Tools for Reducing Service Time in Near-Term Quantum Networks

Tools for Reducing Service Time in Near-Term Quantum Networks Jake Smith1,2,3 , Thomas R. Beauchamp1,2,3 , Scarlett Gauthier1,2,3 , Oumayma Bouchmal1,2,3 , and Stephanie Wehner1,2,3 1

QuTech, Delft University of Technology Kavli Institute of Nanoscience, Delft University of Technology 3 Quantum Computer Science, Electrical Engineering, Mathematics and Computer Science, Delft University of Technology jsssmith@tudelft.nl, t.r.beauchamp@tudelft.nl, s.s.gauthier@tudelft.nl, o.bouchmal@tudelft.nl, s.d.c.wehner@tudelft.nl

arXiv:2608.20954v1 [quant-ph] 21 Aug 2026

2

Abstract—Architectures have been proposed to control entanglement generation in multi-user quantum networks. To allow time for local operations and classical communication at end nodes, these architectures insert fixed separations between consecutive batches of entanglement generation attempts. This reduces network utilization when attempts fail, leaving the network idle during the scheduled separation. To address this limitation, we propose a novel method to reclaim this idle time by shortening the scheduled separation between attempts while respecting hardware constraints. The method uses an analytical execution model to optimize the separation and reduce the total network service time of an application. Evaluations within the Arqon architecture show network service time reductions of up to 42 minutes (7.6%) for single applications and 16–29 minutes (26– 30%) per application when co-scheduled. The approach applies broadly to quantum network architectures that share hardware between entanglement generation and local operations, and the method can be used online by network schedulers. Index Terms—Quantum Internet, entanglement scheduling, minimum separation.

I. I NTRODUCTION Quantum networks generate shared entanglement between end nodes, enabling the execution of applications such as secure remote computation [1], improved clock synchronization [2], precision sensing [3], and others [4], [5]. Application execution at end nodes is organized into an application session, defined by a target number of application instances to be completed. A session describes the application’s demand for entanglement over time, requiring the network to repeatedly generate entanglement in order to serve successive instances. To support the execution of each application instance, the network generates end-to-end entanglement as entangled links. The network computes a schedule that coordinates entanglement generation attempts along a path, organizing them into scheduled blocks of time during which attempts are retried until the required entanglement is produced or the block ends. The network service time is the total time scheduled by the network for a session. The end nodes at either end of the path are not under network control and must actively participate for an entangled This work was funded by the European Union’s Horizon Europe research and innovation programme under grant agreement No. 101102140 – QIA Phase 1. The authors also acknowledge funding from NWO VICI.

link to be generated. Two distinct roles arise within a scheduled block. The network delivers an entangled link when it successfully produces one spanning the interior of the path. The end nodes then consume a delivered link by aligning their own generation attempts with the network’s schedule. After consumption, the end nodes perform local operations and classical communication (LOCC), such as quantum gates, to complete the execution of an application instance. In demonstrated quantum networks, end node hardware participating in consumption also performs LOCC (e.g. [6]–[11]). As a result, consecutive consumptions must be separated by at least the time required for LOCC, an interval we refer to as the minimum separation. The network scheduler must therefore space consecutive deliveries by at least this interval, making schedule computation a constrained scheduling problem. The impact of the minimum separation constraint depends on how the minimum separation compares to the expected interval between successive consumptions. On current hardware, these two timescales are comparable. Demonstrated heralded entanglement generation rates are 2.22 Hz in trappedion systems [6] and 0.022 Hz in solid-state platforms [7], both over 10 km links. Minimum separation times arise from device-level operations such as quantum gates, cooling, and readout [12]–[14], as well as from control-stack overhead including classical processing and communication [15], [16]. These contributions span microseconds to seconds. At the trapped-ion rate, successive consumptions occur every 450 ms on average, comparable to a representative minimum separation of approximately 500 ms, making the separation sufficiently long to constrain every consumption. A conservative scheduling approach assumes that end nodes always require the full minimum separation between consecutively scheduled blocks. Because each block succeeds only probabilistically, many blocks fail to produce an endto-end entangled link. Nevertheless, the minimum separation is scheduled after every block, so following a failure the end nodes remain idle for the full interval despite having no LOCC to perform. This behavior over-provisions the network service time and prolongs session execution time. To reduce this idle time, it is natural to consider shortening the scheduled separation below the minimum. Doing so, however, risks scheduling a block while the end nodes are

still performing LOCC. In this case, the block must be skipped or the current application instance preempted, discarding inprogress computation. Shorter separations therefore reduce network service time but increase the probability that blocks are skipped. Enough skips will prevent the session from completing its target number of instances. This tradeoff admits an optimal scheduled separation that minimizes session execution time while still completing the target instances. In this work, we address this challenge by developing analytical tools that compute the optimal scheduled separation for a given application session. Because the minimum separation constraint arises whenever the end node hardware that participates in consumption also performs LOCC, the proposed tools apply broadly to entanglement delivery scheduling and are not tied to a specific network architecture. We evaluate the tools within the Arqon [17] suite of network control applications, which accepts entanglement demands from end nodes and computes schedules that respect the minimum separation constraint. Arqon commits to delivering the entangled links required by a session with at least a specified service probability, providing a concrete architecture in which to quantify the benefits of an optimized scheduled separation. The contributions of this work are as follows: Analytical optimization of scheduled separation: We develop a novel method to align the network’s delivery schedule with end node consumption by modeling session execution under a reduced scheduled separation as a Markov chain over executed and skipped blocks. From this we formalize a constrained optimization to determine the optimal scheduled separation and derive a closed-form lower bound on the expected execution speedup. The resulting tools are lightweight and can be used by network schedulers at runtime without repeated simulation. • Evaluation under realistic hardware parameters: We evaluate the proposed tools within the Arqon architecture [17] using state-of-the-art and projected trapped-ion entanglement generation parameters. In the state-of-theart regime, the network service time is reduced by up to 42 minutes, a 7.6% reduction corresponding to a singlesession execution speedup of 6.1%. In the projected regime, the network service time is reduced by up to 2.8 hours, a 6.5% reduction and a speedup of 5.1%. We identify the conditions required for positive speedup and validate the analytical predictions using Monte Carlo simulation, confirming that the requested service probabilities are maintained across all tested configurations. • Extension to multi-session scheduling: We extend the analysis to multiple sessions co-scheduled under roundrobin scheduling, showing that this configuration provides additional separation that mitigates or eliminates skip penalties. For two co-scheduled sessions, the network service time is reduced by 16–29 minutes per session, a 26–30% reduction corresponding to execution speedups of 25–27%, substantially exceeding the single-session case while maintaining the required service probability.

