Augmented Backpressure for Decentralized Management of Agentic Networks

arXiv:2608.00914v1 [cs.NI] 2 Aug 2026

Augmented Backpressure for Decentralized Management of Agentic Networks Zuyuan Zhang

Sizhe Tang

Tian Lan

The George Washington University zuyuan.zhang@gwu.edu

The George Washington University s.tang1@gwu.edu

The George Washington University tlan@gwu.edu

Abstract—Agentic foundation-model service networks handle requests spanning retrieval, planning, generation, verification, and tool use. Unlike traditional communication networks, control performance depends on queue dynamics and contextual memory states, including prefix/KV blocks, retrieved contexts, expert warm states, and verified tool outputs. These states arise from execution history and alter service work and downstream successor laws under finite local budgets. Treating them as passive caches or an independent process leaves a queueing-control gap. To this end, we propose Memory-Augmented Backpressure (MABP), a queue–memory control framework for stateful foundationmodel service networks (SFMSNs) that jointly models commodity queues and causal contextual memory dynamics. MABP represents each request by service and state types, estimates memory-dependent work, penalties, and successor probabilities, then reads queues and resident memory each slot, selects feasible routing, transfer, activation, and service actions using a memorydependent pressure score, and retains a budget-feasible subset of resident and newly generated objects. We prove an occupationmeasure capacity outer bound with conditional tightness. We show that modeling contextual memory can strictly increase the stability region through work reduction and transition shaping, establishing a separation between memory-aware and memoryoblivious decisions. We also prove throughput and drift-pluspenalty guarantees for exact frame-MABP with bounded-loss extensions to approximate solvers. Index Terms—Agentic service networks, memory-augmented backpressure, contextual memory, queueing control, foundationmodel serving.

I. I NTRODUCTION Backpressure and max-weight scheduling stabilize stochastic networks by assigning service and routing decisions to actions with large queue-differential reward [1]–[8]. They are among the standard frameworks for decentralized stochastic network control. In canonical formulations, the controller changes routing, service rates, or resource allocations, while service-generated contextual states are either absent, treated as exogenous cache contents, or not deliberately preserved as part of the stabilizing control. This abstraction is increasingly inadequate for agentic foundation-model service networks, where a request may traverse retrieval, planning, generation, expert execution, verification, tool use, and rethinking stages across distributed workers [9]–[18]. These stages leave persistent execution states— retrieved contexts, prefix/KV blocks, expert warm states, activation signatures, and verified tool outputs—that may outlive the request that created them. They are not merely passive

cache files. They are produced by prior executions/services, consume local memory, and can change how later tasks move through the queueing network. Thus the decentralized control problem for agentic service networks must jointly address two coupled dynamics: the visible queueing dynamics of service demand and the contextual memory-state dynamics generated by executions. Existing queue-only backpressure rule can stabilize the visible queues, while leading to suboptimal contextual memory states and causing unmodeled executionstate storage to grow without bound. Proposition 2 gives an illustrative example. This motivates modeling contextual memory as a controlled state variable together with queueing dynamics, mandating an augmented backpressure policy. We introduce a stateful foundation-model service network (SFMSN). Each commodity identifies a request class, a service stage, and a finite state type. Each node maintains both commodity queues and a finite contextual memory state. The transitions are causal: a node inherits memory, transfers or activates existing objects, performs routing and service, observes service-generated objects, and retains a budget-feasible subset for the next slot. This differs from conventional multi-resource queueing models as active memory control is coupled with queueing dynamics and changes the service operator itself. It can reduce work through wi (a, Yi ), alter quality-adjusted penalty through pi (a, Yi ), and, most importantly, change the downstream successor law Pi (c′ | a, Yi ). Two workers with the same queue lengths and nominal compute capacity can therefore attain different performance and create different downstream loads because they hold different contextual memory states. The coupled model creates two new challenges for backpressure based control. First, contextual memory states must be captured as routing to a lightly loaded but state-mismatched worker can increase future verifier or tool queues. Second, memory retention is a delayed control decision. An object generated in slot t cannot be used until a later request, and an object with little immediate value may be necessary to sustain a favorable recurrent memory state. A one-slot cache bonus is therefore insufficient for a theorem-level stability result. Memory-Augmented Backpressure (MABP) prices a service action by the queue it consumes, the downstream queues it is expected to create under the active memory state, and a quality-adjusted penalty. For a service action a at node i with

active memory Yi = Yiact (t), the local service pressure is X ′ ι(a) Wia (t; Yi ) = Qi (t) − Pi (c′ | a, Yi )Qic (t) c′ ∈C

prefix/KV lowers wi

(1)

Ingress

Retrieve

− νpi (a, Yi ). The first term models input-backlog consumption; the second prices expected downstream congestion and depends on the active memory state through Pi (c′ | a, Yi ); and the third controls the penalty–delay tradeoff and depends on memory through pi (a, Yi ). Memory-dependent work wi (a, Yi ) enters the compute constraint, so the scheduler chooses a feasible combination of routing, transfer, activation, and service actions with large total pressure rather than simply adding a cachehit reward. Next, generated and resident objects are evaluated by a retention rule that uses the pre-service queue vector, preserving causality. A slot is one discrete scheduling interval in which multiple queued requests may be routed or served. The exact policy used in the analysis is an H-slot frame oracle that plans over controlled memory states. The one-slot rule used in experiments is its deployable surrogate. This paper goes beyond research on LLM systems that often focus on batching, prefix/KV reuse, phase splitting, cache affinity, and load-aware request placement [14], [16], [17], [19]–[22], rather than the stochastic, multi-stage stability region shaped jointly by queues and contextual memory states. We rigorously analyze the proposed scheme. Theorem 1 gives an occupation-measure outer bound on the stability region and proves conditional tightness under a queuecompatible realization condition, making explicit that recurrent memory states must be generated and sustained causally. We prove that contextual memory can strictly increase the capacity region through two independent mechanisms: reducing service work and reducing downstream verifier/tool creation (in Proposition 1, Theorem 2, and Corollary 1). We also characterize a minimax one-step gap for decisions that observe queues but ignore memory states (in Proposition 3) and quantify throughput for exact frame-MABP(in Theorem 3) and a drift-plus-penalty tradeoff (in Theorem 4). An approximate frame solver using a surrogate is shown to inherit the same guarantees (in Corollary 2). The paper makes three contributions. • Coupled queue–memory model. We introduce SFMSN. It couples queue dynamics with memory-state dynamics as active memory in agentic networks can alter service work, quality-adjusted penalty, and downstream execution. • Stateful capacity and separation theory. We derive occupation-measure constraints that are necessary for stability and conditionally tight under an explicit queuecompatible realization condition. We also prove strict capacity enlargement from work reuse and from memorydependent transition laws, and we give a minimax separation for memory-oblivious decisions. • Memory-Augmented Backpressure. We formulate exact frame-MABP, which plans over causal memory states and achieves throughput and drift-plus-penalty guarantees for

