Biased Backpressure Routing for Multihop Wireless Networks with Heterogeneous Interfaces
arXiv:2609.05844v1 [cs.NI] 5 Sep 2026
Yujun Ming⋆ , Zhongyuan Zhao⋆ , Fikadu Dagefu‡ , Justin Kong‡ , Terrence Moore‡ , Kevin Chan‡ , Ananthram Swami‡ , and Santiago Segarra⋆ ⋆ Rice University, Houston, TX, USA ‡ DEVCOM Army Research Laboratory, Adelphi, MD, USA Abstract—Heterogeneous-interface multihop wireless networks (Het-MuNets) are emerging as a promising paradigm for tactical networks and for infrastructure-light applications such as vehicular communications, wireless backhaul, and nonterrestrial connectivity. To exploit the diverse profiles of heterogeneous communication technologies in penetration, interference, and bandwidth, packet-to-interface assignment must be determined on a per-hop basis, making routing and scheduling highly complex. In this work, we develop a unified framework for joint packet routing, link scheduling, and interface assignment in Het-MuNets with multiple concurrent flows. By modeling packet-to-interface assignment as transmission between virtual subnodes, we transform interface assignment into intra-device virtual routing, which is solved jointly with physical routing and scheduling under a unified multi-layer shortest path-biased Backpressure (SP-BP) scheme. Numerical results demonstrate that the proposed framework outperforms SP-BP operating on other baseline graph models and non-backpressure routing schemes in goodput, latency, and packet delivery rate. Index Terms—Multi-hop routing, heterogeneous wireless network, backpressure routing, queueing networks
I. I NTRODUCTION Heterogeneous-interface multihop wireless networks (HetMuNets) offer a powerful, infrastructure-light paradigm for applications like tactical networking, robotic swarms, wireless backhaul, and non-terrestrial coverage [1]–[4]. By combining diverse communication technologies with complementary physical capabilities, Het-MuNets can simultaneously achieve multiple conflicting objectives that are unsupported by a single technology [5]. For example, a network can achieve high throughput and covertness with mmWave or visible light communications (VLC), while maintaining robust connectivity in long-range and mobile scenarios using the high penetration of sub-GHz bands [6]. It can also balance quality-of-service, energy efficiency and cost for different traffic demands by switching between 5G New Radio (NR) Research was sponsored by the U.S. Army Combat Capabilities Development Command Army Research Laboratory (DEVCOM ARL) Army Research Office and was accomplished under Cooperative Agreement Numbers W911NF-24-2-0008 and W911NF-26-2-A128. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Office or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein. Emails: {yujun.ming, zhongyuan.zhao, segarra}@rice.edu, {justin.h.kong2, terrence.j.moore, fikadu.t.dagefu, ananthram.swami, kevin.s.chan}[email protected]
and co-band protocols (e.g., Wi-Fi and Bluetooth). Although conceptually adjacent to HetNets, multi-RAT systems [2], [7], Multi-Bearer Networks (MBNs) [4], and multi-radio multichannel (MR-MC) networks [1], [8], Het-MuNets emphasize multi-hop packet routing across diverse communication technologies rather than cellular architecture or link-level waveform switching. The primary challenge in Het-MuNets is that interface assignment dictates the topological inputs for routing and scheduling. Joint optimization thus becomes significantly harder with this added degree of freedom, especially under heterogeneous device profiles. Classical multihop routing protocols, such as AODV [9], establish a single path per flow using static additive metrics (e.g., hop count), potentially congesting bottleneck links shared by concurrent flows. Although AODVv2 [10] introduces multi-interface support, interface assignments remain locked after route discovery and cannot adapt to dynamic congestion. To reduce the signaling overhead of global topology tracking, reinforcement learning (RL) has been adopted for on-demand routing in Het-MuNets, including Q-learning for covert routing [11] and a shared deep Q-network that jointly selects the next-hop, interface, and sub-band during route establishment [12]. However, these single-path routing approaches are generally vulnerable to link failures, multiflow resource contention, and real-time congestion dynamics. Backpressure (BP) routing addresses these drawbacks through joint routing and scheduling based on queue differentials (pressure) and supports fully distributed operation with provable queue stability established via Lyapunov drift-pluspenalty (DPP) analysis [13], [14]. By incorporating distancebased biases into pressure calculations, shortest path-biased BP (SP-BP) [13], [15] mitigates the slow startup and randomwalk behavior of classical BP, thus significantly reducing the latency under light-to-moderate loads [16]. Since the biases are queue-agnostic and updated infrequently to capture topological drift, the signaling overhead of SP-BP remains low [15]. However, extending homogeneous SP-BP to HetMuNets remains an open and non-trivial challenge. Closely related schemes include BP for MR-MC and MIMO networks that assume parallel homogeneous interfaces [8], [17]. In this work, we propose multi-layer SP-BP as a multiflow multi-path routing scheme for Het-MuNets. The key innovation is a multi-layer graph model that represents heterogeneous air-interfaces on a device as subnodes connected
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Fig. 1. System diagram of multi-layer BP routing for Het-MuNets: multi-layer graph modeling, queueing system design, and operations.