•

II. R ELATED W ORK The minimum separation constraint poses an explicit scheduling problem only when three conditions hold together. Entanglement must be generated when requested rather than buffered in advance; each end node must manage its own execution, as in quantum internet (QI) applications; and the end node must share hardware between consumption and LOCC, since dedicated hardware for each would let local operations proceed without blocking the next consumption. Demonstrated quantum networks operate in the generatewhen-requested (GWR) paradigm, generating entanglement in response to application demand rather than in advance [6]– [11]. A pre-loaded network instead generates and buffers entanglement independently of requests, and applications draw from the buffer when ready to consume [18]–[20] Each consumption then proceeds when ready and respects the separation by construction. Such architectures, however, require generation rates and coherence times beyond current capabilities [21] and have not been demonstrated. The constraint also depends on which entity schedules end node execution. In distributed quantum computing, a central orchestrator schedules entanglement deliveries, consumption, and local operations jointly [9]–[11], again satisfying the constraint by construction. QI applications, by contrast, partition the application into node-specific programs scheduled independently by each end node. The network schedule coordinates delivery but cannot guarantee the constraint is respected. Even so, most scheduling work for GWR networks serving QI applications terminates the model at the generation of an end-to-end link, abstracting delivery and consumption into a single event [18], [22]–[28], so the local operations that follow consumption on shared hardware never enter the optimization. The constraint therefore has not been observed as a scheduling problem in any demonstration. The networks that both generate on demand and share end node hardware either orchestrate execution centrally, or have not yet performed QI sessions with repeated application instances in which it manifests. This is the regime in which near-term quantum networks will operate and QI applications will be programmed, and recent work has begun to develop end node and network architectures to support it. QNodeOS, implemented by Delle Donne et al. [15], is the first end node runtime enabling platform-independent execution of QI applications written in high-level software. To provide a compatible network architecture, Beauchamp et al. [21] propose a centrally controlled architecture organized around the entanglement packet: a specification of the entangled links required to execute a single application instance, generated within a scheduled interval referred to as a packet generation attempt (PGA). To respect the minimum separation constraint, consecutive PGAs are conservatively separated by the full minimum separation, assuming the worst case in which the packet is generated at the very end of the PGA. This is notably the first architecture to consider how application execution requirements must inform network scheduling.

While the network architecture requires that successive packets be separated by some minimum separation, it does not prescribe how packet deliveries are scheduled or with what reliability. To address this, Arqon [17] builds on this architecture as the first suite of network control applications designed to reliably fulfill accepted demands by periodically computing a network schedule of PGAs. For each accepted demand, a service agreement specifies the number of packets required to complete all instances, and Arqon schedules sufficient PGAs to meet this requirement with the service probability before the expiry time. For each scheduled PGA, Arqon reserves the appropriate network resources on an end-to-end path between the two end nodes executing the application. The Arqon authors further provide a dedicated simulation framework, arqon-sim [29], which simulates schedule execution at the level of individual PGAs: a PGA either succeeds or fails with the packet success probability, and the time at which the packet is generated within a PGA is not modeled. Investigating scheduled separation reduction requires this finer resolution, since minimum separation violations occur only when a packet is generated late enough that LOCC operations extend into the subsequent PGA. We therefore develop a separate, dedicated simulation package that resolves packet generation time within each PGA. III. P RELIMINARIES A. Network Model To optimize the scheduled separation chosen by the network scheduler between consecutive deliveries, we consider a GWR network supporting QI applications. Each end node participates in entanglement generation and performs LOCC using the same hardware. End-to-end entangled links are generated at rate λ with some minimum fidelity, abstracting the network’s interior delivery, the end nodes’ consumption, and any hardware calibration latencies into a single rate parameter. An application App is executed between a pair of networked end nodes N = (N1 , N2 ) and requires N inst instances to complete, subject to a deadline texpiry . These parameters define an application session [21], S = (session_id, N , App, N inst , texpiry ).

(1)

Each session submits a demand to the network specifying the identical packet required for each instance. A packet is generated when the required number of entangled links have been delivered and consumed within a time window w, where w is constrained by the coherence time of the end node’s quantum memory. The network schedules PGAs for each demand, and end nodes participate in all scheduled PGAs. Following [17], we assume each instance’s LOCC begin only after a packet is generated and that end nodes do not interrupt an executing instance to participate in entanglement generation. Successive packet generations not separated by at least the duration of LOCC violate the minimum separation constraint, and we assume the end nodes skip the subsequent PGA. In practice, LOCC also account for hardware calibration

no ingress tgen Original separation (α = 0)

idle (wasted)

PGA i

T ′ = tminsep

E tgen Reduced separation (α > 0)

PGA i+1

ingress α PGA i+1

PGA i T′

E

T ′ = tminsep − αE PGA

ingress α

LOCC

separation

Fig. 1. The ingress α is the fraction of PGA i by which the scheduled separation is shortened, giving T ′ = tminsep − αE. At α = 0 (top) the full minimum separation is kept and the network idles after an early tgen . At α > 0 (bottom) the separation shortens by αE, reclaiming idle time. A packet generated within the shaded ingress region risks a minimum separation violation (Section IV-B).