Generate

direct

Egress

Pi (cv | a, Yi )

verified result reduces Pi (cv | a, Yi )

Verify/Tool

Fig. 1. A stateful service trajectory. Different execution states affect different primitives: prefix/KV state reduces current work, while a reusable verified result can reduce the probability of creating a downstream verifier or tool job.

strictly feasible arrivals. We then give a practical one-slot surrogate based on the same memory-dependent pressure and specify the bounded-loss condition under which an approximate solver preserves the frame guarantees. II. M ODEL This section defines a causal stateful-service model. In each slot, inherited memory is transferred and activated before routing and service; completed service creates successor jobs and memory only for later slots. Thus memory affects work, transition laws, and quality-adjusted penalties without sameslot lookahead. Catalog caching is recovered when objects alter only access cost, while result caching is service-generated execution memory [23]–[26]. A. Network and Stateful Commodities The service system is a finite directed graph G = (N , E). Link (i, j) has capacity Cij and per-unit penalty dcij for commodity c; node i has compute capacity µi and memory budget Bi . Time is slotted, and routing/service amounts are bounded integer quanta, giving a finite approximation to continuous rates. A commodity is c = (r, k, z) ∈ C, where r is the request class, k ∈ {0, . . . , Lr −1} is the service stage, and z ∈ Zr,k is a finite state type encoding, e.g., a prompt cluster, retrieval signature, expert-subset signature, prefix/KV signature, or toolresult signature [10]–[12], [14], [15], [17]. Queue Qci (t) stores unserved units of c at node i. Let I ⊆ N × C be the finite set of admissible node–commodity queues, retained only when a feasible routing or service path can drain them. We set Qci (t) = 0 for (i, c) ∈ / I, and all sums over i, c range over I unless stated otherwise. Let Ht be the pre-decision history containing Q(t), inherited memory Y − (t), and past controls/outcomes, but not slott arrivals or service-generated outcomes. All slot-t controls are Ht -measurable. Exogenous arrivals join Q(t + 1), satisfy Aci (t) ∈ Z+ , E[Aci (t) | Ht ] = λci , and a uniform conditional second-moment bound; write λ = (λci )(i,c)∈I . They are conditionally independent of controlled service and memorygeneration randomness given Ht . B. A Running Example Fig. 1 illustrates the service trajectory. A request is matched with retrieved context, routed to a generator that may hold a compatible prefix/KV or expert state, and then completed

or verified. Each completed stage may generate an object for later requests. Classical backpressure observes queues and nominal capacities; MABP additionally prices how the resident execution state changes service work and downstream job creation. C. Causal Execution-State Memory Let M be the finite set of signature–replica identities. Each object m ∈ M has size bm > 0 and represents a bounded content signature with a replica/version slot or a refreshed local instance. Feasible global memory states are P Y = {(Yi )i∈N : Yi ⊆ M, m∈Yi bm ≤ Bi ∀i, Yi ∩ Yj = ∅ ∀i ̸= j}. Objects with the same content signature may share primitive parameters, but simultaneous replicas have distinct identities. Object m has compatibility indicator χm (c, a) ∈ {0, 1}; for action a, Yi [a] = {m ∈ Yi : χm (ι(a), a) = 1} is the compatible active subset. The primitives wi (a, Yi ), pi (a, Yi ), and Pi (· | a, Yi ) depend on memory only through this subset, which is implemented as a finite compatibility signature indexing profiled primitive tables. At the beginning of slot t, node i inherits Yi− (t). A transfer m decision uses indicators um ij (t) ∈ {0, 1} with uij (t) ≤ 1{m ∈ P − m out Yi (t)}, j:(i,j)∈E uij (t) ≤ 1. With Mi (t) = {m : m in uij (t) = 1 for some j} and Mi (t) = {m : um ji (t) = 1 for some j}, the resident set is Ri (t) = (Yi− (t)\Miout (t))∪ Miin (t). The transfer vector u(t) ∈ U(Y − (t)) enforces transfer bandwidth, topology, destination-budget, global-identity, and any joint data/state link-capacity constraints. The base model is migration; replication can be represented by assigning a copied object a globally free replica identity, without affecting the results below. Resident and active memory satisfy X bm ≤ Bi , Yiact (t) ⊆ Ri (t). (2) m∈Ri (t)

Activation-bandwidth, device-residency, and concurrent-state limits are included in the feasible control set. Slot-t generated memory cannot be used in that slot. After service, completed actions generate random sets Gi (t) ⊆ M respecting global identity constraints; a generated identity must be globally free unless it refreshes an identity already resident at node i. Before observing G(t), the controller selects causal retention maps E(t) = (Ei (t))i∈N , where Ei (t) : 2M → 2M and Ei (t)(A) ⊆ A. For every generated set with positive probability, the retained vector must satisfy the budgets and belong to Y:
Yi+ (t) = Ei (t) Ri (t) ∪ Gi (t) , Yi− (t + 1) = Yi+ (t). (3) Using Ri (t), rather than Yiact (t), keeps inactive resident objects eligible for retention instead of forcing their same-slot eviction.
Let γ(t) = u(t), Y act (t), F (t), S(t), E(t) ∈ Γ(Q(t), Y − (t)) denote the complete feasible one-slot control, including the routing and service variables below. All components are chosen before service outcomes and generated objects are observed; after G(t) is realized, only