by virtual links, and the sub-network of each communication technology as a graph layer (Fig. 1). This approach transforms cross-interface packet assignment into intra-device virtual routing. As a result, interface assignment, packet routing, and link scheduling can be optimized jointly under a unified Backpressure-MaxWeight framework. Furthermore, we introduce virtual link weights and a coupled actual-virtual queueing system to eliminate the intra-device packet looping in the existing graph and queueing-system expansions [8]. Contributions: Our contributions are as follows: • We introduce a multi-layer graph model and a system of
actual-virtual queues for networking in Het-MuNets, enabling fully distributed SP-BP to jointly optimize interface assignment, packet routing, and link scheduling. • We develop a multi-layer SP-BP scheme that solves virtual and physical routing in two stages coupled by virtual queueing states. The intra-device virtual routing adopts a minimum-cost flow (MCF) formulation that can be solved exactly in polynomial time, while the physical routing and scheduling rely on SP-BP for individual interface layers. • Through numerical experiments under mixed streaming and bursty traffic patterns and various network sizes, we show that the proposed multi-layer routing algorithm reduces latency and improves packet delivery and goodput compared with both BP and non-BP baselines. Notation: |·| represents the cardinality of a set. E(·) stands for expectation. Upright bold lower-case symbol, e.g., z, denotes a column vector, and zi denotes the i-th element of vector z. Upright bold upper-case symbol Z denotes a matrix, and Zi,j for its element at row i and column j. Calligraphic upper-case symbol denotes a set, e.g., G for a graph.
II. S YSTEM M ODEL A Het-MuNet can be modeled as a directed multigraph G n = (V 0 , E p ), where a node i ∈ V 0 represents a device, and an edge ep = (im , jm ) ∈ E p indicates that device i can transmit data to device j directly via communication modality m ∈ M, where M is the set of all modalities in the network. There can be multiple edges from i to j. We define communication modalities as wireless technologies with potentially mutually exclusive activations, such as 5G NR, Wi-Fi, Bluetooth, UHF radio, or VLC. We denote the set of available modalities on device i as Mi ⊆ M. To unify modality assignment, packet routing, and link scheduling under the same decision framework, we convert the multigraph G n into a multi-layer graph model G m = (V, E) where E = E p ∪ E v , as illustrated in Fig. 1. We define the set V 0 (reused from G n as it includes all devices) as subnodes in the base layer and the set V m asSsubnodes in the layer of modality m. The vertex set is V = m∈M+ V m , where M+ = {0} ∪ M. On device i, the base node i0 ∈ V 0 serves as the injection point of exogenous arrivals, whereas a modality node im ∈ V m handles endogenous traffic. Any packets destined to device i will be immediately consumed upon arriving at any modality node im ; therefore, G m contains a built-in sink layer, which is omitted for simplicity. A physical modality-link ep = (im , jm ) ∈ E p belongs to layer m. Within a device i, we model packet-modality switching as intra-device virtual links ev ∈ E v , e.g., ev = (i0 , im ) for assigning newly arrived packets to modality m, and ev = (im , im′ ) for assigning packets from modality m to m′ . Note that base node i0 has only outgoing virtual links, i.e., (im , i0 ) ∈ / Ev. We model resource contention between physical links as a conflict graph G c = (E p , H) from G n . Each vertex e ∈ E p
corresponds to a modality-link in G n , while an undirected edge (e1 , e2 ) ∈ H indicates a conflict between links e1 and e2 . Two types of conflicts are considered: 1) interface conflicts from transceiver limitations (e.g., shared RF chains); and 2) interference conflicts from simultaneous co-band or nearby transmissions. For example, two omni-directional links using modality m conflict if their incident nodes are within interference range ρm . Graph G c can be decomposed into c modality layers, {Gem }m∈M , and cross-layer conflict edges. Notice that our fully distributed SP-BP does not require c global knowledge of G c or Gem ; instead, each