# Cooperative Resource Management in Quantum Key Distribution (QKD) Networks
for Semantic Communication

Rakpong Kaewpuang, Minrui Xu, Wei Yang Bryan Lim, Dusit Niyato, , Han Yu,
Jiawen Kang, and Xuemin (Sherman) Shen Rakpong Kaewpuang, Minrui Xu, Dusit
Niyato, and Han Yu are with the School of Computer Science and Engineering,
Nanyang Technological University, Singapore (e-mail:
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>han.yu@ntu.edu.sg).Wei Yang Bryan Lim is with Alibaba Group and Alibaba-NTU
Joint Research Institute, Nanyang Technological University, Singapore (e-mail:
limw0201@e.ntu.edu.sg).Jiawen Kang is with the School of Automation, Guangdong
University of Technology, China (e-mail: kavinkang@gdut.edu.cn).Xuemin Sherman
Shen is with the Department of Electrical and Computer Engineering, University
of Waterloo, Waterloo, ON, Canada, N2L 3G1 (e-mail: sshen@uwaterloo.ca).

###### Abstract

Increasing privacy and security concerns in intelligence-native 6G networks
require quantum key distribution-secured semantic information communication
(QKD-SIC). In QKD-SIC systems, edge devices connected via quantum channels can
efficiently encrypt semantic information from the semantic source, and
securely transmit the encrypted semantic information to the semantic
destination. In this paper, we consider an efficient resource (i.e., QKD and
KM wavelengths) sharing problem to support QKD-SIC systems under the
uncertainty of semantic information generated by edge devices. In such a
system, QKD service providers offer QKD services with different subscription
options to the edge devices. The QKD services are envisioned to follow cloud
computing that has the subscription in the reservation and on-demand options,
i.e., for long and short (immediate) terms, respectively. As such, to reduce
the cost for the edge device users, we propose a QKD resource management
framework for the edge devices communicating semantic information. The
framework is based on a two-stage stochastic optimization model to achieve
optimal QKD deployment. Moreover, to reduce the deployment cost of QKD service
providers, QKD resources in the proposed framework can be utilized based on
efficient QKD-SIC resource management, including semantic information
transmission among edge devices, secret-key provisioning, and cooperation
formation among QKD service providers. In detail, the formulated two-stage
stochastic optimization model can achieve the optimal QKD-SIC resource
deployment while meeting the secret-key requirements for semantic information
transmission of edge devices. Moreover, to share the cost of the QKD resource
pool among cooperative QKD service providers forming a coalition in a fair and
interpretable manner, the proposed framework leverages the concept of Shapley
value from cooperative game theory as a solution. Experimental results
demonstrate that the proposed framework can reduce the deployment cost by
about 40% compared with existing non-cooperative baselines.

###### Index Terms:

Quantum key distribution, semantic information communication, cooperative
resource management, game theory, stochastic programming.

##  I Introduction

As quantum computers become a distinct possibility, they pose new potent
threats to the traditional cryptographic security schemes. This has led to a
revolution in the use of quantum channels for secret key distribution, i.e.,
quantum key distribution (QKD), for the transmission of confidential
information [1, 2, 3, 4]. With the parallel processing of quantum computing,
quantum computers can easily break the process of key exchanging in modern
symmetric cryptography [5]. Fortunately, quantum channels based on the laws of
quantum mechanics, such as Heisenberg’s uncertainty principle and the no-
cloning theorem [6], can perform data transmission, including symmetric key
distribution, with provable security. In addition, QKD can enable information-
theoretic security (ITS) for data transmission in classical data channels via
the one-time pad (OTP) mechanism [7]. In intelligence-native 6G communications
[8], security and privacy are critical requirements in distributed artificial
intelligence (AI) training. In particular, the transmission of essential
feature information in semantic communication makes 6G communications more
demanding for security and privacy in semantic information communication (SIC)
and networking.

Semantic information communication [9, 10, 11] refers to communication
paradigms based on the concept of semantic-meaning transmission. SIC is a
practical application of AI models for exchanging meaningful information
obtained from transmission data with a minimum of communication resources. The
main goal of semantic communication is to extract the meanings or features of
the transmitted data (i.e., semantic information) from a semantic source and
transmit the semantic information to a semantic destination, where it can be
interpreted successfully. In the semantic communication system, we can
consider that the semantic source and destination are semantic providers who
execute intelligent algorithms (e.g., knowledge graphs, deep neural networks)
autonomously. They can be edge devices or sensors with cognitive capabilities.
In such a knowledge-based system, the semantic source and destination can
establish and utilize their background knowledge bases (KBs) through online or
offline training and inference. The KBs are typical models that the semantic
source and destination can learn and validate in advance. For example, images,
speeches, videos and texts are representative KBs. The semantic source uses an
individual KB to extract the semantic information of the raw data, which is
then transmitted to the semantic destination. When the semantic destination
receives the semantic information, it uses its individual KB to interpret
them.

Nevertheless, SIC provides a viable and intelligent way to exchange semantic
information between the semantic source and the semantic destination.
Regarding security, SIC faces confidentiality issues due to the eavesdropping
risk in exchanging semantic information between the semantic source and the
semantic target [12, 13, 14]. To ensure that the transmission of semantic
information is secure, QKD is a promising mechanism to protect public keys
from eavesdropping attacks.

To address the aforementioned issues, we propose a QKD-secured SIC (QKD-SIC)
system to protect the semantic information and public keys from eavesdropping
attacks. We consider the resource management problem that the secret keys are
allocated to support the semantic information of edge devices in SIC. In
addition, we consider that the amount of semantic information generated by
edge devices is unpredictable. To optimize QKD resources when the amount of
semantic information is unpredictable, QKD service providers can cooperate and
establish a global resource pool. In particular, multiple QKD service
providers can share their resources (i.e., QKD and KM wavelengths) in the
pool. Therefore, the resources in the pool can be efficiently utilized by the
QKD service providers. Nevertheless, three important questions arise in the
context of this resource management. Firstly, what is an optimal resource
allocation in the pool of SIC applications to minimize costs and satisfy the
needs of edge devices? Secondly, how can the costs incurred by the resource
pool be shared among cooperative QKD service providers? Thirdly, should QKD
service providers cooperate or not in creating the resource pool? To answer
these questions for QKD service providers, we introduce a decision-making
scheme with the components of resource allocation to SIC applications, cost
management, and collaboration among QKD service providers in a QKD-SIC
environment. The goal of the scheme is to make the best decisions for QKD
service providers as they are rational and interested in minimizing their own
costs. The main contributions of this paper can be summarized as follows:

  * •

We introduce a hierarchical architecture for the QKD-SIC system, where
resource allocation and routing decisions are coupled. To provide sufficient
secret keys for semantic information transmission between virtual service
providers and edge devices, the QKD network manager and controller allocate
QKD resources (i.e., QKD and KM wavelengths) from QKD nodes and determine
routing paths for QKD links.