the preselected maps Ei (t)(·) are evaluated, so retention is adaptive to generated objects but nonanticipative. mem memory cost (t) = PA representative P P is P h m m κij uij (t) + θi m∈Y + (t) bm + (i,j)∈E m∈M i P P i m m where κm ≥ 0 ij , θi , ωi i m∈Gi (t)∩Yi+ (t) ωi , are transfer, holding, and write/update costs. Define h̄mem (y, γ) = E[hmem (t) | Y − (t) = y, γ(t) = γ]. Conditioned on Y − (t) = y and feasible γ, the laws of successor quanta and generated memory are determined by the state–control pair and may be correlated across simultaneous completions. Since M and Y are finite, generation and causal retention induce the controlled memory kernel T(y ′ | y, γ) = Pr{Y − (t + 1) = y ′ | Y − (t) = y, γ(t) = γ}, which depends on Q(t) only through the selected queuefeasible control γ. D. Routing and Service Actions The controller chooses routing amounts Fijc (t) ∈ Z+ P c satisfying c∈C Fij (t) ≤ Cij for each (i, j) ∈ E. Routing preserves commodity labels, and Fijc (t) = 0 unless both (i, c) and (j, c) belong to I; incoming routed traffic becomes available in the next slot. Node i has a finite action set Ai . Each action a ∈ Ai consumes commodity ι(a), and actions with inadmissible input queues are omitted. Under active memory Yiact (t), action a requires work wi (a, Yiact (t)) > 0 per completed unit, with uniformly bounded work values byPfiniteness. Completed service a act amounts Sia (t) ∈ Z+ satisfy (t)) ≤ a∈Ai Si (t)wi (a, Yi µi . Routing and service are jointly queue-feasible:

X j:(i,j)∈E

Fijc (t) +

X

Sia (t) ≤ Qci (t),

(i, c) ∈ I.

a∈Ai :ι(a)=c

(4) A completed unit of action a either exits or creates one downstream commodity c′ with probability Pi (c′ | a, Yiact (t)). With absorbing departure state ∅ and Q∅ i = 0, P ′ P (c | a, Y ) = 1; the support is restricted to ∅ ′ i i c ∈C∪{∅} and commodities c′ with (i, c′ ) ∈ I. A successor job joins ′ Qci (t + 1) and can be routed or served only later. The quality-adjusted service penalty is pi (a, Yi ) = ℓi (a, Yi ) − ηvi (a, Yi ), η ≥ 0, where ℓi is an operator-defined cost or quality-loss term and vi is a reuse/quality credit not already captured by wi . A service-time term in ℓi must not duplicate work represented by wi . Penalties are uniformly bounded and may be negative; η weights the credit, while ν controls the queue–penalty tradeoff. quality-adjusted penalty is h(t) = PThe total P one-slot P P c c act (t))Sia (t) + (i,j)∈E c∈C dij Fij (t) + i a∈Ai pi (a, Yi mem h (t). Its conditional expectation is obtained by replacing hmem (t) with h̄mem (y, γ) and is written as J(y, γ) in Section III. Since feasible routing, service, and memory controls are bounded and primitive penalties are uniformly bounded, h(t) is uniformly bounded above and below over one slot.

E. Queue Dynamics and Stability For each (i, c) ∈ I, X

Qci (t + 1) = Qci (t) −

Fijc (t) −

j:(i,j)∈E

+

X

Sia (t) + Aci (t)

a:ι(a)=c

c Fji (t) +

j:(j,i)∈E

X

X

ξia,c (t),

(5)

a∈Ai

j:(i,j)∈E

where E[ξia,c (t) | Q(t), Y − (t), γ(t)] = Sia (t)Pi (c | a, Yiact (t)). follows from (4). Also, ξia,c (t) ∈ P Nonnegativity a,c a Z+ and c:(i,c)∈I ξi (t) ≤ Si (t) almost surely, so each completion creates at most one successor job. Bounded routing/service quanta and the arrival second-moment assumption then give a uniform conditional second-moment bound on oneslot queue increments. A policy is admissible if it is causal with respect to {Ht } and satisfies all routing, compute, transfer, memory, and queue-feasibility constraints. is strongly stable if PT −1 P The network c E[Q (t)] < ∞. For initial lim supT →∞ T −1 t=0 i (i,c)∈I 0 memory state y ∈ Y, the stateful capacity region Λsfm (y 0 ) is |I| the closure of all λ ∈ R+ for which some admissible policy strongly stabilizes all queues from every finite initial queue vector and Y − (0) = y 0 ; write Λsfm when y 0 is fixed or irrelevant. The no-execution-memory region Λno-mem forces Yi− (t) = Yiact (t) = Yi+ (t) = ∅ for all i, t, evaluates all primitives at empty memory, and discards generated objects. A policy is memory-oblivious if routing and service depend on queue history and fixed system parameters, but not on realized Y − (t), and its service scores are invariant to replacing the realized execution-memory state by any other feasible memory state. III. S TATEFUL C APACITY AND S EPARATIONS This section characterizes capacity under controlled execution memory, which must be causally generated, retained, and revisited. We define an occupation-measure outer region, state its realization condition, and isolate capacity gains from work reuse and successor-law shaping. Fix y 0 . For inherited memory y, let Γ(y) be the finite set of one-slot controls satisfying routing, compute, transfer, activation, and memory constraints before queue truncation, and let Γ(q, y) ⊆ Γ(y) be the queue-feasible subset. A control γ specifies transfer, active memory, routing, service, and retention; Yiγ is the active memory at node i, and T(y ′ | y, γ) is the induced memory kernel. Let Yr (y 0 ) be the states reachable from y 0 . Occupation measures use Γ(y); queue-compatible achievability is imposed by Assumption 1. A stationary memory-state occupation measure is a nonnegative collection x(y, γ), with x(y, γ) = 0 for y ∈ / Yr (y 0 ), satisfying X X x(y, γ) = 1, (6) y∈Y γ∈Γ(y)

X γ∈Γ(y)

x(y, γ) =

X X

x(ỹ, γ̃)T(y | ỹ, γ̃),

The second constraint is the memory-flow balance; it prevents recurrent memory configurations from being treated as static cache placements. For a state–control pair, define the expected net service surplus of queue (i, c) by X X X c gic (y, γ) = Fijc (γ) + Sia (γ) − Fji (γ)

y ∈ Y.

ỹ∈Y γ̃∈Γ(ỹ)

(7)

−

X

a:ι(a)=c

j:(j,i)∈E

Sia (γ)Pi (c | a, Yiγ ).

