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Age-Minimal Transmission in Energy Harvesting Two-hop Networks
Ahmed Arafa, Sennur Ulukus
TL;DR
The paper addresses age minimization in energy-harvesting two-hop networks with fixed, non-zero transmission delays. It solves a single-hop version and transforms the two-hop problem into an equivalent single-hop problem for a combined source-relay node. The resulting policy forwards updates just in time, without waiting packets in the relay buffer.
Problem
The paper asks how to minimize destination age over a communication session when source and relay transmissions face energy causality and relay forwarding also faces data causality.
Source: §I · p. 1Method
It solves a single-hop age-minimization problem and transforms the two-hop problem into that formulation using combined energy-arrival times and adjusted delay and session parameters.
Source: §I · p. 1, §IV · p. 5Results
The optimal two-hop schedule sends each source update just in time for relay forwarding, leaving no packet waiting for service in the relay buffer.
Source: §IV · p. 6, §VI · p. 6Takeaways & Limitations
The single-hop solution balances inter-update times as permitted by energy arrivals and transmission delays.
Source: §VI · p. 6
Abstract
from arXiv · showhide
We consider an energy harvesting two-hop network where a source is communicating to a destination through a relay. During a given communication session time, the source collects measurement updates from a physical phenomenon and sends them to the relay, which then forwards them to the destination. The objective is to send these updates to the destination as timely as possible; namely, such that the total age of information is minimized by the end of the communication session, subject to energy causality constraints at the source and the relay, and data causality constraints at the relay. Both the source and the relay use fixed, yet possibly different, transmission rates. Hence, each update packet incurs fixed non-zero transmission delays. We first solve the single-hop version of this problem, and then show that the two-hop problem is solved by treating the source and relay nodes as one combined node, with some parameter transformations, and solving a single-hop problem between that combined node and the destination.
I. INTRODUCTION
The paper studies offline age minimization in energy-harvesting two-hop networks with fixed, non-zero transmission delays. It reduces the two-hop problem to a single-hop problem by coordinating source and relay transmissions.
I. INTRODUCTION
- Unlike throughput-focused energy-harvesting work – and queueing-based age studies –[21], this setting combines intermittent energy with age minimization,.The paper distinguishes its fixed, non-zero service time from the random or zero service times considered in,.Source: §I · p. 1
- The offline objective is to minimize total destination age within a communication session, subject to source and relay energy causality and relay data causality.Both nodes use fixed communication rates, so each incurs a fixed, non-zero transmission delay.Source: §I · p. 1
- The optimal policy sends updates just in time for relay forwarding, reducing the two-hop problem to a single-hop problem for a combined source-relay node.The paper also solves a single-hop problem with non-zero delays, which it relates to minimum inter-update time constraints.Source: §I · p. 1
II. SYSTEM MODEL AND PROBLEM FORMULATION
The model schedules source and relay updates over a finite session, minimizing the area under the destination age curve while respecting energy, data, and half-duplex constraints.
II. SYSTEM MODEL AND PROBLEM FORMULATION
- Each source and relay update consumes one energy packet, and fixed-rate transmissions take d and d-bar time units on their respective links.The model assumes equal-length update packets and infinite-sized batteries; the number of updates is the smaller of the two energy-arrival counts.Source: §II · p. 2
- Source and relay transmission times must respect their respective energy-arrival times, while relay forwarding cannot precede source delivery.These are the energy-causality and data-causality conditions in the formulation.Source: §II · p. 2
- Half-duplex operation forbids overlapping source and relay transmissions; the paper rules out schedules that leave multiple packets waiting in the relay buffer as age-suboptimal.In the two-packet example, forwarding only the newer update gives lower total age than forwarding both, and avoids spending energy on the stale packet.Source: §II · p. 2
- For three updates, the total-age area is the sum of trapezoids Q1–Q3 and the terminal triangle L.The model defines U(t) as the source timestamp of the latest update received at the destination.Source: §II · pp. 2–3
- Feasibility depends on energy-arrival times, transmission delays, session duration, and the number of energy arrivals.The stated arrival-time conditions ensure later energy arrivals still leave time to receive the remaining updates before the session ends.Source: §II · p. 3
III. SOLUTION BUILDING BLOCK: THE SINGLE-USER CHANNEL
The single-user problem is formulated using inter-update variables that account for positive transmission delay, then solved under energy and minimum-spacing constraints. Its optimum equalizes inter-update times as far as energy causality permits, enforcing minimum spacing where needed and adjusting the final interval.