modality-link only needs to know its conflicting neighbors via channel monitoring and network feedback [15], [17], [18]. We consider a time-slotted orthogonal multiple access system, in which a valid schedule must be a set of nonconflicting links. We denote the instantaneous rate of a modality-link ep ∈ E p in time slot t by r̀ep (t), its longterm rate by rep = E[r̀ep (t)], and the long-term rate vector by r = [rep | ep ∈ E p ], all measured in packets per slot. We refer to all packets destined to device c ∈ V 0 as commodity c. Each device i hosts a separate physical queue (c) (c) Qi for each commodity c. The queue length of Qi , denoted (c) as Qi (t), evolves as X (c) X (c) (c) (c) (c) µep (t)+Ai (t), (1) Qi (t+1) = Qi (t)− µep (t)+ ep ∈Eip,+
ep ∈Eip,−
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where µep (t) ≤ r̀ep (t) is the transmitted packet count, (c) Ai (t) denotes exogenous arrivals, and Eip,+ and Eip,− are the sets of outgoing and incoming physical links of device i, respectively. (c) Each modality subnode im maintains a virtual queue qim (t) counting the physical packets assigned to modality m. Before new arrivals are assigned in slot t, these queues partition the P (c) (c) physical queue: Qi (t) = m∈Mi qim (t). The base node (c) (c) i0 holds only new arrivals, qi0 (t) = Ai (t), and Stage V changes only their modality assignments without creating or removing packets. The decision variables for the numbers of packets to be transmitted on virtual link ev and physical link (c) (c) ep are νev (t) and µep (t), respectively. The virtual queue of commodity c on modality-node im evolves as (c)
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III. B IASED BACKPRESSURE FOR H ET-M U N ETS The proposed multi-layer SP-BP is a fully distributed scheme, where each subnode makes decisions using only information about its own virtual queues and those of its immediate neighbors on G m , together with the instantaneous rates of its incident links. As illustrated in Fig. 2, each slot performs intra-device computation followed by interdevice routing, maximum weight independent set (MWIS) scheduling, and data transmission. Data transmission of slot t is pipelined with the subsequent slot t + 1 intra-device computation. A brief overview of these stages follows: • Stage-V: Virtual Packet Modality Selection. Exogenous packets are enqueued into virtual queues indexed by modality; devices then use an MCF solver to balance workloads by switching packets across modalities (detailed in Section III-B). • Stage-P: Physical Routing and Scheduling. Based on the updated virtual queues, SP-BP routing and scheduling are performed independently and in parallel within each conflict graph layer under conflict-aware transmission constraints (Section III-C). • Data Transmission: Packets are transmitted over scheduled links at the allocated rates. To formally explain these stages, we extend the SP-BP formulation to the multi-layer graph G m . For each subnode im and commodity c, the biased backlog is defined as (c)
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where Bim ≥ 0 is the shortest-path bias from im to destination c computed over G m . The pressure of commodity (c) c on any directed edge e = (v, u) ∈ E is Uvu (t) = (c) (c) Uv (t) − Uu (t), where e may be virtual or physical. As abstract intra-device transfers, virtual links have no intrinsic rates to differentiate connectivity and transmission capabilities of receiving modalities. To break this symmetry, we define the instantaneous rate of each virtual link ev = (in , im ), n, m ∈ M+ , as the average instantaneous rate of the outgoing physical links of the receiving subnode im : X 1 r̀ev (t) = p,+ r̀ep (t), rev = E[r̀ev (t)], (4) |Eim | p p,+ e ∈Eim
where Eip,+ is the outgoing physical-link set defined in m Section II. The virtual link rate in (4) imposes an artificial limit on virtual transmission across modalities. Each edge p e ∈ E is assigned a shortest-path weight: δe = r̄p rmax /re for p e ∈ E p and δe = 1/re for e ∈ E v , where r̄p and rmax are the average and maximum long-term physical-link rates. The bias (c) set {Bv }v∈V,c∈V 0 can be found by all-pairs shortest path (APSP) algorithms [19] on G m with edge weights {δe }e∈E .