(e.g. ion cooling [6] or charge-resonance checks in colorcenter platforms [7]) and software stack overheads such as processing [15] and communication latencies [8]. B. Performance Metrics To quantify the effects of scheduling improvements on network resource provisioning and end node session execution, we define three performance metrics: the reduction in network service time R, the absolute service time savings ∆Tserv , and the session execution speedup S. We first introduce the following quantities. For session S, let E denote the duration of a PGA, tminsep the minimum time an end node requires for LOCC between successive consumptions, and T ′ ≤ tminsep the scheduled separation between consecutive PGAs. Each instance requires exactly one packet, and the network allocates a release time of E +T ′ per instance, following the terminology of Arqon [17]. Let N scheduled denote the total number of release times allocated by the network for a session. Let tgen denote the time at which a packet is generated within a successful PGA, and let τ denote the dimensionless release-time index at which the final required packet is generated, counting both executed and skipped PGAs. The network service time Tserv = N scheduled · (E + T ′ ) is the total duration over which network resources are scheduled for a single session. The reduction in service time at a reduced scheduled separation (T1′ < tminsep ) relative to the baseline at the full minimum separation (T2′ = tminsep ) is R=1−

Tserv,1 , Tserv,2

(2)

with corresponding absolute saving ∆Tserv = R · Tserv,2 . The service time upper bounds the session execution time, Tsession = (τ − 1) · (E + T ′ ) + tgen ,

(3)

since not all allocated release times may be needed to generate the required number of packets. When τ = 1, the first PGA produces the final required packet and Tsession = tgen ≤ E + T ′ . In the ideal case where every scheduled PGA succeeds, τ = N inst and the session completes at

(N inst − 1) · (E + T ′ ) + tgen . The session execution speedup compares Tsession at the reduced and baseline separations: Tsession,1 S = 1− . Tsession,2

(4)

These metrics depend only on the scheduling structure of repeated entanglement deliveries and apply to any network architecture in which persistent demands require successive generation attempts separated by a minimum interval. C. Architecture To develop and evaluate the methods for the constrained scheduling problem, we adopt the demand and scheduling formalism of Arqon [17], which defines end node demands and how their corresponding packet generation probabilities and PGA execution times are calculated. Each session S submits a demand d for entanglement generation to the network, d = (p; tminsep ; texpiry ; N inst ; ϵservice ) d

(5)

with packet p = (w, s, F ) requesting s entangled links of minimum fidelity F to be generated within a window w, and ϵservice defining the service probability (1 − ϵservice ) with d d which the demand must be fulfilled. Arqon constructs periodic schedules over fixed-length scheduling intervals, and each scheduled PGA attempts to generate a packet for end node consumption. To respect the minimum separation constraint, Arqon separates consecutive PGAs scheduled for the same demand by a minimum interval of tminsep . The PGA duration E(p) is the time required to generate a packet with probability at least p, determined by the entanglement generation rate along the end-to-end route: larger p requires more attempts and therefore a longer E(p). The PGA success probability ppacket is then determined by the optimization   ppacket = argmin N SI (p, ϵservice ) E(p) + tminsep , (6) d p

SI

(p, ϵservice ) is the number of PGAs scheduled per d inst

where N scheduling interval for the demand to generate N packets with probability at least (1 − ϵservice ). If all scheduled PGAs d execute, the expected number of packets generated is E[P ] = ppacket N executed ≥ N inst .

(7)

Minimum service is achieved when the number of packets generated is at least N inst , and the service agreement between the network and an accepted demand guarantees that minimum service is provided with probability at least (1 − ϵservice ). d Scheduling improvements must continue to satisfy the service agreement while also reducing the execution time of each session. The definitions of Tserv , Tsession , and session speedup S from Section III-B apply directly with N scheduled = N SI . Because Arqon allocates the same number of PGAs for a given demand regardless of the scheduled separation, N SI is identical for both the original and reduced schedules. Expanding Tserv = N SI · (E + T ′ ) in Eq. (2) shows the ratio N1SI /N2SI cancels, and the service time reduction and session speedup follow from Eqs. (2) and (4).

D. Problem Statement Arqon satisfies the minimum separation constraint by scheduling consecutive PGAs with a separation of tminsep . This assumes the worst-case generation time of E. In practice, packets are frequently generated before E, and the required separation depends on the generation time within the PGA. Arqon’s conservative treatment of this constraint therefore overestimates the required scheduled separation, prolonging session execution time and reserving network resources that could otherwise be allocated to nodes ready to use them. Reducing the scheduled separation below tminsep shortens the release time and recovers this excess capacity. However, a packet generated sufficiently late in the PGA may violate the minimum separation constraint. This work addresses this tradeoff by computing the scheduled separation that minimizes the expected session execution time, subject to the expected packet count reaching N inst within the allocated N SI PGAs. The probabilistic minimum service guarantee is not formally enforced by the optimization and is instead verified empirically by simulation across all tested configurations. IV. M ETHODS The following methods develop the analytical tools for computing the optimal scheduled separation and are publicly available in [30]. A. Simulating Packet Generation We employ a Monte Carlo approach [31] to evaluate whether a packet is generated within a sliding window w for a given demand d. The simulation, described by Algorithm 1, samples end-to-end entangled link generation as a Poisson process with rate λ over the PGA duration E. At each new link generation, the current sliding window is set to the earliest generation still within w of the latest, tracked by the pointer ptrfirst into the arrivals list. If s links accumulate within the window, the packet is generated at time tgen marking the final accumulated link. Otherwise, E expires before s links accumulate and the PGA fails to generate a packet. We compare two parameter regimes of λ and w following recent demonstrations of entanglement distribution between trapped-ion nodes at metropolitan-scale distances. Liu et al. [6] represents the state-of-the-art (SOTA) in rate and coherence time, achieving (λ = 2.22 Hz, tcoh = 550 ms). We also compare this to an optimistic ion-trap regime of (λ = 10 Hz, tcoh = 4 s) [17]. The window w is set to tcoh /2, reserving the remaining coherence time for local quantum operations. The minimum separation is set to tminsep = tcoh /2 + tLOCC , where tLOCC = 0.25 s accounts for local operations and classical communication between nodes [15], [16], and trapped ion cooling latencies [14]. Qubit decoherence after delivery is not modeled, as we assume the window w is chosen to ensure that delivered qubits remain coherent throughout instance execution.