  * •

We formulate and solve the stochastic programming (SP) model to obtain the
optimal resource allocation decisions from a resource pool created by QKD
service providers. The SP model requires the probability distributions of the
random parameters, i.e., the secret key rate (user demand) requirements. This
model can be applied for the cooperative QKD service providers to make
decisions in two stages. In the first stage, the QKD service providers make a
resource reservation decision (i.e., QKD and KM wavelengths) based on the
statistical information about users’ demand. In the second stage, when QKD
service providers know the exact level of users’ demand, they make a decision
to compensate for users’ demand that cannot be satisfied when resources are
insufficient.

  * •

We present a model to allocate the provisioning cost obtained from the
resource pool among QKD service providers in a fair manner. In addition, we
leverage Shapley values to determine fair and interpretable allocation of
provisioning costs.

  * •

We introduce a cooperative game model for QKD service providers to decide
whether they should cooperate and establish the resource pool or not. A stable
cooperation strategy is the solution of the game model, which means that the
rational QKD service providers do not want to change their decisions.

The remainder of this paper is organized as follows: Section II presents
related work. Section III describes the system model, the maritime case study,
and the cost models. Section IV presents the stochastic programming model for
resource allocation in QKD-SIC for a maritime application. Section V presents
cost management model among QKD service providers. Section VI presents the
cooperative game models for optimal cooperation formation for QKD service
providers. Section VII presents the performance evaluation results. Section
VIII concludes the paper.

##  II Related Work

###  II-A Semantic Information Communication

Semantic information communication significantly reduces the need for
transmission resources such as channels by using artificial intelligence to
extract semantic features for the information between source and destination
[15, 16, 17, 18]. The authors in [19] proposed a SIC in the metaverse [20]
between edge devices and the virtual service providers (VSPs) to reduce the
size of transmitted data. In SIC, an edge device (e.g., a smartphone) can
autonomously produce pictures (pre-semantic data) relevant to the interest of
VSPs. The VSP can subscribe to edge devices that the VSP is interested in
transmitting the semantic data to create the metaverse for supporting user
requests, where the user requests are considered as uncertainty. In [19], the
stochastic semantic transmission scheme was proposed to minimize the
transmission cost of VSPs. This scheme was formulated as a two-stage
stochastic integer programming model under the uncertainty of user requests to
achieve an optimal solution. The resource allocation for text SIC was proposed
in [21] to maximize the overall semantic spectral efficiency in terms of
channel utilization and the number of transmitted semantic symbols of all
users. The authors in [21] compared the optimal solution of the proposed
scheme with that of three benchmarks. The simulation results demonstrated that
the performance of the proposed scheme outperformed that of the three
benchmarks. The deep learning-enabled SIC system for speech signals (DeepSC-S)
was proposed in [22] to enhance the recovery accuracy of speech signals.
Particularly, the joint semantic-channel coding was proposed to solve source
distortions and channel effects. The squeeze-and-excitation network was
employed in the proposed DeepSC-S to learn and extract speech semantic
information. In experimental results, the performance of the proposed DeepSC-S
was compared with the traditional communication system and the system with an
additional feature coding for speech transmission. In addition, the proposed
DeepSC-S was applied to telephone systems and multimedia transmission systems.
However, current SIC systems only leverage AI models to transmit necessary
semantic information between the source encoder and the destination decoder,
while the security issues in SIC systems are largely overlooked.

###  II-B The Quantum Internet

With the ability to provide information-theoretic security for communications,
QKD and QKD-secured systems are expected to protect various 6G communications
such as optical, satellite, maritime, and their cross-layer communications
[23, 24, 8, 25]. Unlike traditional QKD protocols, such as Bennett-
Brassard-1984 (BB84) [26], Bennett-Brassard-Mermin-1992 (BBM92) [1], and
Grosshans-Grangier-2002 (GG02) [1], modern QKD protocols based on MDI
technology can provide longer distribution distance and stronger security
without assuming trusted relays [27]. To minimize the cost of deploying QKD
resources in MDI-QKD, the authors in [28] proposed a static linear programming
model and the CO-QBN algorithm to efficiently manage QKD networks. The authors
in [29] proposed the resource allocation scheme for quantum-secured space-air-
ground integrated networks (SAGIN) in which QKD services protect secure
communications between space, aerial, and ground nodes by exchanging secret
keys in quantum channels. In [29], the authors formulated and solved the
stochastic programming model to achieve the optimal solution for resource
allocation (i.e., QKD and KM wavelengths) and routing under the uncertainties
of secret-key rate requirements and weather conditions.

In [13], the authors proposed the QKD-based secure federated learning (FL)
scheme to support the FL model encryption against eavesdropping attacks on FL
networks. Particularly, the resource allocation scheme for QKD resources
(i.e., wavelengths) to support FL networks was proposed to minimize the
overall deployment cost under the uncertainty of the number of FL workers
associated with the secret-key rate requirements. The resource allocation
scheme was formulated based on the two-stage SP to achieve the optimal
solution under the uncertainty of the number of FL workers. The concept of QKD
over SAGIN was proposed to achieve secure communications with global coverage
and reconfigurable [30], [25]. The authors [25] proposed the QKD service
provisioning framework to establish secure communications in quantum-secured
SAGIN. The two-stage SP was formulated to minimize the provisioning cost under
the uncertainty of secret-key rates. In practice, the framework was applied to
support metaverse applications to achieve the optimal solution. The authors
[14] proposed the hierarchical architecture for QKD-secured federated learning
systems. In [14], the QKD resource management was formulated, which is based
on the two-stage SP, to minimize the deployment cost under the uncertainty of
the secret-key rate requirements from FL applications. [31], [32] introduced
systems to provide QKD services via free space to secure applications to
satisfy security requirements. The authors in [32] demonstrated the quantum
satellite (i.e., Micius) that successfully distributes the secret key (i.e.,
with the rate of 0.12 bits per second) between two ground nodes via free
space. The drone-based entanglement distributions were proposed in [33], [34].
In [34], the mobile quantum communications via entanglement distribution
demonstrated that the proposed system can tolerate various weather conditions
(i.e., daytime, clear nights, and rainy nights). Compared to fiber- and
satellite-based quantum communications, drone-based entanglement distribution
offers the advantages of unparalleled mobility, flexibility, and
reconfigurability.

###  II-C Game-based Resource Allocation

The authors in [35] proposed a game-based resource allocation framework for
SIC system, where the semantic content of a message to be transmitted over a
noisy channel was taken into consideration to optimize the performance of a
communication system. In addition, the impact of social influence on how the
messages were translated by considering the influence of the agent in the
communication network. The agent had an influence on the receiver, i.e., how
to decode the received messages by providing side information. The influence
of the agent can be adversarial or helpful to the communicating parties.
Therefore, the authors considered the SIC problem with social influence as a
Bayesian game with incomplete information and investigated the conditions
under which a Bayesian Nash equilibrium exists.

Nevertheless, the existing works in the literature do not consider the
uncertainty of secret-key requirements from SIC. In addition, the works in the
literature do not consider the problem of optimizing the resource allocation
to semantic applications, cost management, and cooperation formation for QKD
service providers under uncertainty.