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Fig. 3. An exemplary directed acyclic graph Di (t) in the MCF formulation of Stage V with |V 0 | = 2 commodities and |Mi | = 2 link modalities.
A. Coupled Virtual and Physical Stages The MaxWeight formulation of SP-BP is given by [13], [15], [17] µ⋆ (t), ν ⋆ (t) = argmax Wpt + Wvt , (5) (µ,ν)∈Π(t)
where Π(t) is the feasible joint action set defined by the linkrate, backlog-availability, and conflict-graph constrints, and Wp and Wv are defined as XX XX (c) t Wpt = µ(c) νe(c)(t) Ue(c)(t). e (t) Ue (t), Wv = e∈E p c∈V 0
e∈E v c∈V 0
(6) Solving (5) is challenging because it couples conflict-free virtual links with conflict-aware physical links. Therefore, we approximate it in two stages: Stage V jointly assigns exogenous and endogenous packets to modalities by solving XX ν̃e(c)(t) Ue(c)(t). (7) ν(t) = argmax ν̃(t)∈Πν (t) e∈E v
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B. Intra-device Virtual Routing Since virtual link sets Eiv and Ejv are always disjoint for i ̸= j, (7) is equivalent to independently solving subproblems on individual devices. Because virtual links are conflict-free, we formulate Stage V in (7) for device i as an MCF problem on a directed acyclic graph (DAG) Di (t), as illustrated in Fig. 3. Each virtual commodity queue on modality-node im (c) (c) serves as a flow source Sm that supplies qim (t) packets for (c) 0 m ∈ M+ i , c ∈ V . A pooling node Wm is created for each 0 m ∈ Mi , c ∈ V , and a transfer node Lm,m′ is created for
each pair m, m′ ∈ Mi and m ̸= m′ , to enforce the virtual link (c) (c) (c) rate constraint. The sources S0 , Sm are connected to Wm , which is subsequently connected to Lm,m′ . The pooling and transfer nodes are all connected to the sink Te that absorbs all flows. The arc costs and capacities are as follows: Injection arcs (c) (c) (c) (c) S0 → Wm have capacity Ai (t) and cost −Ui0 im (t). (c) (c) (c) Queue arcs Sm → Wm have capacity qim (t) and zero (c) cost. Retention arcs Wm → Te have zero cost and infinite (c) capacity, while switching arcs Wm → Lm,m′ have cost (c) −Uev (t) for ev = (im , im′ ). Finally, each Lm,m′ → Te arc has capacity r̀ev (t) and zero cost, so that the assignments respect the virtual link rate constraint. Thus, minimizing the total flow cost maximizes the objective in (7), subject to backlog availability and the one-switch-per-slot structure. The network simplex MCF solver [20] then determines in(c) jection flows νi0 im (t) for assigning new arrivals to a modality (c) subnode, and switching flows νev (t) for assigning queued packets from one modality to another. C. Stage P and Data Transmission In Stage P, SP-BP [15] is executed in parallel on disjoint layers of the multi-layer connectivity graph G m and multilayer conflict graph G c . Specifically, each modality without cross-modality conflicts forms a separate layer, whereas mutually conflicting modalities share a joint layer containing their cross-modality conflict edges. The procedure uses the (c) intermediate virtual queues {q̂v (t)} and consists of commodity selection, preliminary rate allocation, MaxWeight link scheduling, and final rate assignment as detailed in [15]. The c NP-hard MWIS problem on Gem in MaxWeight scheduling can be approximated by distributed heuristics such as the local greedy scheduler (LGS) [21] and GCN-LGS [18]. Stages V and P only manipulate virtual queues without moving any packets in the physical queues; data packets are only dequeued for transmission after Stage P (Fig. 2) based (c) on the final {µv (t)}v∈V,c∈V 0 , avoiding intra-device looping. D. Complexity Analysis We denote the number of commodities as |V 0 |, the number of communication modalities on device i as Mi = |Mi |, c = maxi∈V 0 {Mi }, and Dn , Dc as the maximum with M node degrees on G n , G c . Table I summarizes the per-slot local complexity under distributed execution, where the networkwide complexity is governed by the maximum local term. Stage V solves an MCF on a DAG Di (t) with O(Mi2 |V 0 |)
