We study queue-based activation protocols in random-access networks. The network is modeled as an interference graph. Each node of the graph represents a server with a queue. Packets arrive at the nodes as independent Poisson processes and have independent exponentially distributed sizes. Each node can be either active or inactive. When a node is active, it deactivates at unit rate. When a node is inactive, it activates at a rate that depends on its current queue length, provided none of its neighboring nodes is active. Thus, two nodes that are connected by a bond cannot be active simultaneously. This situation arises in random-access wireless networks where, due to interference, servers that are close to each other cannot use the same frequency band. In the limit as the queue lengths at the nodes become very large, we compute the transition time between the two states where one half of the network is active and the other half is inactive. We compare the transition time with that of a network in which the activation rates are not controlled by the queue length but are externally driven, a situation that was dealt with in an earlier paper. Namely, we first sandwich the transition time between that of two networks in which the activation rates are small perturbations of a certain prescribed function of the mean queue length. After that we show that, as the perturbation tends to zero, the two transition times become asymptotically similar. We focus on a complete bipartite network: we identify the scale of the transition time in terms of the model parameters and we show that its law on the scale of its mean has a trichotomy depending on the aggressiveness of the activation rates. Our aim in future work is to use similar comparison techniques for more general bipartite networks and for more complicated queue-based activation protocols.

中文翻译：

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随机接入网络中基于队列的激活协议的过渡时间渐近

我们研究随机访问网络中基于队列的激活协议。该网络被建模为干扰图。图中的每个节点代表一个带有队列的服务器。数据包作为独立的泊松过程到达节点，并具有独立的指数分布大小。每个节点可以是活动的或非活动的。当节点处于活动状态时，它会以单位速率停用。当一个节点处于非活动状态时，它会以取决于其当前队列长度的速率激活，前提是其相邻节点均未处于活动状态。因此，通过绑定连接的两个节点不能同时处于活动状态。这种情况出现在随机接入无线网络中，由于干扰，彼此靠近的服务器不能使用相同的频段。在节点的队列长度变得非常大的极限中，我们计算网络的一半处于活动状态而另一半处于非活动状态的两种状态之间的转换时间。我们将转换时间与激活率不受队列长度控制而是由外部驱动的网络的转换时间进行比较，这种情况在之前的论文中已经处理过。也就是说，我们首先将两个网络的转换时间夹在中间，其中激活率是平均队列长度的某个规定函数的小扰动。之后我们证明，随着扰动趋于零，两个过渡时间变得渐近相似。我们专注于一个完整的二分网络：我们根据模型参数确定过渡时间的尺度，并且我们表明其均值尺度上的规律具有三分法，具体取决于激活率的积极性。我们未来工作的目标是将类似的比较技术用于更一般的二分网络和更复杂的基于队列的激活协议。