##  III System Model and Case Study

Figure 1: The QKD-SIC architecture.

We propose a three-layer QKD-SIC architecture illustrated in Fig. 1. The
architecture consists of the QKD layer, the QKD management and control layer,
and the semantic layer. In the QKD layer, there are three types of nodes and
two types of links. The three types of nodes are QKD nodes, trusted relays,
and untrusted relays. The two types of links are key management (KM) links and
QKD links. We assume that QKD nodes are co-located with SIC nodes, which can
be edge devices or quantum secure link users. The links (i.e., QKD, KM, and
optical links) are multiplexed within a single fiber by MUX/DEMUX components
[28]. Therefore, the network topology for the QKD layer is the same as that
for the semantic layer. Let $\mathcal{N}$ and $\mathcal{E}$ denote the set of
SIC/QKD nodes and the set of fiber links between SIC/QKD nodes, respectively.

A QKD node supplies secret keys to its co-located SIC node. The trusted relay
and the untrusted relay are located between the QKD nodes. The QKD node
consists of a global key server (GKS), a local key manager (LKM), a security
infrastructure (SI), and components of trusted/untrusted relays. A trusted
relay consists of two or multiple transmitters of MDI-QKD (MDI-Txs), LKM, and
SI, while an untrusted relay consists of one or multiple receivers of MDI-QKD
(MDI-Rxs). An MDI-QRx must be located between two MDI-QTxs to generate local
secret keys. An LKM is used to obtain and store the local secret keys from the
connected MDI-QTxs, and LKM performs secret key propagation via the one-time-
pad (OTP) method [28] to generate global secret keys between QKD nodes. The
SIs are applied to ensure that trusted relays work in a secure manner. In the
QKD layer, the QKD links are used to connect between MDI-QTxs and MDI-QRxs,
while the KM links are used to connect between LKMs. The QKD link contains
quantum and classical channels, while the KM link contains a classical channel
implemented by wavelengths.

In the semantic layer, an SIC node can be an edge device (e.g., a smartphone
and surveillance camera) or a virtual service operator (i.e., semantic
receiver). Each edge device in different areas has different views for
locations, as shown in Fig. 1. The edge device can capture the pictures and
extract the semantic information (i.e., snapshots) from the pictures. Then,
the edge device transmits the semantic information via optical fibers to the
respective virtual service operator (VSO), which uses this data to create the
virtual environment for a particular application, for example, transportation
and education. Before the semantic information transmission between the edge
device and the VSO happens, the VSO has to subscribe to the respective edge
devices.

When the VSO subscribes to the respective edge devices, the respective edge
devices will request secret keys from the QKD manager based on their security
requirements for encrypting their semantic information. When the QKD manager
receives the request, the QKD manager responds by checking the available
secret keys. Suppose the secret keys are available to support the request, and
thus the QKD manager will order the QKD controller to select the suitable QKD
nodes to provide the secret keys in a suitable format for semantic information
encryption. If the secret keys are unavailable to support the request, the
edge devices and the VSO will wait until the secret keys are available. Then,
the edge devices encrypt the semantic information via the secret keys and
transmit the encrypted semantic information to the semantic receiver. The
amount of semantic information (i.e., snapshots) of each edge device directly
depends on the available secret-key resources in the QKD layer as well as the
security requests in the semantic layer. Therefore, it has a trade-off between
the QKD resources and the requests from edge devices.

###  III-A Cooperation Among QKD Service Providers and Pooling of QKD
Resources

Figure 2: Components of the decision-making framework for QKD service
providers.

To minimize the deployment costs, the QKD service providers can cooperate and
establish a QKD resource pool that comprises QKD and KM wavelengths to support
the security demand. The set of cooperative QKD service providers that agree
to cooperate and establish the QKD resource pool is represented by
$\mathbb{C}$. It could have many coalitions. Therefore, let
$\Upsilon=\\{{\mathbb{C}}_{1},{\mathbb{C}}_{2},{\mathbb{C}}_{3},\ldots,{\mathbb{C}}_{|\Upsilon|}\\}$
denote a set of all coalitions (i.e., cooperation structure or coalitional
structure) where ${\mathbb{C}}_{i}\in\Upsilon$. Each coalition $i$
($\mathbb{C}_{i}$) has its own QKD resource pool that is created by the
cooperative QKD service providers in the coalition $i$. Given a coalition
${\mathbb{C}}$, $W^{\mathrm{qkd}}_{i,j}(\mathbb{C})$ denotes available QKD
wavelengths on the QKD link between SIC/QKD nodes $i$ and $j$ in the
corresponding resource pool. $W^{\mathrm{kml}}_{i,j}(\mathbb{C})$ denotes
available KM wavelengths on the KM link between SIC/QKD nodes $i$ and $j$ in
the corresponding resource pool. The cost generated from the QKD resource pool
will be shared by all cooperative QKD service providers in the coalition.

With the collaboration among the QKD service providers, we introduce a
framework that comprises three main components.

  * •

_QKD resource allocation for QKD service providers:_ With cooperative QKD
service providers, the objective of QKD resource allocation is to procure the
optimal numbers of QKD and KM wavelengths while the deployment cost of the
cooperative QKD service providers is minimized. Hence, we formulate an
optimization problem based on SP. The probability distributions of random
parameters in the SP model are taken into consideration.

  * •

_Cost management:_ Given a coalition of QKD service providers and the optimal
QKD resource allocation, we apply the concept of Shapley value to obtain the
share of deployment cost for each cooperative QKD service providers.

  * •

_Coalition formation:_ We analyze the stable cooperation strategy among QKD
service providers by using the cooperation formation model. The dynamics of
coalition formation are analyzed by Markov chain models.

Figure 2 illustrates the interactions among the three components. The
coalition formation is first performed to achieve the coalition structure and
then the coalition structure is applied by the QKD resource allocation. The
optimal QKD resource allocation of the SP model is used by the cost management
in which the Shapley value is applied to allocate the incurred deployment cost
to QKD service providers. The QKD service providers alter their coalition
formation strategies accordingly when they obtain their costs.

###  III-B The Network Model

In the QKD-SIC model, let $\mathcal{F}$ and
$f(s_{f},d_{f},P_{f}(\cdot))\in\mathcal{F}$ denote the set of QKD-SIC chain
requests and a QKD-SIC chain request of SIC nodes, respectively. $s_{f}$ and
$d_{f}$ are the source node and destination node of QKD-SIC chain request $f$,
respectively. Let $K_{D}$ denote the maximum achievable secret-key rate at a
distance $D$. The number of parallel QKD-SIC links $P_{f}(\cdot)$ to satisfy
the security demand (secret-key rate) of SIC transmission between $s_{f}$ and
$d_{f}$ is expressed as follows:

$P_{f}(\tilde{\omega})=\Big{\lceil}\frac{\tilde{\kappa}_{f}}{K_{D}}\Big{\rceil}$
(1)

where $f$ is the QKD-SIC chain request and $\tilde{\kappa}_{f}$ is a random
variable of secret-key rate requirement of request $f$. $D$ is a distance
between two connected MDI-QTxs, which can be expressed in as follows:

$D\approx 2\phi$ (2)

where $\phi$ is the distance between the MDI-QRx connecting to the MDI-Tx.