Under our earlier assumption that a minimum separation violation causes the end node to skip the subsequent PGA, pmv (α) is also the probability that an end node skips the next PGA. The distribution of packet generation times within a PGA admits no closed-form expression, so pmv (α) cannot be evaluated analytically. The Markov chain session model in the following subsection requires pmv at many values of α during the constrained optimization, which would be prohibitively expensive to evaluate by fresh Monte Carlo simulation, particularly at runtime by a network scheduler. To avoid this cost, we precompute pmv into a lookup table by sweeping ppacket and α with the Monte Carlo simulator of Algorithm 1. Each table is computed once per physical configuration (λ, w, s) and reused across all demands sharing those parameters. For each configuration, we collect N = 104 successful trials via rejection sampling, and pmv is the fraction in which the packet is generated within the violation zone. Once the table is available, the Markov chain model and the optimization over α reduce to table lookups and closed-form expressions, enabling lightweight runtime evaluation within the network scheduler.

Algorithm 1: Packet generation within sliding window Input: λ, tstart , E, w, s Output: Success flag and tgen , the time s links accumulate within w Initialize t ← tstart , ptrfirst ← nil, arrivals ← [ ]; while t < E do t ← t + Exp(1/λ); if t ≥ E then break; Append t to arrivals; for each ti ∈ arrivals do if t − ti < w then ptrfirst ← index of ti ; break; if |arrivals| − ptrfirst ≥ s then tgen ← t; return true, tgen ; return false, nil

C. Modeling Session Performance with Reduced Scheduled Separation

B. Relaxing a Demand’s Scheduled Separation We consider reducing the scheduled separation between consecutive PGAs from the demand’s minimum separation tminsep to T ′ ≤ tminsep , described by Figure 1. A minimum separation violation occurs when the interval between packet generation in PGA i and the start of PGA i+1 is less than the time required for LOCC tminsep , formally T ′ + (E − tgen ) < tminsep . We restrict our analysis to the single-skip regime, in which at most one consecutive PGA is ever skipped. In the worst case, a packet generated at time E of PGA i triggers a skip of PGA i+1, and the time from that packet to the start of PGA i+2 is 2T ′ + E. Reducing T ′ further would leave insufficient time for PGA i+2 and trigger cascading skips. The single-skip guarantee therefore requires 2T ′ + E ≥ tminsep ,

(8)

yielding a lower bound on the reduced separation of (tminsep − E)/2 ≤ T ′ . Since T ′ is a scheduled time interval and therefore non-negative, the admissible range is   tminsep − E max 0, ≤ T ′ ≤ tminsep . (9) 2 At the upper bound, the original schedule is recovered. At the lower bound, the single-skip constraint (8) ensures the worstcase violation skips only PGA i+1. Further reduction would permit cascading skips. We parameterize the reduction of the scheduled separation by the PGA ingress α = (tminsep − T ′ )/E ∈ [0, 1]. A packet generated within the violation zone ((1 − α)E, E] causes a minimum separation violation. The conditional probability of this event given successful packet generation is pmv (α) = P[tgen ∈ ((1 − α)E, E] | tgen ≤ E].

(10)

To evaluate session performance under reduced scheduled separation, we simulate sessions by executing N SI scheduled PGAs subject to the skip behavior introduced in Section IV-B, in which a minimum separation violation causes the end node to skip the subsequent PGA. Although demands are generally long-lived and served across multiple scheduling intervals until texpiry , our analysis focuses on a single session, which isolates the tradeoff between reduced separation and skip penalties. For each executed PGA, sampled link generations determine whether a packet is generated and whether tgen falls in the violation zone ((1 − α)E, E]. The session terminates when N inst packets have been generated or when all N SI PGAs are exhausted. For completed sessions, Tsession takes the form of Eq. (3), with τ the release time in which the final packet is generated. Sessions that do not generate N inst packets within the allocation are assigned Tsession = N SI · (E + T ′ (α)). We estimate the expected session times as sample means over all trials, both completed and incomplete. The average simulated speedup at a given ingress is   E (τ (α)−1)(E + T ′ (α)) + tgen (α)  . (11) S(α) = 1 −  E (τ (0)−1)(E + tminsep ) + tgen (0) We define S(α) as one minus the ratio of expected session execution times, not the expectation of the per-session speedup of Eq. (4). We report the former to match the mean-based quantities predicted by the Markov chain model developed below. To predict session performance without repeated simulation, we introduce a Markov chain for the expected number of PGAs executed out of the total scheduled. We introduce several quantities to define the state space. Let Nk be the cumulative number of executed PGAs after k scheduled steps. Each scheduled step is labeled as either executed (G) or

skipped due to a minimum separation violation at the preceding step (G), and Sk ∈ {G, G} denotes this label at step k. The session is then modeled as the Markov chain Mk = (Nk , Sk ) on the state space N × {G, G} ∪ {(0, G)}. The transition structure reflects the skip rule directly. In state G, the scheduled PGA is executed, and it triggers a skip of the next step only if both hold: a packet is generated, which occurs with probability ppacket , and its generation time falls in the violation zone, which occurs with conditional probability pmv . The transition G → G therefore occurs with unconditional probability pr = ppacket · pmv . Oherwise the state remains in G. From G, the state returns deterministically to G, since a skipped step triggers no minimum separation violation, and the single-skip regime of Eq. (8) prevents any repeated ones. The sub-chain Sk forms a reduced Markov chain on the state space {G, G}, which describes the transitions between executed and skipped steps and can be used to compute the k-step transition probabilities. Its transition matrix is   1 − pr pr PS = , (12) 1 0 where rows and columns are ordered (G, G), from which the k-step transition probabilities follow by induction. In particular, the probability of being in the skip state G at step k, starting from G, is fk (pr ) = pr

k−1 X

(−pr )i .