III. SOLUTION BUILDING BLOCK: THE SINGLE-USER CHANNEL
- The change of variables accounts for positive transmission delay d, extending the zero-delay single-user results of to finite transmission rates.The section uses the single-user solution as a building block for the two-hop problem.Source: §III · p. 3
- The single-user formulation minimizes age through a convex problem in inter-update variables, with energy-causality and minimum service-time constraints.The transformed variables are dependent through a sum constraint involving T and Nd.Source: §III · p. 3
- The optimal inter-update sequence from index 2 onward is non-increasing; the first interval can be either smaller or larger than the second under distinct necessary conditions.The first interval exceeds the second only if x*_1 = s_1 + d, while it is smaller only if an interval from index 2 onward equals 2d.Source: §III · p. 4
- For Nd ≤ T < (N + 1)d, the optimum sends updates back-to-back at the minimum inter-update time that permits all updates to be received by time T.Nd is the least session duration for which the problem is feasible.Source: §III · p. 4
- The optimal inter-update times are equalized as far as energy causality permits; when this violates minimum-spacing constraints, the affected intervals are set to their minimum and the final interval is adjusted.The Inter-Update Balancing algorithm equalizes the variables subject to energy causality.Source: §III · p. 5
IV. TWO-HOP NETWORK: SOLUTION OF PROBLEM (9)
The two-hop optimum is obtained from the single-hop solution by transforming the energy-arrival times, transmission delay, and session time. This corresponds to sending each source update just as the relay is ready to forward, leaving no packet waiting in the relay buffer.
IV. TWO-HOP NETWORK: SOLUTION OF PROBLEM (9)
- The two-hop optimum follows the single-hop solution after replacing s_i with max{ s̄_i, s_i + d }, d with d + d̄, and T with T + d.These substitutions are specified by Theorem 3.Source: §IV · p. 5
- The source should send each update just in time for relay forwarding, so no update waits for service in the relay buffer.The theorem’s proof sets source transmission times to relay transmission times minus d, making data causality tight.Source: §IV · pp. 5–6
- The combined-node view generalizes to multiple relays, with each node sending when the next node is ready to forward.The combined node receives energy at times max{ s̄_i, s_i + d }.Source: §IV · p. 6
V. NUMERICAL RESULTS
The numerical examples apply the transformed single-hop problem to two sets of source and relay energy-arrival times. They illustrate optimal schedules in both session-time regimes described by the theorems.
V. NUMERICAL RESULTS
- [6.5, 6.5, 6, 6, 6, 4] is the optimal inter-update schedule for the first example after transformation to d = 3 and T = 20.The source and relay energy arrivals are [2, 6, 7, 11, 13] and [1, 4, 9, 10, 15], respectively.Source: §V · p. 6
- The examples use different solution regimes: transformed T = 20 ≥ 18 in the first and T = 17 < 18 in the second.The first example uses Theorem 2; the second uses Theorem 1 at T = 17 and Theorem 2 after T is increased to 18.Source: §V · p. 6
- [5, 6, 6, 6, 6, 3] is the optimal schedule for the second example with transformed T = 17 < 18.When T is increased to 18, the reported schedule is [5, 6, 6, 6, 6, 5].Source: §V · p. 6
VI. CONCLUSIONS
The paper proposes age-minimal policies for energy-harvesting two-hop networks with fixed transmission delays. Its policy sends updates just in time for relay forwarding and reduces the problem to a single-hop problem solved by balancing inter-update times within energy and delay constraints.
VI. CONCLUSIONS
- The proposed policies target age minimization in energy-harvesting two-hop networks with fixed transmission delays.The setting includes a source, relay, and destination.Source: §VI · p. 6
- The relay buffer should have no packet waiting for service; the source sends each update just as the relay is ready to forward.This just-in-time schedule permits the source and relay to be treated as a combined node.Source: §VI · p. 6
- The resulting single-hop problem is solved by balancing inter-update times as energy-arrival times and transmission delays allow.The two-hop reduction uses a combined source-relay node communicating with the destination.Source: §VI · p. 6