###  III-C The Cost Model

We adopt the costs of the QKD network components and the links from [28] to
support the deployment of QKD with hybrid trusted/untrusted relays on the
existing optical backbone network. The QKD network components include MDI-
QTxs, MDI-QRxs, LKMs, SIs, and multiplexing/demultiplexing (MUX/DEMUX)
components. The links include QKD links and KM links. The components and links
can be described as follows:

####  III-C1 MDI-QTxs and MDI-QRxs

The MDI-QKD process requires two MDI-QTxs and one MDI-QRx and therefore the
number of MDI-QTxs ($A^{f}_{\mathrm{tx}}(\cdot)$) and MDI-QRxs
($A^{f}_{\mathrm{rx}}(\cdot)$), which satisfy the QKD-SIC chain request $f$,
are expressed in (3) and (4), respectively.

$\displaystyle
A^{f}_{\mathrm{tx}}(\tilde{\omega})=\sum_{i\in\mathcal{N}_{f}}\sum_{j\in\mathcal{N}_{f}}2P_{f}(\tilde{\omega})\Big{\lceil}\frac{e_{(i,j)}}{D}\Big{\rceil}$
(3) $\displaystyle
A^{f}_{\mathrm{rx}}(\tilde{\omega})=\sum_{i\in\mathcal{N}_{f}}\sum_{j\in\mathcal{N}_{f}}P_{f}(\tilde{\omega})\Big{\lceil}\frac{e_{(i,j)}}{D}\Big{\rceil}$
(4)

$e_{(i,j)}$ is the distance of physical fiber link between nodes $i$ and $j$.
$\mathcal{N}_{f}$ is set of nodes having the fiber links on the route of
request $f$.

####  III-C2 Local key managers (LKMs)

For the QKD-SIC chain request $f$, the required number of LKMs
($A^{f}_{\mathrm{km}}$) is expressed as follows:

$\displaystyle
A^{f}_{\mathrm{km}}=\sum_{i\in\mathcal{N}_{f}}\sum_{j\in\mathcal{N}_{f}}\Big{\lceil}\frac{e_{(i,j)}}{D}+1\Big{\rceil}.$
(5)

####  III-C3 Security Infrastructures (SI)

The required number of SIs ($A^{f}_{\mathrm{si}}$) to satisfy the QKD-SIC
chain request $f$ is expressed as follows:

$\displaystyle
A^{f}_{\mathrm{si}}=\sum_{i\in\mathcal{N}_{f}}\sum_{j\in\mathcal{N}_{f}}\Big{\lceil}\frac{e_{(i,j)}}{D}-1\Big{\rceil}.$
(6)

####  III-C4 MUX/DEMUX components

The required number of MUX/DEMUX component pairs ($A^{f}_{\mathrm{md}}$) for
the QKD-SIC chain request $f$ is expressed as follows:

$\displaystyle
A^{f}_{\mathrm{md}}=\sum_{i\in\mathcal{N}_{f}}\sum_{j\in\mathcal{N}_{f}}\Big{\lceil}\frac{e_{(i,j)}}{D}\Big{\rceil}+\sum_{i\in\mathcal{N}_{f}}\sum_{j\in\mathcal{N}_{f}}\Big{\lceil}\frac{e_{(i,j)}}{D}-1\Big{\rceil}.$
(7)

####  III-C5 QKD and KM links

Three wavelengths and one wavelength are occupied by the QKD link and the KM
link, respectively [36]. The link cost for the QKD-SIC chain request $f$ is
expressed as follows:

$\displaystyle
A^{f}_{\mathrm{ch}}(\tilde{\omega})=\sum_{i\in\mathcal{N}_{f}}\sum_{j\in\mathcal{N}_{f}}(3P_{f}(\tilde{\omega})e_{(i,j)}+e_{(i,j)})$
(8)

where the physical lengths of QKD links and KM links are denoted by
$3P_{f}(\tilde{\omega})e_{(i,j)}$ and $e_{(i,j)}$, respectively.

###  III-D The Cooperation Cost

We consider the cooperation cost of QKD service providers when the providers
decide to share the resources (i.e., QKD and KM wavelengths) in the resource
pool. The costs of QKD and KM wavelengths to be shared in the pool can be
expressed in (9) and (10), respectively.

$\displaystyle C^{\mathrm{qkd}}(s)=W^{\mathrm{qkd}}_{s}C^{\mathrm{qkd}}_{s}$
(9) $\displaystyle
C^{\mathrm{kmw}}(s)=W^{\mathrm{kmw}}_{s}C^{\mathrm{kmw}}_{s}$ (10)

$W^{\mathrm{qkd}}_{s}$ and $W^{\mathrm{kmw}}_{s}$ are the number of QKD
wavelengths and the number of KM shared in the pool by QKD service provider
$s$, respectively. $C^{\mathrm{qkd}}_{s}$ and $C^{\mathrm{kmw}}_{s}$ are the
costs of shared QKD and KM wavelengths to be charged by QKD service provider
$s$, respectively. In addition, let $C^{\mathrm{coc}}_{s}$ denote the fixed
cooperation cost that will be charged when QKD service provider $s$ is in
cooperation. This cost can incur due to the communication and computation
overhead and other security management tasks, which is typically incurred when
independent providers interact with each other.

###  III-E The Case Study: Maritime Transportation Surveillance over QKD

In the case study of maritime transportation surveillance in QKD-SIC, we give
an overview of maritime transportation and highlight the semantic requirements
(i.e., semantic information) of maritime transportation applications.

Figure 3: An example of a maritime transportation surveillance semantic
information transfer.

Maritime transportation, where goods or people are transported via sea routes,
is introduced as a case study. In this example, we consider maritime
transportation from the perspective of a virtual service operator (VSO), i.e.,
semantic receiver, who participates in the virtual environment [9], [20]. It
is challenging for new employees to practically sail ships since they are
unfamiliar with locations and procedures. This may cause accidents with
employees and boatmen. Therefore, a virtual environment for maritime
application is a promising solution. The VSO can create a virtual environment
of maritime applications for practicing employees.

In Fig. 3, there are two VSOs and two edge devices in the system. We consider
that the VSOs are maritime shipping companies to provide the services of
autonomous maritime ships. The VSO can set up simulated environments in the
virtual environment in which the VSO can deploy their employee to use the
virtual environment and experiment with the autonomous maritime ships. To
create the maritime virtual environment, the VSO has to subscribe to edge
devices to request the semantic information. Before subscribing to edge
devices, the VSO uses bidirectional encoder representations from transformers
(BERT) [37] with the average similarity score of the category of each semantic
information that generated by edge devices. For example, in Fig. 3, edge
devices send the sample data and the average category similarity scores to
VSOs. When VSOs receive the sample data and the average category similarity
scores, VSOs extract the semantic information (i.e., snapshots of boats) in
categories and then compare their interest (i.e., boats) with the similarity
scores. As shown in Fig. 3, the average similarity score of semantic
information of edge device 1 is the highest score (i.e., 0.95). Therefore VSOs
subscribe edge device 1 to transmit the semantic information.