(13)

i=0

Conditioning Nk on Sk−1 yields the recurrence E[Nk ] = E[Nk−1 ] + 1 − fk−1 (pr ),

(14)

where fk is as in (13). This resolves to the closed form E[Nk ] =

k−1 X

(k − i)(−pr )i .

πG = (15)

i=0

The expected number of packets generated after k scheduled steps is then E[Pk ] = ppacket · E[Nk ], since each executed PGA independently produces a packet with probability ppacket . We define k ∗ as the smallest k such that E[Pk ] ≥ N inst . An outline of the full proof is provided in Appendix VII-A. In any individual session, however, variance in both the number of executed PGAs and the per-PGA packet outcomes causes τ , the release time the session completes, to differ from k ∗ . Because Pk has positive variance around its mean, a fraction of simulated sessions complete before step k ∗ , such that E[τ ] ≤ k ∗ . The quantity k ∗ (α) · (E + T ′ (α)) therefore provides a conservative upper bound on E[Tsession (α)]. The minimum session speedup is then S̃(α∗ , k ∗ ) = 1 −

k ∗ (α∗ ) E + T ′ (α∗ ) · , k ∗ (0) E + tminsep

Three constraints bound the admissible PGA ingress α and define the feasible region of the optimization. First, the scheduled separation cannot be reduced below zero, since there is no remaining interval to shorten. This geometric constraint requires T ′ (α) ≥ 0, equivalently α ≤ tminsep /E. Second, the single-skip constraint (8) requires T ′ (α) ≥ max(0, (tminsep − E)/2), ensuring that the worst-case violation skips at most one consecutive PGA. When E < tminsep , this prevents T ′ from being reduced to zero. Finally, the service feasibility constraint requires k ∗ (α) ≤ N SI , ensuring the expected packet count reaches N inst within the allocated PGAs. We define α∗ as the ingress that minimizes the Markov session time Tsession (α) = k ∗ (α) · (E + T ′ (α)) over this feasible region. We remark that k ∗ enforces an expectation-based analog of the probabilistic service guarantee defined in Eq. (7). While minimum service requires that N inst packets are generated with probability at least (1 − ϵservice ), the Markov chain tracks only d the expected packet count and does not bound the probability that an individual session falls short. The service feasibility constraint k ∗ (α) ≤ N SI is therefore necessary but not sufficient for the probabilistic guarantee, and we verify empirically with simulation that minimum service is maintained at the optimal ingress across all tested configurations. While the speedup lower bound in Eq. (16) can be evaluated numerically from k ∗ , it does not reveal the structure of the tradeoff between reduced release time and skip penalties. To decompose the speedup into scheduling quantities, we derive a closed-form approximation of the ratio k ∗ (α∗ )/k ∗ (0) using the stationary distribution of the reduced Markov chain. The stationary distribution π = (πG , πG ) satisfies πPS = π and πG + πG = 1, which yields

(16)

which lower bounds the average simulation speedup S(α∗ ) ≥ S̃(α∗ , k ∗ ).

1 , 1 + pr

πG =

pr . 1 + pr

(17)

In steady state, a fraction πG of scheduled PGAs result in an executed PGA. Since each executed PGA produces a packet with probability ppacket regardless of the skip dynamics, the number of executions required to generate N inst packets is the same at any ingress. In the stationary limit, the number of scheduled steps at ingress α∗ therefore satisfies k ∗ (α∗ ) = k ∗ (0), 1 + pr (α∗ )

(18)

and k ∗ (α∗ ) = k ∗ (0) · (1 + pr (α∗ )). The relation is exact in the stationary limit and applies to finite sessions once they are long enough for the chain to converge, as is the case for all tested configurations with N inst = 103 averaged over 103 iterations. For shorter sessions, Eq. (18) should be replaced by numerical evaluation. Substituting Eq. (18) into Eq. (16) yields a closed-form speedup expressed only by the release time reduction R(α∗ ) and the unconditional skip probability pr (α∗ ). Because N SI is held fixed across scheduled separations, the service time reduction in Eq. (2) simplifies to the release time reduc-

State-of-the-Art Regime

101

100

100

0.0

0.2

0.4 0.6 ppacket [a.u.]

0.8

1.0

tgen/w E/w d10 d20

102 E/w [a.u.]

101

102 tgen / w [a.u.]

102 E/w [a.u.]

tgen / w [a.u.]

102

Optimistic Regime

tgen/w E/w d2 d4

101

101

100

100

0.0

0.2

0.4 0.6 ppacket [a.u.]

0.8

1.0

Fig. 2. Average packet generation time (solid, left axis) and PGA duration (dashed, right axis) as a function of packet generation probability ppacket for the state-of-the-art regime (left; λ = 2.22 Hz, w = 275 ms) and optimistic regime (right; λ = 10 Hz, w = 2 s). Both quantities are normalized by the window length √ w to enable comparison across demands with different PGA durations. Demands are labeled by their requested link count s, with z-scores z = (s − λw)/ λw of 1.8 and 4.3 for d2 and d4 (SOTA), and −2.2 and 0 for d10 and d20 (optimistic). Shaded regions indicate ± one standard deviation over 104 trials per point. All demands use N inst = 1 and ϵservice = 10−5 . d

tion R(α) = 1 − (E + T ′ (α))/(E + tminsep ), and the stationary speedup becomes

S̃(α∗ ) = 1 − 1 + pr (α∗ ) · 1 − R(α∗ )
= R(α∗ ) − pr (α∗ ) · 1 − R(α∗ ) . (19) | {z } | {z } release time reduction

skip penalty

∗

The first term, R(α ), captures the fraction of session time savings if no PGAs were skipped. The second term captures the fraction of that savings lost to the additional PGAs executed to compensate for skipped PGAs. The speedup is positive only when the release time reduction exceeds the skip penalty. Evaluating S̃(α∗ ) requires no simulation, only the pmv (α∗ ) lookup table and T ′ (α∗ ) = tminsep − α∗ E computed from demand parameters. Determining the optimal α∗ requires evaluating the service feasibility constraint k ∗ (α) ≤ N SI through the Markov chain recurrence, since the decomposition alone cannot predict whether the session completes in time. V. E VALUATION We evaluate the proposed methods in three stages. We first characterize the distribution of packet generation times within a PGA and the resulting lookup tables of minimum separation violation probabilities that drive the optimization, then quantify the single-session speedup, and finally extend to round-robin scheduling of two co-scheduled sessions. A. Simulating Packet Generation Time Distributions To evaluate scheduled separation relaxation, the distribution of packet generation times within a PGA must first be understood, since packets arriving near the end of the PGA are more likely to cause minimum separation violations. Demands ds are indexed by the requested packet’s link count s. Fig. 2 shows d2 and d4 in the SOTA regime and d10 and d20 in the optimistic regime. Fig. 2 plots tgen /w against ppacket , with E/w on the secondary axis; normalization by w enables comparison across