(8)

a∈Ai

Whenever γ(t) is queue-feasible, E[Qci (t + 1) − Qci (t) | Q(t), Y − (t), γ(t)] = λci − gic (Y − (t), γ(t)). expectedPone-slot is J(y, γ) = P penalty PThe conditional P γ a c c (γ)p + S (γ)d F i (a, Yi ) + ijP i∈N a∈Ai i (i,j)∈E c∈C ij mem ¯ h̄ (y, γ). Let J(x) = y,γ x(y, γ)J(y, γ), and assume perslot penalties are uniformly bounded by Jmin and Jmax . An occupation measure supports λ with slack δ ≥ 0 if X X ḡic (x) ≜ x(y, γ)gic (y, γ) ≥ λci + δ, (i, c) ∈ I. y∈Y γ∈Γ(y)

(9) Definition 1 (Occupation-measure outer region). Λocc (y 0 ) is the closure of all arrival vectors satisfying (6)–(9) for some occupation measure supported on Yr (y 0 ). It is strictly feasible if the inequalities in (9) hold with a uniform δ > 0. The occupation region need not be achievable: an emptyqueue truncation can change both current service surplus and the future memory state. The next assumption certifies a queue-compatible realization of the occupation measure. Assumption 1 (Queue-compatible realization with bounded transient loss). For every reachable occupation measure x supporting λ with positive slack, there exist Hx ∈ Z+ and Lx ≥ 0 such that, for every H ≥ Hx , there are finite CH,x , DH,x ≥ 0 and a causal H-slot rule with γ(τ ) ∈ Γ(Q(τ ), Y − (τ )). P Writing q · v = (i,c)∈I qic vic , uniformly over frozen queues q and reachable initial memory states y, initialization with Q(0) = q and Y − (0) = y satisfies "H−1 # X − E q · g(Y (τ ), γ(τ )) ≥ H q · ḡ(x) − Lx ∥q∥1 − CH,x , τ =0

(10) "H−1 # X − ¯ + DH,x . E J(Y (τ ), γ(τ )) ≤ H J(x)

(11)

τ =0

Assumption 1 is stronger than finiteness or unichainness: it excludes memory frequencies that collapse under queue truncation. The frame proof therefore requires Hδ > Lx . Theorem 1 (Occupation outer bound and conditional tightness). Under finite memory and control sets and bounded conditional second moments, every strongly stabilizable arrival vector from initial memory y 0 belongs to Λocc (y 0 ). If strictly feasible vectors are dense in Λocc (y 0 ) and every strictly feasible vector is supported by a reachable occupation measure satisfying Assumption 1, then Λsfm (y 0 ) = Λocc (y 0 ).

ProofPsketch. For necessity, form xT (y, γ) = T −1 t<T Pr{Y − (t) = y, γ(t) = γ}. Compactness gives a convergent subsequence, and strong stability gives a further subsequence with E[∥Q(Tn )∥1 ]/Tn → 0. TelescopingPmemory indicators yields (7); telescoping (5) yields Tn−1 t<Tn E[gic (Y − (t), γ(t))] → λci , hence (9) with zero slack and support only on reachable memory states. For sufficiency, fix a strictly feasible x with slack δ > 0. Assumption 1 supplies a queue-compatible H-slot comparator. Choosing H with Hδ > Lx , Theorem 3 gives negative linear frame drift outside a bounded set while (11) controls penalties. Thus all strictly feasible vectors covered by the assumption are stabilizable; density and closure give equality. The next results isolate two independent capacity gains from causally retained execution memory: reduced work and changed successor generation. Proposition 1 (Work-reuse strict containment). For the empty initial memory state y ∅ , Λno-mem ⊆ Λsfm (y ∅ ), and the containment can be strict in a single-stage network. Proof sketch. Containment follows by discarding all generated objects and emulating the no-memory system. For strictness, take one server with compute capacity κw ≥ 2. Without memory, each request consumes κw units, so the service rate is 1; after one completed request generates and retains a compatible object m, each later request consumes one unit, so the service rate is κw . The setup phase is finite and does not affect strong stability, hence every λ ∈ (1, κw ) separates the regions. The second gain does not rely on reducing service work. The next example keeps all work requirements fixed and separates the regions only through memory-dependent successor generation. Theorem 2 (Transition-law strict containment). There exists an SFMSN with empty initial memory in which wi (a, Yi ) is identical for all memory states but Λno-mem ⊊ Λsfm (y ∅ ) because the successor law Pi (c′ | a, Yi ) is memory-dependent. Proof sketch. Let the upstream server have capacity 2, the verifier capacity 1, and the link capacity at least 2. Work is one unit at each node for all memory states. Without memory, every upstream completion creates a verifier job, so stability requires λ < 1. With a retained object m, generated after the first upstream completion, later upstream completions depart directly; the verifier sees only finite transient load while the upstream server stabilizes every λ < 2. Thus any λ ∈ (1, 2) proves strict containment without changing wi (a, Yi ). Corollary 1 (Probabilistic verifier reduction). Let the upstream and verifier completion capacities be µu and µv , and assume the upstream–verifier link has capacity at least µu . If the first upstream completion deterministically generates and retains m, and Pu (cv | a, ∅) = πh , Pu (cv | a, {m}) = πl , with 0 ≤ πl < πh ≤ 1, then the no-memory system

requires λ < min{µu , µv /πh }, whereas retaining m permits λ < min{µu , µv /πl }, with µv /0 = ∞. Proof sketch. The upstream constraint is λ < µu . The verifier receives rate πh λ without memory and πl λ after the finite memory-generation transient. Combining these with the verifier capacity gives the two intervals. The improvement is strict exactly when the reduced verifier-arrival rate relaxes the active bottleneck.