The VSO can create and provide a real-time virtual environment for maritime
applications to support its users. The virtual environment of maritime
applications can interact with edge devices by transmitting semantic
information back and forth since the VSO can build the details of the virtual
environment. The edge devices can be considered to be surveillance cameras or
smartphones that are located in different areas. The edge devices and VSO are
connected with the optical fibers. The edge devices generate the semantic
information by using the semantic segmentation modules. In the semantic
segmentation module, the pre-trained machine learning models are applied,
e.g., you only look once (YOLO) [38]. The snapshots from edge devices are
considered to be the semantic information of interest to the VSO. When the VSO
receives the semantic information, the VSO uses this semantic information to
build the virtual environment of maritime applications for users.

Therefore, to prevent eavesdropping attacks, the QKD is promising for
providing proven secure key distribution schemes for the semantic information
transmitting between edge devices and the VSO by facilitating public key and
data encryption. In this paper, we consider that the number of secret-key rate
requirements in the QKD layer directly depends on the number of snapshots
captured by edge devices. However, the number of snapshots of each edge device
can not be known in advance since the number of snapshots which are generated
by the semantic segmentation module is relevant to the interest of VSO.
Therefore, the number of snapshots of each device becomes an uncertain
parameter.

Figure 4: The two-stage stochastic programming.

The QKD resource allocation for maritime transportation applications from the
QKD resource pool that is created by QKD service providers in coalition
$\mathbb{C}$ is formulated as the two stages of the SP model. Figure 4
illustrates the two stages of the SP model for maritime transportation
applications given the QKD resource pool that is conceived by cooperative QKD
service providers in coalition $\mathbb{C}$. In the first stage, decisions of
provisioning QKD and KM wavelengths, and a decision of the routing according
to the semantic requirements (i.e., from the semantic source node to
destination node) are made. In the second stage, when the semantic
requirements (i.e., the number of snapshots generated by an edge device) are
observed, the QKD and KM wavelengths that are provisioned in the first stage
will be allocated to satisfy the semantic requirements. If the numbers of
reserved QKD and KM wavelengths in the first stage are not enough, the QKD and
KM wavelengths in the on-demand phase are provisioned.

##  IV Optimization Formulation

In this section, we first describe sets, constants, and decision variables of
the SP model. We next present the SP model from the QKD resource pool created
by the cooperative QKD service providers in coalition $\mathbb{C}$. Finally,
we present the deterministic equivalent formulation to achieve the solution of
the SP model.

###  IV-A Model Description

The optimization model is based on the two-stage SP [39]. The sets and
constants in the optimization are defined as follows:

  * •

${\mathcal{N}}$ denotes a set of all nodes in the network.

  * •

${\mathcal{O}}_{n}$ denotes a set of outgoing links from node
$n\in{\mathcal{N}}$.

  * •

${\mathcal{I}}_{n}$ denotes a set of incoming links to node
$n\in{\mathcal{N}}$.

  * •

${\mathcal{F}}$ denotes a set of QKD-SIC chain requests in the network.

  * •

$W^{\mathrm{qkd}}_{f}$ denotes 3 wavelengths of QKD link of
$f\in{\mathcal{F}}$ satisfying the secrete-key rate [28].

  * •

$W^{\mathrm{kml}}_{f}$ denotes 1 wavelength of KM link of $f\in{\mathcal{F}}$
satisfying the secrete-key rate [28].

  * •

$B^{\mathrm{eng}}_{n,f}$ denotes the amount of energy required for
transferring traffic of request $f\in{\mathcal{F}}$ through node
$n\in{\mathcal{N}}$.

We denote $S_{f}\in{\mathcal{N}}$ and $D_{f}\in{\mathcal{N}}$ as the source
node and the destination node of request $f$, respectively. The decision
variables of the network are as follows:

  * •

$x_{i,j,f}$ is a binary variable indicating whether request
$f\in{\mathcal{F}}$ will take a route with the link from node
$i\in{\mathcal{N}}$ to node $j\in{\mathcal{N}}$ or not.

  * •

$y^{\mathrm{r}}_{i,n,f}$ and $z^{\mathrm{r}}_{i,n,f}$ are non-negative
variables indicating the wavelength channels for QKD and KM links in the
reservation phases, respectively.

  * •

$y^{\mathrm{e}}_{i,n,f}$ and $z^{\mathrm{e}}_{i,n,f}$ are non-negative
variables indicating the utilized wavelength channels for QKD and KM links,
respectively.

  * •

$y^{\mathrm{o}}_{i,n,f}$ and $z^{\mathrm{o}}_{i,n,f}$ are non-negative
variables indicating the wavelength channels for QKD and KM links in the on-
demand phases, respectively.

###  IV-B Stochastic Programming Formulation

We first propose a QKD resource allocation scheme based on a two-stage SP
model among cooperative QKD service providers in coalition structure
$\mathbb{C}$. The resource allocation decisions in the proposed scheme are
conducted in two stages according to the result of the SP model.