√ demands. The z-score z = (s − λw)/ λw indicates within how many standard deviations the requested link count lies from the expected number of arrivals per window. When z ≫ 0, a single window is unlikely to generate enough links, so the PGA spans many overlapping windows (E ≫ w). Generation times then spread across a substantial portion of E, as seen for d2 at z = 1.8 and d4 at z = 4.3. When z ≤ 0, a single window suffices (E ≤ w) and packets are likely generated within roughly one window scan (d10 at z = −2.2). B. Demand Lookup Tables Since no closed-form expression exists for the packet generation time within a PGA, the conditional violation probability pmv must be obtained by sampling. Fig. 3 presents pmv as a function of α and ppacket for one representative configuration from each regime: d2 (z = 1.8) in the SOTA regime and d10 (z = −2.2) in the optimistic regime. The two surfaces exhibit qualitatively distinct contour structures. In the SOTA regime (E ≫ w), pmv contours disperse along the α axis, allowing the ingress to encompass a substantial fraction of the PGA before minimum separation violations dominate. In the optimistic regime (E ≤ w), the contours concentrate at low α and pmv rises steeply with even small ingress fractions. Increasing ppacket at fixed α reduces pmv in both regimes, since a longer PGA shifts generation times earlier. These surfaces confirm that the z-score governs pmv ’s sensitivity to ingress: low-z demands reach high minimum separation violation probabilities at smaller ingress and therefore require more conservative separation reductions. Each surface is computed once for a given physical configuration (λ, w, s) and stored as a lookup table that the Markov chain session model queries during optimization. Because the table is precomputed, evaluating pmv at any candidate (α, ppacket ) requires only interpolation rather than repeated simulation. In regimes where λw is large, a unit√change in s produces only a small shift in z-score (∆z = 1/ λw), suggesting that tables

State-of-the-Art Regime 0.3 0.4

0.96

0.1 0.2 0.3 0. 0.45 0.6 0.7 0.8 0.9

0.2

0.5 0.6 0.7 0.8 0.9

0.1

Optimistic Regime 0.96 0.84

0.8

0.84

0.8

0.48 0.4

0.36 0.24

0.2

0.6

0.60

pmv [a.u.]

0.60

0.72

ppacket [a.u.]

0.6

pmv [a.u.]

ppacket [a.u.]

0.72

0.48 0.4

0.36 0.24

0.2

0.12 0.0

0.2

0.4

0.6

[a.u.]

0.8

1.0

0.00

0.12 0.0

0.2

0.4

0.6

[a.u.]

0.8

1.0

0.00

Fig. 3. Probability pmv of a minimum separation violation, conditioned on successful packet generation. The surfaces are shown as a function of PGA ingress α and packet generation probability ppacket . Two configurations are compared: requested links s = 2 in the state-of-the-art regime (left; λ = 2.22 Hz, w = 275 ms, z = 1.8) and s = 10 in the optimistic regime (right; λ = 10 Hz, w = 2 s, z = −2.2). Each grid point is estimated from 104 successful trials obtained by rejection sampling. White contours are spaced at intervals of 0.1. These surfaces are computed once per physical configuration (λ, w, s) and stored as lookup tables for use by the Markov chain session model.

computed for nearby s values may provide reasonable approximations without recomputation. Quantifying the interpolation error across configurations is left to future work. C. Single Session Speedup To evaluate session performance under reduced scheduled separation, demands ds from each regime are configured with N inst = 103 and ϵservice = 0.5, and the associated sessions d are simulated over 103 trials. We restrict the analysis to feasible demands with z-scores below 5, namely d2 , d3 , d4 in the SOTA regime and d10 , d20 , d28 , d34 in the optimistic regime. Of these, d10 yields no speedup at any feasible ingress, since its high ppacket = 0.951 causes the skip penalty to dominate the admissible range. The remaining six demands achieve positive speedup; for each, Table I reports the optimal ingress α∗ , the Markov minimum speedup bound S̃(α∗ , k ∗ ), and the sample mean simulated speedup S(α∗ ) along with its 95% confidence interval over 103 trials. Across these configurations, Tsession (α) is strictly decreasing in α over the feasible region, so α∗ coincides with the largest feasible ingress. We remark that the stationary approximation S̃(α∗ ) agrees with S̃(α∗ , k ∗ ) to within 0.05 percentage points across all configurations, confirming that the chain converges well within the session length at N inst = 103 . The Markov bound S̃(α∗ , k ∗ ) underestimates the simulated speedup S(α∗ ) by 0.1 to 0.7 percentage points, remaining tight enough to be predictive without repeated simulation. Whether any speedup exists depends on ppacket . Since pr = packet mv p ·p (α), a high ppacket makes the skip penalty large at every α, and when ppacket is near unity the penalty wipes out any gain across the admissible range, as observed for d10 . When speedup is possible, its size is controlled by z through R(α∗ ).