IV. S IMULATION R ESULTS We evaluate the proposed multi-layer SP-BP on simulated wireless networks. For each |V 0 | ∈ {20, 30, . . . , 100}, we generate 10 independent random topologies in a 2D square with node density 8/π. We consider three concurrently operating communication modalities (5 GHz, 2.4 GHz, and 900 MHz UHF) over the same physical nodes. Each modality’s communication range induces a distinct connected layer of G m , and its interference radius determines the modalityc . Cross-modality interference is specific conflict graph Gem absent. The interference radii satisfy ρ5G < ρ2.4G < ρUHF , and larger values yield higher conflict degrees and fewer transmission opportunities. We assign modality layer m a total rate budget of 26wm Em , where Em is its number of undirected edges and [w0 , w1 , w2 ] = [0.5, 0.35, 0.15]. To prevent any modality from dominating spatially, this budget is distributed among (m) its Em edges as re = 26Em wm pe , where p(m) ∼ Dirichlet(0.81Em ). Each undirected edge e is then converted into a pair of directed modality-links with equal long-term rates, e.g., r(im ,jm ) = r(jm ,im ) = re . Although symmetric rates are used for simplicity, our routing scheme fully supports asymmetric cases. Finally, to simulate fading, real-time link rates are sampled at each slot as r̀ep (t) ∼ N (rep , 2), truncated within rep ± 9. For each network topology, we evaluate 10 flow configurations, each containing a number of sourcedestination pairs sampled uniformly between ⌊0.3|V 0 |⌋ and ⌈0.5|V 0 |⌉. Each combination of network topology and flow configuration defines a test instance, which is simulated for T = 1000 slots under each routing algorithm, yielding 100 test instances per network size and traffic setting. We compare our multi-layer framework with a total of seven baselines grouped in two categories, based on whether
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arcs via network simplex, resulting in a time complexity of O(Mi3 |V 0 |2 log(Mi |V 0 |)) [20]. For SP-BP routing, computing biased pressure weights and selecting the best commodity requires O(|V 0 |Dn ) operations per slot. In LGS [21], each physical link compares its weight with those of its contention neighbors, leading to a time complexity of O(Dc ) per round. Signaling complexity measures the number of distributed control rounds. SP-BP routing performs one neighbor queuestate exchange per slot, giving O(1). LGS requires distributed coordination among conflicting links and converges in O(log |E p |) rounds [21]. The bias matrix is reused until the topology changes. With recomputation period τB , distributed APSP [19] requires Õ(|V|/τB ) amortized rounds, where Õ(·) suppresses polylogarithmic factors. Message-size complexity measures information per sigc) per-modality naling round. SP-BP routing sends O(|V 0 |M queue counters per neighbor exchange. With differential updates, only changed counters are sent, reducing overhead c) link weights and in practice. LGS messages carry O(M scheduling states. Each APSP round propagates one bias value per commodity, giving O(|V 0 |).
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Fig. 4. Average network goodput (average packets delivered network-wide per slot) versus per-flow injection rate λ in 100-node MWNs with streaming traffic; shading denotes 95% bootstrap confidence intervals for the mean.