$\displaystyle v(\mathbb{C})$
$\displaystyle=\min\sum_{f\in{\mathcal{F}}}\sum_{n\in{\mathcal{N}}}\sum_{i\in{\mathcal{I}}_{n}}\Big{(}B^{\mathrm{eng}}_{n,f}x_{i,n,f}$
(11)
$\displaystyle+\big{(}\frac{1}{3}(A^{f}_{\mathrm{tx}}(\bar{\omega})\beta^{\mathrm{r}}_{\mathrm{tx}}+A^{f}_{\mathrm{rx}}(\bar{\omega})\beta^{\mathrm{r}}_{\mathrm{rx}})\big{)}y^{\mathrm{r}}_{i,n,f}$
$\displaystyle+(A^{f}_{\mathrm{km}}\beta^{\mathrm{r}}_{\mathrm{km}}+A^{f}_{\mathrm{si}}\beta^{\mathrm{r}}_{\mathrm{si}}+A^{f}_{\mathrm{md}}\beta^{\mathrm{r}}_{\mathrm{md}})z^{\mathrm{r}}_{i,n,f}$
$\displaystyle+e_{(i,n)}(y^{\mathrm{r}}_{i,n,f}+z^{\mathrm{r}}_{i,n,f})\beta^{\mathrm{r}}_{\mathrm{ch}}\Big{)}$
$\displaystyle+{\mathbb{E}}\left[{\mathscr{C}}_{n}(x_{i,n,f},y^{\mathrm{r}}_{i,n,f},z^{\mathrm{r}}_{i,n,f},\tilde{\omega})\right],$
where
$\displaystyle{\mathscr{C}}_{n}(y^{\mathrm{r}}_{i,n,f},z^{\mathrm{r}}_{i,n,f},\tilde{\omega})$
(12)
$\displaystyle=\min_{x_{i,n,f},y^{\mathrm{r}}_{i,n,f},z^{\mathrm{r}}_{i,n,f}}\sum_{f\in{\mathcal{F}}}\sum_{n\in{\mathcal{N}}}\sum_{i\in{\mathcal{I}}_{n}}$
$\displaystyle\Big{(}\big{(}\frac{1}{3}(A^{f}_{\mathrm{tx}}(\tilde{\omega})\beta^{\mathrm{e}}_{\mathrm{tx}}+A^{f}_{\mathrm{rx}}(\tilde{\omega})\beta^{\mathrm{e}}_{\mathrm{rx}})\big{)}y^{\mathrm{e}}_{i,n,f,\tilde{\omega}}$
$\displaystyle+(A^{f}_{\mathrm{km}}\beta^{\mathrm{e}}_{\mathrm{km}}+A^{f}_{\mathrm{si}}\beta^{\mathrm{e}}_{\mathrm{si}}+A^{f}_{\mathrm{md}}\beta^{\mathrm{e}}_{\mathrm{md}})z^{\mathrm{e}}_{i,n,f,\tilde{\omega}}$
$\displaystyle+e_{(i,n)}(y^{\mathrm{e}}_{i,n,f,\tilde{\omega}}+z^{\mathrm{e}}_{i,n,f,\tilde{\omega}})\beta^{\mathrm{e}}_{\mathrm{ch}}$
$\displaystyle+\big{(}\frac{1}{3}(A^{f}_{\mathrm{tx}}(\tilde{\omega})\beta^{\mathrm{o}}_{\mathrm{tx}}+A^{f}_{\mathrm{rx}}(\tilde{\omega})\beta^{\mathrm{o}}_{\mathrm{rx}})\big{)}y^{\mathrm{o}}_{i,n,f,\tilde{\omega}}$
$\displaystyle+(A^{f}_{\mathrm{km}}\beta^{\mathrm{o}}_{\mathrm{km}}+A^{f}_{\mathrm{si}}\beta^{\mathrm{o}}_{\mathrm{si}}+A^{f}_{\mathrm{md}}\beta^{\mathrm{o}}_{\mathrm{md}})z^{\mathrm{o}}_{i,n,f,\tilde{\omega}}$
$\displaystyle+e_{(i,n)}(y^{\mathrm{o}}_{i,n,f,\tilde{\omega}}+z^{\mathrm{o}}_{i,n,f,\tilde{\omega}})\beta^{\mathrm{o}}_{\mathrm{ch}}\Big{)}$
s.t.
$\displaystyle\sum_{j^{\prime}\in{\mathcal{O}}_{S_{f}}}x_{S_{f},j^{\prime},f}-\sum_{i^{\prime}\in{\mathcal{I}}_{S_{f}}}x_{i^{\prime},S_{f},f}=1,f\in{\mathcal{F}}$
(26)
$\displaystyle\sum_{i^{\prime}\in{\mathcal{I}}_{D_{f}}}x_{i^{\prime},D_{f},f}-\sum_{j^{\prime}\in{\mathcal{O}}_{D_{f}}}x_{D_{f},j^{\prime},f}=1,f\in{\mathcal{F}}$
$\displaystyle\sum_{j^{\prime}\in{\mathcal{O}}_{n}}x_{n,j^{\prime},f}-\sum_{i^{\prime}\in{\mathcal{I}}_{n}}x_{i^{\prime},n,f}=0,$
$\displaystyle f\in{\mathcal{F}},n\in{\mathcal{N}}\setminus\\{S_{f},D_{f}\\}$
$\displaystyle\sum_{j^{\prime}\in{\mathcal{O}}_{n}}x_{n,j^{\prime},f}\leq
1,n\in{\mathcal{N}},f\in{\mathcal{F}}$
$\displaystyle\sum_{f\in{\mathcal{F}}}\big{(}y^{\mathrm{e}}_{i,j,f,\tilde{\omega}}x_{i,j,f}\big{)}\leq
W^{\mathrm{qkd}}_{i,j}(\mathbb{C}),i,j\in{\mathcal{N}}$
$\displaystyle\sum_{f\in{\mathcal{F}}}(z^{\mathrm{e}}_{i,j,f,\tilde{\omega}}x_{i,j,f})\leq
W^{\mathrm{kml}}_{i,j}(\mathbb{C}),i,j\in{\mathcal{N}}$ $\displaystyle
y^{\mathrm{e}}_{i,n,f,\tilde{\omega}}x_{i,j,f}\leq
y^{\mathrm{r}}_{i,n,f}x_{i,j,f},i,n\in{\mathcal{N}},f\in{\mathcal{F}}$
$\displaystyle z^{\mathrm{e}}_{i,n,f,\tilde{\omega}}x_{i,j,f}\leq
z^{\mathrm{r}}_{i,n,f}x_{i,j,f},i,n\in{\mathcal{N}},f\in{\mathcal{F}}$
$\displaystyle(y^{\mathrm{e}}_{i,n,f,\tilde{\omega}}x_{i,j,f})+y^{\mathrm{o}}_{i,n,f,\tilde{\omega}}\geq
P_{f}({\tilde{\omega}})W^{\mathrm{qkd}}_{f}x_{i,n,f},$ $\displaystyle
i,n\in{\mathcal{N}},f\in{\mathcal{F}}$
$\displaystyle(z^{\mathrm{e}}_{i,n,f,\tilde{\omega}}x_{i,j,f})+z^{\mathrm{o}}_{i,n,f,\tilde{\omega}}\geq
P_{f}({\tilde{\omega}})W^{\mathrm{kml}}_{f}x_{i,n,f},$ $\displaystyle
i,n\in{\mathcal{N}},f\in{\mathcal{F}}$ $\displaystyle
W^{\mathrm{qkd}}(\mathbb{C})_{i,j}\leq
W^{\mathrm{qkd}}_{\mathrm{max}},i,j\in{\mathcal{N}}$ $\displaystyle
W^{\mathrm{kml}}(\mathbb{C})_{i,j}\leq
W^{\mathrm{kml}}_{\mathrm{max}},i,j\in{\mathcal{N}}$ $\displaystyle
x_{i,n,f}\in\\{0,1\\},i,n\in{\mathcal{N}},f\in{\mathcal{F}}$ $\displaystyle
y^{\mathrm{r}}_{i,n,f},y^{\mathrm{e}}_{i,n,f,\tilde{\omega}},y^{\mathrm{o}}_{i,n,f,\tilde{\omega}},z^{\mathrm{r}}_{i,n,f},z^{\mathrm{e}}_{i,n,f,\tilde{\omega}},$
$\displaystyle
z^{\mathrm{o}}_{i,n,f,\tilde{\omega}}\in\\{0,1,2,\dots\\},i,n\in{\mathcal{N}},f\in{\mathcal{F}}$

In the proposed system, QKD resources (i.e., QKD and KM wavelengths) are
allocated by QKD service providers to SIC applications in the QKD network. In
the first stage, QKD resources are allocated in the resource pool of QKD
nodes. The actual number of QKD resources in the resource pool is determined
in the second stage. The SP model considers the uncertainties (i.e., the
number of _scenarios_), including the number of QKD devices required by the
SIC nodes. In the QKD network, the secret-key rates are uncertain. Let
$\tilde{\kappa}_{f}$ denote a random variable representing the secret key rate
required for the QKD-SIC chain request $f$. Let $\varphi$ denote the size of
the scenario spaces $\mathcal{K}_{f}$.