The geometric ceiling on R(α∗ ) is tminsep /(E + tminsep ), which shrinks with z as E grows relative to tminsep . However, the ceiling also becomes more reachable as z grows: longer PGAs spread generation times out, so pmv rises more slowly with α and the optimizer can push α∗ further before the skip penalty dominates. Near z = 0 (d20 ), the ceiling is large at 42.2%, but pmv rises quickly and holds R(α∗ ) to 2.8%, yielding a speedup of at least 0.8%. At the opposite extreme (d4 ), the ceiling shrinks to 7.6% but is reached by R(α∗ ), yielding a speedup of at least 6.1%. Across all configurations, simulations confirm that at least N inst packets are generated with probability 1 − ϵservice d at α∗ , though the analytical framework does not guarantee service delivery. Formalizing the relationship between α∗ and ϵservice is left to future work. d D. Multi-Session Speedup When multiple sessions share the network by round-robin scheduling, the PGAs of other sessions provide natural separation between consecutive same-session PGAs. For sesP sion i, this separation is gi = Ti′ + j̸=i (Ej + Tj′ ). When gi ≥ tminsep , no minimum separation violations occur and Ti′ i can be fully eliminated. Otherwise, αi∗ still sets the scheduled separation through Ti′ = tminsep − αi∗ Ei . However, only the i minsep ∗ residual αeff,i = (ti − gi )/Ei ≤ αi∗ contributes to ∗ violations through the violation zone ((1 − αeff,i )Ei , Ei ]. We restrict our analysis to n = 2 co-scheduled sessions, which eliminates skip penalties for nearly all demand configurations demonstrating single-session speedup, since the no-violation condition reduces to E ≥ tminsep . The analysis extends to larger n by replacing the single intervening PGA with the sum of the other n − 1 intervening PGA durations.

TABLE I S INGLE -S ESSION PERFORMANCE AT OPTIMAL INGRESS α∗

TABLE II M ULTI - SESSION PERFORMANCE AT OPTIMAL INGRESS α∗i

Session

z

α∗

S̃(α∗ , k∗ ) [%]

S(α∗ ) [%]

R(α∗ ) [%]

∆Tserv

Session

z

α∗i

S̃i (α∗i , k∗ ) [%]

S i [%]

Ri [%]

∆Tserv,i

d2 d3 d4

1.8 3.1 4.3

0.063 0.095 0.082

2.0 5.2 6.1

2.7 ± 0.1 5.8 ± 0.2 6.2 ± 0.2

4.4 7.9 7.6

2.3 min 11.1 min 41.9 min

d20 d2

0.0 1.8

0.442 0.431

25.2 26.7

25.4 ± 0.1 26.7 ± 0.1

25.6 30.1

29.2 min 15.8 min

d20 d28 d34

0.0 1.8 3.1

0.049 0.062 0.069

0.8 2.5 5.1

1.1 ± 0.1 3.1 ± 0.1 5.3 ± 0.2

2.8 4.9 6.5

3.2 min 18.1 min 2.8 hr

We consider co-scheduling demand d20 with demand d2 under round-robin. The round-robin cycle has duration ′ E20 + T20 + E2 + T2′ , with each Ti′ = tminsep − αi∗ Ei . The i cycle reduction Rcyc is the fractional reduction of the optimized cycle relative to the baseline cycle in which every session uses its full minimum separation. We fix α2∗ = 0.431 at its geometric maximum, which eliminates T2′ ; this is jointly ∗ optimal because αeff,2 = 0 leaves no skip penalty. Optimizing ∗ d20 then yields α20 = 0.442, giving an inter-PGA separation ∗ of 2.1 s (tminsep = 2.25 s) and a residual αeff,20 = 0.047. 20 Two distinct mechanisms drive the multi-session speedup. First, the round-robin cycle replaces the single-session release time E + T ′ , and the release time reduction R(α∗ ) is correspondingly replaced by the cycle reduction Rcyc defined above. Second, only the residual ingress contributes to minimum separation violations, so the skip probability pr,i is evaluated ∗ at αeff,i rather than at αi∗ . Combining both effects, the persession speedup admits the two analytical forms developed for the single-session case: the Markov lower bound on speedup S̃i (αi∗ , k ∗ ) analogous to Eq. (16), and its closed-form ∗ stationary approximation S̃i (αeff,i ) analogous to Eq. (19). The Markov bound is what Table II reports, and is computed by evaluating the chain of Section IV-C with transition probability ∗ pr,i (αeff,i ) to obtain ki∗ , then substituting into the multisession analog of Eq. (16). The stationary approximation, obtained from the Markov bound by substituting the stationary approximation of Eq. (18), gives a closed-form expression for S̃i measured against the baseline cycle:

∗ ∗ S̃i (αeff,i ) ≈ 1 − 1 + pr,i (αeff,i ) · 1 − Rcyc
∗ = Rcyc − pr,i (αeff,i ) · 1 − Rcyc , (20) |{z} {z } | cycle reduction

skip penalty

with analogous decomposition of scheduling quantities as Eq. (19). We remark that the approximation agrees with the Markov bound to within 0.01 percentage points. The results of co-scheduling are reported in Table II, where each row corresponds to one session. The first three columns, αi∗ , S̃i (αi∗ , k ∗ ), and S i , give the optimal ingress, the Markov minimum speedup bound, and the sample mean simulated speedup with its 95% confidence interval over 103 trials. The remaining columns, Ri and ∆Tserv,i , give the release time reduction session i would achieve if run alone at αi∗ , computed against the baseline Ei + tminsep as in Table I. i