IV. M EMORY-AUGMENTED BACKPRESSURE This section derives the queue controller from Section III. The key score is the memory-augmented backpressure weight Φ, whose net-service form matches gic . Since retention may affect only future service, the theorem-level policy is an exact finite-frame oracle; the experiments use a one-slot surrogate that inherits guarantees only under Corollary 2. A queue-only rule can stabilize visible demand while leaving execution memory uncontrolled: Proposition 2 (Queue-stable but unmanaged-memory growth). A queue-only rule can strongly stabilize a single-server request queue while the expected number of unmanaged servicegenerated objects grows linearly. Proof sketch. With Bernoulli arrivals 0 < λ < 1, unit service, and independent object generation probability ρ > 0 per completion, rate conservation gives E[M (T )]/T → ρλ. Thus queue stability alone does not control unmanaged memory; SFMSN enforces (2)–(3). The local reason is that visible queues and nominal capacities may be identical while compatible memory differs. Proposition 3 (Minimax decision-state separation). Consider one unit of demand that must be assigned to exactly one of two branches with identical visible queues and nominal capacities, and let K > 1. In one instance branch 1 has service penalty 1 and branch 2 has penalty K; in the paired instance the penalties are reversed because the resident-memory states are reversed. Any randomized rule whose split is invariant to these paired memory states has worst-case expected one-step penalty at least (K + 1)/2, whereas a memory-aware rule pays 1. Equivalently, the worst-case additive gap is at least (K − 1)/2. Proof sketch. Let α be the probability of assigning the unit to branch 1. The two expected penalties are α + (1 − α)K and αK + (1 − α). Their maximum is minimized at α = 1/2, where it equals (K + 1)/2. A memory-aware rule observes which branch is compatible and assigns the unit to that branch, incurring penalty 1. For queue vector Q, inherited memory y, and one-slot

control γ, define Φ(Q, y, γ) =

X X

Fijc Qci − Qcj − νdcij

(i,j)∈E c∈C

+

ι(a)

X X

Sia Qi

− νpi (a, Yiγ )

X

Pi (c′ | a, Yiγ )Qci

i∈N a∈Ai

−

X X

Sia

=

(12)

c′ ∈C

i∈N a∈Ai mem

−X ν h̄

′

(y, γ)

Qci gic (y, γ) − νJ(y, γ).

(13) (14)

(i,c)∈I

The second form is used in the proofs: Φ rewards expected net queue drainage and subtracts weighted routing, service, and memory penalties. A one-slot maximizer prices active memory but can undervalue retention whose benefit appears later. Let H be a frame length, tn = nH, Qn = Q(tn ), and Y0 = − Y (tn ). For τ = 0, . . . , H − 1, write Yn (τ ) = Y − (tn + τ ) and γn (τ ) = γ(tn + τ ). A causal frame policy uses only the realized queue–memory history inside the frame; the physical queues evolve normally, but the objective freezes Qn as the linear weight. Define "H−1 # X
n ,Y0 Φ Qn , Yn (τ ), γn (τ ) , ΦH (Qn , Y0 , π) = EQ π τ =0

(15) where the expectation covers arrivals, service outcomes, generated objects, and internal randomization.

Algorithm 1 Exact Frame-Based MABP Oracle Require: H, ν, arrival law, service-outcome kernels, and memory-transition kernels 1: for each frame boundary tn = nH do 2: Observe Q(tn ) and Y − (tn ). 3: Compute a causal feasible H-slot policy maximizing ΦH (Q(tn ), Y − (tn ), π). 4: Execute it for H slots, updating queues and memory by (5) and (3). 5: end for Algorithm 2 Practical One-Slot MABP Surrogate Require: ν, primitive profiles or estimators, and queue/memory telemetry 1: for t = 0, 1, . . . do 2: Observe Q(t), Y − (t), and feasible transfer candidates. 3: Solve (16) for routing, transfer, activation, and service. 4: Execute the pre-service decisions; observe successor jobs and generated objects Gi (t). 5: for all nodes i do 6: Set Mcand (t) = Ri (t) ∪ Gi (t). i 7: Evaluate the precommitted rule (18), or its greedy marginal version, to obtain feasible Zi . 8: Set Yi− (t + 1) = Zi . 9: end for 10: Add exogenous arrivals and update Q(t + 1) by (5). 11: end for

Definition 2 (Frame-MABP). Exact frame-MABP maximizes (15) over causal H-slot policies whose realized actions satisfy γ(tn +τ ) ∈ Γ(Q(tn +τ ), Y − (tn +τ )) on every sample path, executes the maximizer for the frame, and replans at tn+1 . The case H = 1 is myopic; H > 1 values retained memory through later service operators.

Generated objects are unknown before service, so retention is handled by the post-service proxy below. For resident set Ri , let Yiact (Ri ) be the feasible active subsets. Define the frozen-queue node value

A. Practical One-Slot Surrogate

Ψi (Ri ; qi ) :=

The exact frame oracle is analytic, so experiments use a one-slot surrogate: it preserves the instantaneous Φ weights and approximates delayed retention by a frozen-queue local value. It is not automatically throughput-optimal; the guarantee requires the bounded-loss condition in Corollary 2. Routing and service use Wijc (t) = Qci (t) − Qcj (t) − P ′ ι(a) νdcij , Wia (qi ; Yi ) = qi − c′ ∈C Pi (c′ | a, Yi )qic − νpi (a, Yi ). Let ζ(t) = (F (t), u(t), R(t), Y act (t), S(t)), and let Γpre (Q(t), Y − (t)) collect the pre-service tuples satisfying migration-source, residency, memory-budget, link-capacity, compute-capacity, queue-feasibility, activation, and transferlink constraints. The pre-service decision is X X max Fijc (t)Wijc (t) pre − ζ(t)∈Γ

(Q(t),Y

(t))

(i,j)∈E c∈C

+

X X

Sia (t)Wia Qi (t); Yiact (t)

i∈N a∈Ai

−ν

X

X

(i,j)∈E m∈M

m κm ij uij (t).

(16)

X

max

|A | Yi ∈Yiact (Ri ), s∈Z+ i a∈A i

s.t.

X

sa Wia (qi ; Yi )

sa wi (a, Yi ) ≤ µi ,

a

(17) X

sa ≤ qic ,

(i, c) ∈ I.

a:ι(a)=c

After service, the candidate set is Mcand (t) = Ri (t) ∪ Gi (t). i A precommitted retention rule is evaluated on the realized candidates using frozen Qi (t): Vi (Z; t) = Ψi (Z; Qi (t)) − νθi

X m∈Z

bm − ν

X

ωim ,

m∈Z∩Gi (t)

(18) over feasible Z ⊆ Mcand (t). Exact proxy retention maxii mizes (18); a greedy implementation uses ∆mem (m | Z; t) = i Vi (Z ∪ {m}; t) − Vi (Z; t). This proxy estimates next-slot service value only, so its gap to the exact H-slot oracle must be uniformly bounded in queue magnitude to preserve the theorem-level guarantee.