$\displaystyle\tilde{\omega}\in\big{\\{}\tilde{\kappa}_{f}\big{|}\tilde{\kappa}_{f}\in\mathcal{K}_{f}\big{\\}},$
$\displaystyle\mathcal{K}_{f}=\\{\tilde{\kappa}_{f_{1}},\tilde{\kappa}_{f_{2}},\tilde{\kappa}_{f_{3}},\dots,\tilde{\kappa}_{f_{\varphi}}\\}.$
(27)

The objective of the SP model for QKD resource allocation under uncertainty
expressed in (11) is to minimize the total cost of deploying QKD resources,
including MDI-QTxs, MDI-QRxs, LKMs, SIs, MUX /DEMUX components, and QKD and KM
links.

The constraint (26) ensures that the number of outgoing routes is larger than
the number of incoming routes when the node is the source node $S_{f}$ of the
QKD-SIC chain request $f$. The constraint (26) ensures that the number of
incoming routes is larger than the number of outgoing routes if the node is
the destination node $D_{f}$ of the QKD-SIC chain request $f$. The constraint
(26) ensures that the number of outgoing routes must equal the number of
incoming routes if the node is the intermediate node of the QKD-SIC chain
request $f$. The constraint (26) ensures that there is no loop for any QKD-SIC
chain request, meaning that there is only one outgoing route for the QKD-SIC
chain request of any node. The constraint (26) ensures that the QKD
wavelengths of all QKD-SIC chain requests of any node must not exceed the
available QKD wavelengths in the QKD resource pool (i.e.,
$W^{\mathrm{qkd}}_{i,j}(\mathbb{C})$). The constraint (26) ensures that the KM
wavelengths of all QKD-SIC chain requests from any node must not exceed the
available KM wavelengths in the QKD resource pool (i.e.,
$W^{\mathrm{kml}}_{i,j}(\mathbb{C})$).

The constraint (26) ensures that the number of QKD wavelengths utilized is
less than or equal to the number of QKD wavelengths reserved. The constraint
(26) ensures that the number of KM wavelengths utilized is less than or equal
to the number of KM wavelengths reserved. The constraint (26) ensures that the
number of QKD wavelengths used plus the number of on-demand QKD wavelengths
must satisfy the security requirements (i.e., secret-key rates). The
constraint (26) ensures that the number of KM wavelengths used plus the number
of on-demand KM wavelengths must satisfy the requirements. The constraints
(26) and (26) ensure that the QKD and KM wavelength resource pools do not
exceed the maximum available QKD and wavelength resource pools, respectively.
The constraints (26) and (26) are the binary and integer variables,
respectively.

