⚡ electrical engineering

Analysis and Design of Spare Strategy for Large-Scale Satellite Constellation Using Direct Insertion under (r,q) Policy

This paper proposes a Markov chain-based framework to analyze and optimize the cost-effective spare management strategy for large-scale satellite constellations using a direct insertion approach under an (r,q) policy.

Original authors: Seungyeop Han, Zachary Grieser, Shoji Yoshikawa, Takumi Noro, Takumi Suda, Koki Ho

Published 2026-05-21

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Seungyeop Han, Zachary Grieser, Shoji Yoshikawa, Takumi Noro, Takumi Suda, Koki Ho

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a massive fleet of thousands of satellites orbiting Earth, working together like a giant, high-tech net to provide internet or communication services. Just like any machine, these satellites sometimes break. When one breaks, the whole network gets a little weaker. To keep the net strong, the company needs a plan for having "spare" satellites ready to jump in and fix the hole immediately.

This paper is about designing the most efficient way to manage those spare satellites. The authors, a team from Georgia Tech and Mitsubishi Electric, created a mathematical "crystal ball" to predict exactly how many spares to keep and when to order new ones to save money while keeping the network safe.

Here is a breakdown of their approach using simple analogies:

1. The Two Ways to Restock: The "Taxi" vs. The "Bus"

The paper compares two ways to get spare satellites into orbit:

The Indirect Strategy (The Bus): You send a huge rocket full of spares into a "parking orbit" (a waiting lane). They wait there until the Earth's gravity naturally drifts them into the right position to join the main fleet. This is cheap per satellite, but it takes a long time because the "bus" moves slowly.
The Direct Strategy (The Taxi): This is what the paper focuses on. You use a smaller rocket to fly a spare satellite directly to the exact spot where a satellite broke. It's like calling a taxi instead of waiting for a bus. It's faster and gets the job done immediately, but it costs more per trip.

2. The "Gas Station" Rule: The (r, q) Policy

To manage the spares, the authors use a classic inventory rule called the (r, q) policy. Think of it like a gas station or a coffee shop:

The Reorder Point (r): Imagine you have a coffee shop. You decide that when your coffee bean supply drops to 10 bags, you must order more. You don't wait until you run out; you act early.
The Order Quantity (q): When you hit that 10-bag mark, you don't just order one bag. You order a big batch, say 5 bags, to bring your stock back up to a comfortable level.

In the satellite world, the "stock" is the number of spare satellites sitting in the orbit. When the number of spares drops to a certain level ( $r$ ), they launch a new batch ( $q$ ) to refill the tank.

3. The "Crystal Ball": Markov Chains

The hardest part of this problem is that satellites break randomly (like a car engine failing on a rainy day), and rockets take different amounts of time to launch (like traffic delays).

The authors used a mathematical tool called a Markov Chain. Imagine a board game where you roll dice to move.

The Dice: One die represents "Did a satellite break today?" Another die represents "Did the new rocket arrive today?"
The Board: The squares on the board represent how many spare satellites you currently have.
The Crystal Ball: Instead of playing the game once, the math calculates the average outcome of playing this game millions of times. It predicts exactly how often you will run out of spares and how much money you will spend over a long period.

This allows them to see the future without actually waiting 20 years to see what happens.

4. The Goal: The Perfect Balance

The authors built a computer model to find the "Goldilocks" solution.

If you order too few spares, you risk the network failing (bad for business).
If you order too many, you waste millions of dollars building and launching satellites you don't need.

They ran a simulation using a real-world "mega-constellation" (a huge network of satellites) to find the perfect numbers for $r$ and $q$ .

5. What They Found

Speed vs. Cost: The "Direct Strategy" (Taxi) is very fast. Even though it costs more per launch, the math showed that for this specific setup, it was actually the cheapest way to run the whole operation because it kept the network running smoothly without expensive downtime.
The "Full Truck" Trick: They discovered that even if you only need a few satellites, it's often cheaper to buy the entire rocket launch (the full truck) rather than paying for just a few seats (rideshare), because the discount on the full truck is so big.
Super Fast Math: Their new method is incredibly fast. A traditional computer simulation would take hours to predict the future of these satellites. Their "Markov Chain" method does it in less than a millisecond. This means engineers can test thousands of different plans in the time it takes to brew a cup of coffee.

Summary

In short, this paper gives satellite companies a new, super-fast calculator. It helps them figure out exactly how many backup satellites to keep in orbit and when to launch them, ensuring they don't waste money but also don't leave their customers without service. They proved that for big satellite networks, sending spares directly to the broken spot is a winning strategy.

Technical Summary: Analysis and Design of Spare Strategy for Large-Scale Satellite Constellation Using Direct Insertion under (𝑟, 𝑞) Policy

Problem Statement
Large-scale Low Earth Orbit (LEO) satellite constellations require robust spare-management policies to maintain performance amidst satellite failures. While on-orbit repair is often prohibitively expensive, launching replacement satellites is increasingly cost-effective due to declining launch costs and mass-production efficiencies. Two primary replenishment strategies exist: indirect (batching spares into parking orbits for later drift-based transfer) and direct (immediate insertion into constellation planes). This paper focuses on the direct spare strategy, addressing the need for a rigorous, efficient analytical framework to model, evaluate, and optimize the $(r, q)$ inventory policy (reorder point $r$ , order quantity $q$ ) specific to direct insertion. Unlike previous studies that may rely on computationally intensive simulations, this work seeks to provide a fast, accurate analytical method suitable for design optimization.