V. S TABILITY AND C OST G UARANTEES This section converts the frozen queue-weight objective into a finite-frame Lyapunov bound, uses Assumption 1 as a slack comparator, and derives the penalty–backlog tradeoff. Assumption 2 (Bounded finite primitives). The commodity, action, memory-state, and one-slot feasible-control sets are finite. Arrivals have uniformly bounded conditional second moments, and routing amounts, service amounts, memory costs, and service penalties are uniformly bounded under every feasible control. P Throughout, let L(Q) = 12 (i,c)∈I (Qci )2 , Q · v = P c c − (i,c)∈I Qi vi , and En [·] = E[· | Q(tn ), Y (tn )]. At tn = nH, define ∆H (tn ) = En [L(Q(tn + H)) − L(Q(tn ))]. By Assumption 2, there is a finite BH , independent of Q(tn ) and Y − (tn ), such that every causal H-slot policy satisfies ∆H (tn ) ≤ BH + HQ(tn ) · λ   H−1 X X Qci (tn )gic (Yn (τ ), γn (τ )) . − En  τ =0 (i,c)∈I

(19) This is the only Lyapunov estimate used below; memory enters only through gic . Also, E[h(t) | Y − (t), γ(t)] = J(Y − (t), γ(t)). Theorem 3 (Throughput of exact frame-MABP). Suppose Assumptions 2 and 1 hold. Fix ν ≥ 0 and an arrival vector λ supported by a reachable occupation measure x with uniform slack δ > 0. If H ≥ Hx and Hδ > Lx , then exact frame-MABP strongly stabilizes all queues. Moreover, for hsome finite BH,x,ν , independent of Q(tn ) and Yi − (tn ), PH−1 En L(Q(tn + H)) − L(Q(tn )) + ν τ =0 h(tn + τ ) ≤
BH,x,ν − Hδ − Lx ∥Q(tn )∥1 . Proof sketch. Adding the conditional expected penalty to (19) and using Φ(Q,h y, γ) = Q · g(y, i γ) − νJ(y, γ) gives PH−1 ∆H (tn ) + νEn ≤ BH + HQ(tn ) · τ =0 h(tn + τ ) λ − ΦH (Q(tn ), Y − (tn ), πMABP ). Let πx be the queuecompatible realization of x. Exact optimality and (10)– (11) give ΦH (Q(tn ), Y − (tn ), πMABP ) ≥ HQ(tn ) · ḡ(x) − c c ¯ Lx ∥Q(tn )∥1 −CH,x −ν(H J(x)+D H,x ). Since ḡi (x) ≥ λi +δ, substitution yields the stated drift bound after absorbing constants into BH,x,ν . Because Hδ > Lx and h(t) is bounded below, the frame drift is negative outside a bounded set. The standard frame Lyapunov criterion gives strong stability at frame boundaries, and bounded intra-frame increments extend it to all slots. To quantify the penalty–backlog tradeoff, compare exact frame-MABP with the best realizable slackδ occupation measure. ForP any policy π, define J¯π = lim supT →∞ T1 t<T E[hπ (t)], Q̄π = P 1 lim supT →∞ T t<T E[∥Qπ (t)∥1 ]. For fixed δ > 0, let ¯ Jδ⋆ be the infimum of J(x) over reachable occupation

measures that support λ with slack at least δ and satisfy Assumption 1. Theorem 4 (Frame drift-plus-penalty). Suppose ν > 0, and assume that the infimum defining Jδ⋆ is attained by a reachable occupation measure x⋆δ satisfying Assumption 1. Let Hδ , Lδ , CH,δ , and DH,δ be the realization constants associated with x⋆δ . For every fixed H ≥ Hδ satisfying Hδ > Lδ , exact frameMABP satisfies BH + CH,δ DH,δ + , J¯MABP ≤ Jδ⋆ + H νH Q̄MABP = O(ν) + O(1),

(20)

where BH is the constant in (19). The hidden constants may depend on H, δ, Lδ , Jmin , and the primitive bounds, but not on ν. Proof sketch. Repeating hthe comparison ini Theorem 3 with PH−1 x⋆δ gives ∆H (tn ) + νEn h(tn + τ ) ≤ BH + CH,δ +
τ =0 ⋆ ν(HJδ +DH,δ )− Hδ−Lδ ∥Q(tn )∥1 . Summing over frames, telescoping L(Q), dropping the nonpositive backlog term, dividing by νH, and taking T → ∞ gives the penalty bound. For backlog, retain the negative backlog P term and use h(t) ≥ Jmin , which yields lim supN
→∞ N1 n<N E[∥Q(tn )∥1 ] ≤ BH +CH,δ +ν H(Jδ⋆ −Jmin )+DH,δ Hδ−Lδ

. Thus the frame-boundary backlog is O(ν) + O(1), and bounded intra-frame increments extend the same scaling to Q̄MABP . Corollary 2 (Additive frame approximation). Fix ν. Suppose that, for every reachable Q, Y , an approximate H-slot solver app satisfies ΦH (Q, Y, π e) ≥ supπ ΦH (Q, Y, π) − CH , where the supremum is over causal, physically feasible frame policies app and CH is finite and uniform in Q, Y . Then the drift inequalities in Theorems 3 and 4 only gain the queue-independent term app CH . Hence throughput is preserved whenever Hδ > Lx , and app B +CH,δ +CH DH,δ J¯πe ≤ Jδ⋆ + H + H νH . Proof sketch. Replace the exact optimality step ΦH (πMABP ) ≥ ΦH (πx ) by the additive inequality app above. The loss CH does not scale with ∥Q∥1 , so it only changes the constant term and leaves the negative backlog coefficient unchanged. Remark 1 (Scope of the practical rule). Algorithm 2 is not automatically throughput-optimal. It inherits the frame theorem only when its accumulated ΦH -loss is uniformly bounded in queue magnitude; generic greedy retention or errors in Pi need not satisfy this condition. VI. E VALUATION We evaluate Algorithm 2, not the exact frame oracle. Unless stated otherwise, MABP uses greedy feasible max-weight decisions in (16) and marginal-value retention in (18). OracleSur applies near-exact local search with known primitives, measuring the one-slot optimization gap rather than a capacity upper bound.