This framing allows direct comparison with each session’s standalone performance in Section V-C. Co-scheduling raises the minimum session speedup S̃i from 0.8% to 25.2% for d20 , and from 2.0% to 26.7% for d2 . The co-scheduled PGA fills most of the round-robin cycle on its own, so each session’s residual ingress is small relative to the ∗ ∗ single-session case (αeff,20 = 0.047, αeff,2 = 0). The Markov bound S̃i underestimates the simulated speedup S i by at most 0.2 percentage points, narrower than in the single-session case due to the reduced skip penalty. VI. D ISCUSSION The evaluation yields two practical takeaways for entanglement generation scheduling. Firstly, the single-session results reveal that scheduled separation reduction yields no benefit when packets are produced with near-certainty within a scheduled block. In this regime, minimum separation violations become very likely, the resulting skips accumulate, and execution time grows faster than the separation reduction recovers. Schedulers can use this criterion to evaluate whether scheduled separation reduction is indicated for a given demand. Secondly, co-scheduled sessions provide natural separation between a session’s consumptions, absorbing time that would otherwise be reserved for the scheduled separation. This enables larger scheduled separation reductions. Within both state-of-the-art and projected hardware regimes, the co-scheduling effect can fully eliminate the scheduled separation: when the duration of co-scheduled blocks exceeds a session’s minimum separation, no minimum separation violations are possible. The scheduler then allocates the minimum scheduled time, and the session correspondingly achieves the maximum reduction in network service time and maximum execution speedup. Whether this condition holds in practice depends on how the scheduler allocates resources across demands. Network schedulers may partition scheduled resources into non-interacting regions, as Arqon does through its path partition; within such a region a session may have few or no co-scheduled partners, and the single-session optimization determines the largest feasible separation reduction. Conversely, when multiple sessions share the same resources, the duration of intervening co-scheduled blocks is more likely to exceed a session’s minimum separation, enabling its scheduled separation to be fully eliminated. Several directions remain regarding improvements to the analytical methods and demonstrations of the scheduled separation optimization in quantum network testbeds. Firstly, within the Arqon architecture, accepted sessions receive service agreements committing the scheduler to produce the

required packets before expiry with at least the specified service probability. Monte Carlo simulations confirm that the service commitment is met under the optimized separation, but the framework does not formally guarantee it. Establishing this guarantee analytically would convert the empirical observation into a formal property of the scheduler. Secondly, the Markov lower bound on the expected session execution speedup is conservative. Because packet generation times have variance, individual sessions on average complete earlier than the bound predicts. Accounting for this variance would tighten the bound while preserving its closed form. For network schedulers that enter service agreements with sessions, extending the framework to stricter operating regimes raises two further questions. The evaluation uses a relaxed service probability to isolate the effect of scheduled separation reduction. Higher service probabilities require the scheduler to allocate additional blocks to compensate for minimum separation violations. Jointly optimizing the separation and the block allocation in this regime remains open. This joint optimization could also incorporate the session’s expiry, guaranteeing the service time meets the deadline. Although developed within the Arqon architecture, our methods apply to any architecture in which end nodes share hardware between entanglement generation and LOCC. Alternative network schedulers or end node behaviors may yield further gains in scheduling or session execution efficiency. For example, adaptive scheduling could allocate additional blocks in regimes where minimum separation violations are likely, providing benefit in cases where uniform reduction yields none. End node policies could also begin consumption mid-block as soon as LOCC operations complete, rather than skipping the block entirely. ACKNOWLEDGMENT Claude Opus 4.6 co-developed simulations running Alg. 1.

It is then equivalent to show that  1 − fk (x) P (x)k = 1 − gk (x)

We proceed by induction. For k = 1, we have f1 (x) = x and g1 (x) = 0, so P (x)1 is of the desired form. Assume the results holds for n = k. Then    1 − fk (x) fk (x) 1 − x x P (x)k+1 = P (x)k P (x) = 1 − gk (x) gk (x) 1 0   1 − x(1 − fk (x)) x(1 − fk (x)) = 1 − x(1 − gk (x)) x(1 − gk (x)) By direct calculation one can show x(1 − fk (x)) = fk+1 (x), x(1 − gk (x)) = gk+1 (x) and the result follows. Claim 2. E[Nk ] = E[Nk−1 ] + 1 − fk−1 (pr) Proof. E[Nk ] =

X

=

X

Let Mk be a Markov chain defined on the state space N × {G, G} ∪ {(0, G)}, with transition probabilities of Eq. 12, and let us write Mk = (Nk , Sk ). We want to calculate E[Nk ]. Claim 1 (k-step transition matrix of Sk ). Let PS be the transition matrix for the Markov chain Sk . Then the k-step transition matrix for Sk , PSk is given by   1 − fk (pr) fk (pr) PSk = (21) 1 − gk (pr) gk (pr) where (−x)i

gk (x) = x

i=0

Proof. We can write PS = P (pr), where   1−x x P (x) = . 1 0

n P[Nk = n | Nk−1 = m] P[Nk−1 = m]

n,m

=

X  n P[Nk = n | Nk−1 = m, Sk−1 = G] P[Sk−1 = G] n,m

+ P[Nk = n | Nk−1 = m, Sk−1 = Ḡ] P[Sk−1 = Ḡ] × P[Nk−1 = m].



(24)

Now, we have that  1 P[Nk = n|Nk−1 = m, Sk−1 = G] = 0  1 P[Nk = n|Nk−1 = m, Sk−1 = G] = 0

n=m+1 else n=m else

and so from (24), we get  X E[Nk ] = m P[Sk−1 = G] + (m + 1) P[Sk−1 = Ḡ] m

× P[Nk−1 = m].

A. Expected Number of Executed PGAs

k−1 X

n P[Nk = n]

n

VII. A PPENDICES

fk (x) = x

 fk (x) . gk (x)

k−2 X

(−x)i .

(22)

i=0

(23)

From the earlier claim, we have that P[Sk−1 = G] = 1 − P[Sk−1 = G] = fk−1 (pr). Therefore, we have E[Nk ] =

X

=

X

mfk−1 (pr) + (1 + m)(1 − fk−1 (pr)) P[Nk−1 = m]

m

mP[Nk−1 = m] + (1 − fk−1 (pr))

m

X

P[Nk−1 = m]

m

= E[Nk−1 ] + 1 − fk−1 (pr)

Claim 3. E[Nk ] =

Pk−1

i i=0 (k − i)(−pr)

Proof. We proceed by induction. By inspection of the corresponding Markov chain in Figure ??, we can see that given we start in state (0, G), we mustPhave E[N1 ] = 1. From the 0 desired form, we have E[N1 ] = i=0 (k − i)(−pr)i = 1. So the base case holds. Let us now assume that the assumption holds for n = k. Then, using the recurrence relation from Claim 2, we have that E[Nk+1 ] = E[Nk ] + 1 − fk+1 and the result follows by direct calculation.

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