A General Class of Throughput Optimal Routing Policies in Multi-hop Wireless Networks

This paper considers the problem of throughput optimal routing/scheduling in a multi-hop constrained queueing network with random connectivity whose special case includes opportunistic multi-hop wireless networks and input-queued switch fabrics. The main challenge in the design of throughput optimal routing policies is closely related to identifying appropriate and universal Lyapunov functions with negative expected drift. The few well-known throughput optimal policies in the literature are constructed using simple quadratic or exponential Lyapunov functions of the queue backlogs and as such they seek to balance the queue backlogs across network independent of the topology. By considering a class of continuous, differentiable, and piece-wise quadratic Lyapunov functions, this paper provides a large class of throughput optimal routing policies. The proposed class of Lyapunov functions allow for the routing policy to control the traffic along short paths for a large portion of state-space while ensuring a negative expected drift. This structure enables the design of a large class of routing policies. In particular, and in addition to recovering the throughput optimality of the well known backpressure routing policy, an opportunistic routing policy with congestion diversity is proved to be throughput optimal.

💡 Research Summary

This paper tackles the classic problem of designing routing and scheduling policies that are throughput‑optimal in multi‑hop constrained queueing networks with random connectivity. The model encompasses opportunistic multi‑hop wireless networks as well as input‑queued switch fabrics. The central difficulty lies in finding a Lyapunov function whose expected drift is negative for all non‑zero queue states, thereby guaranteeing stability for any arrival rate vector that lies inside the network’s capacity region.

Traditional throughput‑optimal policies such as Backpressure rely on simple quadratic or exponential Lyapunov functions that treat the network as a homogeneous entity: they try to equalize queue backlogs regardless of topology, path length, or channel state. The authors propose a far richer class of Lyapunov functions: continuous, differentiable, and piece‑wise quadratic. Formally, the state space is partitioned into a finite number of regions 𝔅₁,…,𝔅_K, and on each region the Lyapunov function takes the form

 V(q) = ½ qᵀ H_k q + c_kᵀ q + d_k, q ∈ 𝔅_k,

where H_k is a positive‑semidefinite matrix, c_k a vector, and d_k a scalar. By allowing the matrices H_k and vectors c_k to depend on the region, the Lyapunov function can embed information about short paths, congestion levels, or channel quality.

The main theoretical contribution is a set of sufficient conditions under which any Lyapunov function from this class yields a negative expected drift, i.e.,

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