≈

occupation boundary 

Memory budget 

()

>

1500

1000

500

()

1.0

0.8

40%

0.6

30%

0.4

20% 0.2

Stable-seed fraction

50% <

2000

Total backlog

TABLE I M IXED - WORKFLOW RESULTS . S TABLE LOAD IS THE LARGEST STABLE TESTED LOAD ; OTHER METRICS ARE MEASURED AT ρ0 = 0.85. E NTRIES REPORT MEAN ± STANDARD DEVIATION OVER REPEATED RUNS . H IGHER IS BETTER FOR STABLE LOAD AND HIT RATIO ; LOWER IS BETTER OTHERWISE .

10%

SP JSQ BP Cost-BP GreedyCache-BP GNN scheduler MABP Oracle-Sur

Stable load 0.5 0.5 0.6 0.6 0.7 0.8 0.9 1.0

Penalty

P95 delay

Hit ratio

Churn

3.86±0.00 2249.03±187.23 0.05±0.00 0.06±0.00 3.85±0.00 1688.81±195.79 0.06±0.00 0.05±0.00 3.86±0.00 1295.90±98.58 0.06±0.00 0.06±0.00 3.69±0.00 1219.89±98.19 0.10±0.00 0.06±0.00 3.52±0.00 932.02±56.95 0.32±0.00 0.07±0.00 3.41±0.00 562.21±20.22 0.33±0.00 0.06±0.00 3.12±0.00 129.13±16.91 0.43±0.00 0.04±0.00 3.05±0.00 88.76±32.27 0.46±0.00 0.04±0.00

0 0

B. Simulation Results Table I separates stability from fixed-load stress performance. MABP reaches stable load 0.9, exceeding GNN scheduler, GreedyCache-BP, and BP by 0.1, 0.2, and 0.3, respectively. At ρ0 = 0.85, it reduces P95 delay by 90.0% over BP and 77.0% over GNN scheduler, reduces penalty by 19.2% and 8.5%, respectively, and stays within 2.3% of Oracle-Sur.

200

300

400

500

5%

600

0.2

0.4

Time slot

0.6

0.8

Offered load 

1.0

1.2

0.0

(a) Backlog trace below/near/above

(b) Occupation boundary versus memory

boundary

budget

Fig. 2. Stability and capacity-region diagnostics. (a) The below-boundary load keeps the backlog controlled, while near-boundary and above-boundary loads produce visibly larger backlog growth. (b) The occupation-measure outer boundary expands as the memory budget increases.

One-step penalty gap

A. Methodology a) Workloads and parameters.: The simulator instantiates finite RAG, MoE, and agentic SFMSNs whose commodities and objects follow Section II; it is not a trace replay of Qwen/Qwen2.5-7B-Instruct. We use a two-server separation case, a synthetic occupation-boundary network, and a mixed RAG–MoE–agentic graph. Normalized load ρ scales Poisson arrivals, and Bi (β) = βBimax , where Bimax is the eligible-object footprint. Defaults are horizon 600, ρ0 = 0.85, β = 0.25, reuse skew 1.0, transition noise 0.10, migration cost 2.0, noncached factor K = 4, and ν = 5. Sweeps use ρ ∈ {0.1, . . . , 1.2}, β ∈ {0.05, 0.10, 0.20, 0.30, 0.40, 0.50}, K ∈ {1, 2, 4, 8, 16}, ν ∈ {0, 1, 2, 5, 10, 20}, and Zipf skew in {0.5, 1.0, 1.5, 2.0}. b) Policies and baselines.: All policies share the same feasibility constraints and realized primitives; only ranking scores differ. SP uses shortest-path cost, JSQ the smallest eligible queue, BP memory-oblivious differentials, Cost-BP an added reference cost, and GreedyCache-BP popularity-based retention. GNN uses the same telemetry as MABP before the common feasibility projection. Memory-oblivious scores ignore Yi in ranking but obey state-dependent feasibility and may obtain passive hits; their successor law is averaged over profiled memory states. Learn-MABP uses frozen estimates of wi , pi , and Pi , with unseen pairs using no-reuse primitives. c) Metrics.: We discard the first 20% of slots and report mean ± standard deviation. Stability requires post-warm-up backlog slope below 0.035 and mean backlog below 120; stable load is the largest tested load passing both. Fixed-load metrics at ρ0 = 0.85 may place weaker baselines outside their stable region. Penalty is per served request, P95 delay uses completed requests, hit ratio counts compatible memoryassisted actions, churn counts insertions/evictions, and completion ratio is served work divided by arrivals.

100

7

( − )/ reference

350

6

Mean total backlog

Policy

5 4 3 2 1 0

1

2

4

8

16

Noncached penalty factor

(a) Memory-oblivious separation versus K

= =

=

300

=

250

= =

200

3.05

3.10

3.15

3.20

3.25

Quality-adjusted penalty / request

(b) Penalty–backlog tradeoff versus ν

Fig. 3. Mechanism-level tradeoffs. (a) The one-step BP–MABP penalty gap grows with the noncached penalty factor K. (b) Increasing ν moves the operating point toward lower penalty and larger backlog.

The higher hit ratio with lower churn supports selective state reuse rather than aggressive replacement. Fig. 2 validates the stability mechanism. The controlled trace separates below-, near-, and above-boundary regimes at loads 0.66, 0.96, and 1.14, with only the below-boundary case keeping backlog controlled. The occupation boundary rises from ρocc = 0.947 at β = 0.05 to 1.086 at β = 0.50, and the empirical stable grid load reaches 1.0 once β ≥ 0.20. Fig. 3 tests the mechanism-level predictions. In the paired separation instance of Proposition 3, the BP-minus-MABP one-step penalty gap is near zero at K = 1 and grows to 1.46, 3.50, and 7.52 at K = 4, 8, 16, matching the scaling toward (K − 1)/2 = 7.5. Increasing ν from 0 to 20 lowers penalty from 3.24 to 3.02, while mean backlog increases from 209.4 to 335.6, as expected for the DPP tradeoff. Fig. 4 shows where the penalty gain comes from: MABP reduces compute work from 2.116 under BP and 1.846 under GNN scheduler to 1.628, lowers transmission cost from 0.251 to 0.204, and concentrates retention on semantically compatible node–object pairs. Fig. 5 confirms that the mixed workflow is the hardest tail setting, with P95 delay 205.18 versus 113.04 for RAG and 69.96 for MoE. The penalty gain over Cost-BP increases with both memory budget and reuse skew, from 10.2% to 17.6% across β and from 12.2% to 16.5% across skew. Fig. 6 isolates the practical policy terms at ρ = 0.86, β = 0.25, K = 4, and ν = 5. Removing downstream transition pressure, reuse credit, or marginal-value retention raises P95 delay to 587.45, 281.25, and 307.10, respectively; LearnMABP gives 277.04, while full MABP gives 165.26 and the

TABLE II R EAL MULTI -GPU LLM- SYSTEM RESULTS . E NTRIES REPORT MEAN ± THE TWO - SIDED 95% S TUDENT-t CONFIDENCE - INTERVAL HALF - WIDTH ACROSS INDEPENDENT RUNS ; BOLD MARKS THE BEST POINT ESTIMATE . Metric

1.0 0.5

Aff.