SP-BP is employed. All baselines employ the same local greedy scheduler [21] to approximate MWIS scheduling. SP-BP operating on baseline graph models: • Unimodal (5G, 2.4G, UHF): ablation baselines that restrict all network traffic to one specific modality. • Unimodal (Aggregation): a single-layer graph variant in which the link rate between a node pair is aggregated across all modalities in the original Het-MuNet. • Siloed layers: a simplified version of the proposed algorithm, in which each packet is assigned at injection to a single graph layer based on the current biased backlog (c) Uim (t) and remains on that layer for its entire route. Non-BP routing with decoupled MaxWeight scheduling: • SP-only: a delay-weighted shortest-path routing policy operating on a collapsed single-layer graph that retains only the highest-rate modality for each physical link. • MCF-Prob: a static probabilistic forwarding policy derived by solving an offline multi-commodity MCF problem over the physical links of our layered topology, where each link capacity is set to be the link rate divided by conflict degree. A. Network Goodput Analysis The impact of the routing schemes on network capacity is evaluated under streaming flows with Poisson arrivals at identical rates λ ∈ [0.3, 12]. As shown in Fig. 4, the proposed multi-layer SP-BP achieves the highest mean goodput across all evaluated injection rates, with its advantage widening under higher load. Siloed layers also continues to improve under higher load, but stays below multi-layer SP-BP because it lacks dynamic cross-modality switching. In contrast, SPonly and MCF-Prob increase much more slowly, whereas the unimodal baselines saturate at substantially lower goodput. This saturation occurs under excessive load because the MWIS scheduler prioritizes links near source nodes while starving those near destinations [15]. In particular, Unimodal (Aggregation) peaks near λ = 3 and then gradually declines. B. Performance under Mixed Traffic In this experiment, the traffic is a mixture of streaming and bursty flows, assigned independently with equal probability.
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(d) Bursty delay
Fig. 5. Routing performance under low-to-medium mixed traffic: (a), (c) packet delivery ratio and (b), (d) end-to-end delay for streaming (a), (b) and bursty (c), (d) traffic. Undelivered packets are assigned a delay of T ; shading denotes 95% bootstrap confidence intervals for the mean.
Streaming flows model continuous data injection with Poisson arrivals at a fixed rate λs ∼ U (0.2, 1.0), whereas bursty flows model short-lived, high-intensity traffic with Poisson arrivals at rate λb ∼ U (6.6, 33.0) for t < 30 and none thereafter. Fig. 5 shows that multi-layer SP-BP achieves the best tradeoff between delivery and delay under streaming traffic, maintaining delivery rates above 0.99 across all tested network sizes while yielding the lowest end-to-end delay. Although MCF-Prob closely approaches it in small networks, its static flow split cannot adapt as the network grows, reducing delivery to approximately 0.86 at |V 0 | = 100 and forcing longer routes with higher delay. In contrast, unimodal schemes and Siloed layers show a faster decline in delivery and higher delay. Under bursty traffic, similar scaling trends hold, with the proposed method maintaining robust performance across network sizes. The relative ordering of Siloed layers and Aggregation reverses across traffic types. Aggregation performs better under bursty traffic by pooling modal capacities, whereas Siloed layers benefits from parallel transmissions under streaming traffic. V. C ONCLUSIONS We developed a fully distributed, multi-layer SP-BP scheme that formally couples dynamic packet-to-interface assignment with packet routing and link scheduling in HetMuNets. In interference-constrained environments, this joint optimization significantly improves resource utilization and manages inter-flow interference, yielding superior goodput, latency, and packet delivery rates compared with static forwarding policies or siloed SP-BP baselines. Ultimately, our approach successfully navigates the tightly coupled decision space of Het-MuNets to unlock their full potential. Future work will explore advanced bias formulations to further optimize energy efficiency, reduce radio footprints, and enhance resilience to mobility and link failures. R EFERENCES [1] A. A. Al Islam, M. J. Islam, N. Nurain, and V. Raghunathan, “Channel assignment techniques for multi-radio wireless mesh networks: A survey,” IEEE Commun. Surveys & Tutorials, vol. 18, no. 2, pp. 988– 1017, 2015. [2] I. F. Akyildiz, A. Kak, and S. Nie, “6G and beyond: The future of wireless communications systems,” IEEE Access, vol. 8, pp. 133995– 134030, 2020. [3] A. Kott, A. Swami, and B. J. West, “The internet of battle things,” Computer, vol. 49, no. 12, pp. 70–75, 2016.
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