the line topology as in a saturated system some nodes receive a throughput lower than $2/5$.

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