GNN-Sched.

MABP-Route

6.14±0.06 2.61±0.19 73.37±2.02 1.94±0.06 0.985±0.007 0.972±0.006 1.179±0.015 0.00±0.00 42.93±9.03 1.00±0.00

6.10±0.10 2.64±0.22 71.77±2.08 2.08±0.14 0.986±0.003 0.908±0.010 1.196±0.032 0.00±0.00 48.25±12.61 1.00±0.00

6.21±0.08 2.82±0.33 70.53±0.86 1.93±0.14 0.965±0.003 0.998±0.014 1.056±0.051 0.00±0.00 33.74±12.06 1.00±0.00

1.0

memory

1.5

BP 6.04±0.07 2.41±0.17 71.62±1.49 2.18±0.03 0.993±0.003 0.884±0.007 1.268±0.025 0.00±0.00 31.68±6.18 1.00±0.00

retrieved context 0.8

rerank list kv prefix

0.6

expert warm

0.4

tool result

0.2

verified output

0.0

BP

P d. t-BP che he-B Cos N-S Cac GN edy Gre

BP

MA

Ora

e e d be tor MoE-A MoE-B erat n Em AG S Ge R

ur cle-S

(a) Penalty-component breakdown

l Too

0.0

rify Ve

Service node

(b) Learned memory-placement heatmap

8

penalty

hit ratio

7

P95 delay

completion

6 5 4 3 2 1

BP

t-BP

Cos

ABP ABP ABP ABP P-M it-M m-M rn-M Ref red Me Lea Pop NoC

BP

MA

Fig. 6. Component ablations normalized to BP. For lower-is-better metrics the score is xBP /x, and for higher-is-better metrics the score is x/xBP .

0.18

150

100

50

40% 0.16

30% 0.14

20%

0.12

10%

0 RAG

MoE

RAG--MoE agentic

(a) Tail-delay distribution by workload

5% 0.50

0.10

0.75

1.00

1.25

1.50

1.75

Penalty reduction vs. Cost-BP

50%

200

Memory budget 

Instantaneous delay (slots)

Fig. 4. Cost composition and memory placement. (a) MABP reduces compute work, transmission cost, and memory cost on the mixed workload. (b) Retention scores concentrate on semantically compatible node–object pairs rather than uniformly caching all objects.

Normalized score vs. BP

transmission

2.0

JSQ 6.05±0.10 2.25±0.15 71.82±2.40 2.05±0.18 0.994±0.006 0.884±0.023 1.278±0.066 0.00±0.00 38.64±4.31 1.00±0.00

Retention score

compute work

2.5

Execution-state object

Cost per completed request

Throughput (req/s) SLO goodput (req/s) P95 TTFT (ms) P95 E2E latency (s) GPU seconds/request KV hit ratio Downstream jobs/request Drop rate (%) P95 scheduler time (µs) Mean attempts

2.00

Reuse skew

(b) Memory-budget/reuse-skew sensitivity

Fig. 5. Tail behavior and sensitivity. (a) The mixed workflow has the heaviest tail because it combines RAG, MoE, and agentic transition stages. (b) The MABP gain grows as either memory budget or reuse skew increases.

lowest penalty 3.12. These terms are therefore complementary. C. Real Multi-GPU LLM-System Evaluation We also evaluate a routing-only deployment, MABP-Route, in a real four-GPU vLLM system serving Qwen/Qwen2.5-7B-Instruct through an OpenAIcompatible chat interface. An external asynchronous dispatcher routes a tool-heavy RAG, MoE-style, and agentic tool/verification workload with reusable prefix/KV and downstream tool-result keys. Because vLLM exposes prefix/KV reuse but not full object-level eviction, MABP-Route controls routing while leaving low-level KV eviction to the engine. All policies share the same request stream, model, worker pool, concurrency limit, and timeout configuration: interarrival time 0.13 s, at most 12 in-flight requests, timeout 180 s, one retry, temperature 0, maximum output length 72, and SLO 1.7 s. JSQ uses in-flight queue length; BP uses queue pressure with a completion tie-breaker; Aff. favors prefix

locality; GNN-Sched. uses the same telemetry as MABP with a learned graph-score rule; MABP-Route combines queue pressure, prefix/KV reuse, downstream tool-state reuse, and overload damping. Table II shows that MABP-Route attains the best point estimates for SLO goodput, E2E latency, GPU work, KV hit ratio, and downstream jobs, with zero drops and one attempt per request for all policies. Relative to GNN-Sched., it improves SLO goodput by 6.8%, reduces P95 E2E latency by 7.2%, reduces GPU seconds/request by 2.1%, raises KV hit ratio from 0.908 to 0.998, and reduces downstream jobs/request by 11.7%. The real-system margins are smaller than in simulation because all policies share the same vLLM cache manager and safe concurrency limit, and some confidence intervals overlap; we therefore interpret them as mechanism-consistent gains rather than broad statistical dominance. VII. C ONCLUSION This paper develops MABP for stateful foundation-model service networks, where reusable execution states affect not only local work and penalty but also downstream successor generation. By modeling memory as a controlled network state, we derive an occupation-measure capacity outer bound, identify when memory enlarges the stabilizable region, prove memory-aware/memory-oblivious separation, and give throughput and drift-plus-penalty guarantees for exact frameMABP with bounded-loss approximate extensions. Experiments on mixed RAG–MoE–agentic workloads and a real multi-GPU LLM serving system show that the practical oneslot surrogate improves stability, tail delay, state reuse, and SLO goodput.

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