Methodology
The authors propose a discrete-time Markov chain-based framework to model the stochastic dynamics of satellite failures and launch vehicle (LV) lead times. The methodology proceeds through the following steps:

System Modeling:
- State Definition: The number of satellites (operational plus spares) in a single orbital plane is modeled as a state $X_k \in \{0, 1, \dots, N_{sat}\}$ .
- Failure Dynamics: Satellite failures follow a Poisson process with rate $\lambda_{sat}$ . A transition matrix $P_f$ is constructed to model state decreases, assuming immediate replacement of failed operational satellites by spares and that spares themselves do not fail.
- Lead Time Dynamics: The ground-resupply lead time is modeled as a shifted exponential distribution ( $T \sim \text{Exp}(\mu_{lv}) + \tau_{lv}$ ), where $\tau_{lv}$ is a constant processing delay.
- Policy Logic: The $(r, q)$ policy is enforced: if the stock level drops to or below $r$ , an order for $q$ spares is placed.
Analytical Derivation:
The core contribution is the derivation of the stationary state distribution over a full replenishment cycle ( $\pi_{rc}$ ) by decomposing the cycle into two distinct periods:
- Inter-Order (IO) Period: The time from a replenishment arrival until the stock level hits the reorder point $r$ . The authors derive the expected state distribution $\pi_{io}$ and the duration $k_{io}$ using projection matrices ( $C_r^+$ ) to isolate states above $r$ .
- Lead-Time (LT) Period: The time from placing an order until the new spares arrive. The expected distribution $\pi_{lt}$ and duration $k_{lt}$ are derived using the lead-time probability density and the failure matrix $P_f$ .
- Cycle Aggregation: The overall stationary distribution $\pi_{rc}$ is computed as a weighted linear combination of $\pi_{io}$ and $\pi_{lt}$ , normalized by their respective expected durations.
Optimization Framework:
Using the derived $\pi_{rc}$ , the authors formulate an optimization problem to minimize the total expected operating cost ( $C_{total}$ ), defined as the sum of manufacturing, holding, and launch costs. The problem is subject to:
- Resilience Constraint: The expected shortage ( $S$ ) must not exceed a user-defined threshold $\epsilon$ .
- Payload Constraint: The total mass of the order ( $q \times m_{sat}$ ) must not exceed the launch vehicle's payload capacity.
- Decision Variables: The integer values for order quantity $q$ and reorder point $r$ .

Key Contributions

Direct Strategy Analysis: The paper provides a detailed analytical model specifically for the direct spare replenishment strategy, extending previous work that focused on indirect strategies.
Non-Iterative Solution: Unlike the indirect strategy analysis which may require iterative procedures, the direct strategy analysis presented here is non-iterative. It solves for the stationary distribution via matrix operations, significantly reducing computational complexity.
Cost and Resilience Metrics: The framework explicitly quantifies the trade-off between operational costs (manufacturing, holding, launch) and system resilience (expected shortage), enabling data-driven design decisions.
Validation: The analytical model is validated against Monte Carlo simulations across 100 test cases with varying parameters (failure rates, lead times, order sizes).

Results

Accuracy and Efficiency: Numerical validation demonstrates that the analytical method achieves a maximum relative error of less than 1% compared to 20-year Monte Carlo simulations (averaged over 1,000 runs). The analytical method computes results in under one millisecond, whereas simulations require several hours, confirming its suitability as an inner loop for optimization.
Optimization Findings:
- In a baseline scenario (failure rate $\lambda_{sat} = 0.05$ , shortage threshold $\epsilon = 0.25$ ), the optimal policy was found to be $q=2$ and $r=39$ .
- Rideshare vs. Full Contract: The analysis reveals that reserving the full launch vehicle is generally optimal, even when rideshare opportunities are available, due to the specific cost structures assumed (a 20% premium for per-unit pricing). Rideshare only becomes cost-effective in a very low-failure regime ( $\lambda_{sat} \in [0.001, 0.008]$ ).
- Sensitivity: As the satellite failure rate increases, the optimal reorder point $r$ increases (e.g., from 39 to 42) to maintain the required resilience level, while the total cost rises accordingly.

Significance and Claims
The paper claims that the proposed Markov chain-based framework offers a fast and accurate tool for the analysis and design of spare-management policies for large-scale constellations. Its primary significance lies in enabling design optimization by providing a computationally efficient alternative to simulation-heavy approaches. The authors state that the optimized results for the direct strategy serve as a counterpart for comparison with the indirect strategy in a companion study. Furthermore, the framework is presented as extensible to other constellation configurations (e.g., heterogeneous agents) and alternative replenishment policies, though the current work focuses strictly on the direct strategy under an $(r, q)$ policy.