具有动态替代路由的损失网络的亚稳定性
Metastability in Loss Networks with Dynamic Alternative Routing
摘要：
考虑用链路互连的$N$站，每个容量$K$，形成一个完整的图。呼叫到达每个链路的速率为$λ$，离开链路的速率为$1$。如果一个呼叫到达连接站$x$和$y$的链路$x y$，它是满负荷的，那么第三个站$z$被均匀地随机选择，并且该呼叫试图通过$z$路由：如果两个链路$x z$和$z y$都有空闲容量，那么该呼叫在这两个链路上同时保持;否则该呼叫丢失。我们分析了这个模型的近似。我们严格地表明，有三个阶段，根据交通强度$α:= λ/K$：对于$α\in (0,α_c) \cup (1,\infty)$，系统的混合时间对数的链接数$n := \binom N2$ ;对于$α\in (α_c,1)$，系统的混合时间指数的$n$，链接数。这里，$α_c := \tfrac13 (5 \sqrt{10} - 13) \approx 0.937$是具有简单解释的明确临界阈值。我们还考虑允许多次重新路由尝试。这对整体行为几乎没有影响;它不会消除亚稳态阶段。最后，我们添加中继预留：在这种情况下，预留一定数量的$σ$电路;只有当至少有$σ+1$电路可用时，才接受重新路由尝试。我们表明，如果选择足够大的$σ$，仅取决于$α$，而不是$K$或$n$，那么亚稳相被删除。
英文摘要：
Consider $N$ stations interconnected with links, each of capacity $K$, forming a complete graph. Calls arrive to each link at rate $λ$ and depart at rate $1$. If a call arrives to a link $x y$, connecting stations $x$ and $y$, which is at capacity, then a third station $z$ is chosen uniformly at random and the call is attempted to be routed via $z$: if both links $x z$ and $z y$ have spare capacity, then the call is held simultaneously on these two; otherwise the call is lost. We analyse an approximation of this model. We show rigorously that there are three phases according to the traffic intensity $α:= λ/K$: for $α\in (0,α_c) \cup (1,\infty)$, the system has mixing time logarithmic in the number of links $n := \binom N2$; for $α\in (α_c,1)$ the system has mixing time exponential in $n$, the number of links. Here $α_c := \tfrac13 (5 \sqrt{10} - 13) \approx 0.937$ is an explicit critical threshold with a simple interpretation. We also consider allowing multiple rerouting attempts. This has little effect on the overall behaviour; it does not remove the metastability phase. Finally, we add trunk reservation: in this, some number $σ$ of circuits are reserved; a rerouting attempt is only accepted if at least $σ+1$ circuits are available. We show that if $σ$ is chosen sufficiently large, depending only on $α$, not $K$ or $n$, then the metastability phase is removed.
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