###  IV-C Deterministic Equivalent Formulation

$\displaystyle v(\mathbb{C})$
$\displaystyle=\min_{x_{i,n,f},y^{\mathrm{r}}_{i,n,f},z^{\mathrm{r}}_{i,n,f}}\sum_{f\in{\mathcal{F}}}\sum_{n\in{\mathcal{N}}}\sum_{i\in{\mathcal{I}}_{n}}\Big{(}B^{\mathrm{eng}}_{n,f}x_{i,n,f}$
(28)
$\displaystyle+\big{(}\frac{1}{3}(A^{f}_{\mathrm{tx}}(\bar{\omega})\beta^{\mathrm{r}}_{\mathrm{tx}}+A^{f}_{\mathrm{rx}}(\bar{\omega})\beta^{\mathrm{r}}_{\mathrm{rx}})\big{)}y^{\mathrm{r}}_{i,n,f}$
$\displaystyle+(A^{f}_{\mathrm{km}}\beta^{\mathrm{r}}_{\mathrm{km}}+A^{f}_{\mathrm{si}}\beta^{\mathrm{r}}_{\mathrm{si}}+A^{f}_{\mathrm{md}}\beta^{\mathrm{r}}_{\mathrm{md}})z^{\mathrm{r}}_{i,n,f}$
$\displaystyle+e_{(i,n)}(y^{\mathrm{r}}_{i,n,f}+z^{\mathrm{r}}_{i,n,f})\beta^{\mathrm{r}}_{\mathrm{ch}}\Big{)}$
$\displaystyle+\sum_{f\in{\mathcal{F}}}\mathbb{P}_{f}(\omega)\sum_{n\in{\mathcal{N}}}\sum_{i\in{\mathcal{I}}_{n}}$
$\displaystyle\Big{(}\big{(}\frac{1}{3}(A^{f}_{\mathrm{tx}}(\omega)\beta^{\mathrm{e}}_{\mathrm{tx}}+A^{f}_{\mathrm{rx}}(\omega)\beta^{\mathrm{e}}_{\mathrm{rx}})\big{)}y^{\mathrm{e}}_{i,n,f,\omega}$
$\displaystyle+(A^{f}_{\mathrm{km}}\beta^{\mathrm{e}}_{\mathrm{km}}+A^{f}_{\mathrm{si}}\beta^{\mathrm{e}}_{\mathrm{si}}+A^{f}_{\mathrm{md}}\beta^{\mathrm{e}}_{\mathrm{md}})z^{\mathrm{e}}_{i,n,f,\omega}$
$\displaystyle+e_{(i,n)}(y^{\mathrm{e}}_{i,n,f,\omega}+z^{\mathrm{e}}_{i,n,f,\omega})\beta^{\mathrm{e}}_{\mathrm{ch}}$
$\displaystyle+\big{(}\frac{1}{3}(A^{f}_{\mathrm{tx}}(\omega)\beta^{\mathrm{o}}_{\mathrm{tx}}+A^{f}_{\mathrm{rx}}(\omega)\beta^{\mathrm{o}}_{\mathrm{rx}})\big{)}y^{\mathrm{o}}_{i,n,f,\omega}$
$\displaystyle+(A^{f}_{\mathrm{km}}\beta^{\mathrm{o}}_{\mathrm{km}}+A^{f}_{\mathrm{si}}\beta^{\mathrm{o}}_{\mathrm{si}}+A^{f}_{\mathrm{md}}\beta^{\mathrm{o}}_{\mathrm{md}})z^{\mathrm{o}}_{i,n,f,\omega}$
$\displaystyle+e_{(i,n)}(y^{\mathrm{o}}_{i,n,f,\omega}+z^{\mathrm{o}}_{i,n,f,\omega})\beta^{\mathrm{o}}_{\mathrm{ch}}\Big{)}$
s.t.
$\displaystyle\sum_{j^{\prime}\in{\mathcal{O}}_{S_{f}}}x_{S_{f},j^{\prime},f}-\sum_{i^{\prime}\in{\mathcal{I}}_{S_{f}}}x_{i^{\prime},S_{f},f}=1,f\in{\mathcal{F}}$
(42)
$\displaystyle\sum_{i^{\prime}\in{\mathcal{I}}_{D_{f}}}x_{i^{\prime},D_{f},f}-\sum_{j^{\prime}\in{\mathcal{O}}_{D_{f}}}x_{D_{f},j^{\prime},f}=1,f\in{\mathcal{F}}$
$\displaystyle\sum_{j^{\prime}\in{\mathcal{O}}_{n}}x_{n,j^{\prime},f}-\sum_{i^{\prime}\in{\mathcal{I}}_{n}}x_{i^{\prime},n,f}=0,$
$\displaystyle f\in{\mathcal{F}},n\in{\mathcal{N}}\setminus\\{S_{f},D_{f}\\}$
$\displaystyle\sum_{j^{\prime}\in{\mathcal{O}}_{n}}x_{n,j^{\prime},f}\leq
1,n\in{\mathcal{N}},f\in{\mathcal{F}}$
$\displaystyle\sum_{f\in{\mathcal{F}}}\big{(}y^{\mathrm{e}}_{i,j,f,\omega}x_{i,j,f}\big{)}\leq
W^{\mathrm{qkd}}_{i,j}(\mathbb{C}),$ $\displaystyle
i,j\in{\mathcal{N}},\forall\omega\in\Omega_{f}$
$\displaystyle\sum_{f\in{\mathcal{F}}}(z^{\mathrm{e}}_{i,j,f,\omega}x_{i,j,f})\leq
W^{\mathrm{kml}}_{i,j}(\mathbb{C}),$ $\displaystyle
i,j\in{\mathcal{N}},\forall\omega\in\Omega_{f}$ $\displaystyle
y^{\mathrm{e}}_{i,n,f,\omega}x_{i,j,f}\leq y^{\mathrm{r}}_{i,n,f}x_{i,j,f},$
$\displaystyle
i,n\in{\mathcal{N}},f\in{\mathcal{F}},\forall\omega\in\Omega_{f}$
$\displaystyle z^{\mathrm{e}}_{i,n,f,\omega}x_{i,j,f}\leq
z^{\mathrm{r}}_{i,n,f}x_{i,j,f},$ $\displaystyle
i,n\in{\mathcal{N}},f\in{\mathcal{F}},\forall\omega\in\Omega_{f}$
$\displaystyle(y^{\mathrm{e}}_{i,n,f,\omega}x_{i,j,f})+y^{\mathrm{o}}_{i,n,f,\omega}\geq$
$\displaystyle\sum_{\omega\in\Omega_{f}}(P_{f}({\omega})W^{\mathrm{qkd}}_{f}x_{i,n,f}),i,n\in{\mathcal{N}},f\in{\mathcal{F}}$
$\displaystyle(z^{\mathrm{e}}_{i,n,f,\omega}x_{i,j,f})+z^{\mathrm{o}}_{i,n,f,\omega}\geq$
$\displaystyle\sum_{\omega\in\Omega_{f}}(P_{f}({\omega})W^{\mathrm{kml}}_{f}x_{i,n,f}),i,n\in{\mathcal{N}},f\in{\mathcal{F}}$
$\displaystyle W^{\mathrm{qkd}}_{i,j}(\mathbb{C})\leq
W^{\mathrm{qkd}}_{\mathrm{max}},i,j\in{\mathcal{N}}$ $\displaystyle
W^{\mathrm{kml}}_{i,j}(\mathbb{C})\leq
W^{\mathrm{kml}}_{\mathrm{max}},i,j\in{\mathcal{N}}$ $\displaystyle
x_{i,n,f}\in\\{0,1\\},i,n\in{\mathcal{N}},f\in{\mathcal{F}}$ $\displaystyle
y^{\mathrm{r}}_{i,n,f},y^{\mathrm{e}}_{i,n,f,\omega},y^{\mathrm{o}}_{i,n,f,\omega},z^{\mathrm{r}}_{i,n,f},z^{\mathrm{e}}_{i,n,f,\omega},z^{\mathrm{o}}_{i,n,f,\omega}\in$
$\displaystyle\\{0,1,2,\dots\\},i,n\in{\mathcal{N}},f\in{\mathcal{F}},\forall\omega\in\Omega_{f}$

The SP with the random variable $\tilde{\omega}$ in (11) - (26) can be
transformed into the deterministic equivalence problem [39] as expressed in
(28) - (42). The random variable $\tilde{\omega}$ can be represented by a
scenario. The scenario is a realization of a random variable, which is denoted
by $\omega$. The value of the random variable can be taken from a set of
scenarios. Let $\Psi$ and $\Omega_{f}$ as defined in (43) and (44),
respectively, be the set of all scenarios of each requirement (i.e., a
scenario space) and the set of all scenarios of request $f$, respectively. Let
$K$ denote the maximum required secret-key rate of the request $f$.

$\displaystyle\Psi={\displaystyle\prod_{f\in\mathcal{F}}}\Omega_{f}=\Omega_{1}\times\Omega_{2}\times\cdots\times\Omega_{|\mathcal{F}|}$
(43) $\displaystyle\Omega_{f}=\\{0,1,2,\dots,K\\}$ (44)

The expectation $\mathbb{E}[\cdot]$ of the SP in (11) can be represented by
the weighted sum of scenarios and their probabilities
$\mathbb{P}_{f}(\omega)$.

The objective function (28) is to minimize the deployment cost. The decision
variables $y^{\mathrm{e}}_{i,n,f,\omega}$, $y^{\mathrm{o}}_{i,n,f,\omega}$,
$z^{\mathrm{e}}_{i,n,f,\omega}$, and $z^{\mathrm{o}}_{i,n,f,\omega}$ are under
$\omega$ (i.e., $\omega\in\Omega$) which implies that the values of demands
are available when $\omega$ is observed. The constraints (42), (42), (42), and
(42) are the routing constraints, which have the same meanings as (26), (26),
(26), and (26), respectively. The constraints (42) and (42) guarantee that the
utilization of QKD and KM wavelengths does not exceed the QKD and KM
wavelength capacity, which are similar to (26) and (26). The constraints (42)
and (42) guarantee that the expended QKD and KM wavelengths respectively must
not exceed the reserved QKD and KM wavelengths, which have the same meanings
as (26) and (26). The constraints (42) and (42) guarantee that all demands are
satisfied, which are similar to (26) and (26), respectively. The constraints
(42) and (42) are the capacity constraints of QKD and KM wavelength resource
pools which have the same meanings as (26) and (26). The constraints in (42)
and (42) indicate that all decision variables are non-negative integers.

##  V Cost Management Among QKD Service Providers

When the QKD resource allocation for maritime transportation applications is
performed, the cooperative QKD service providers in coalition $\mathbb{C}$
share the deployment cost created from satisfying the semantic requirements.
In this section, we present the cost management to apply the concept of
Shapley value to determine the deployment cost-sharing that each of
cooperative QKD service providers will obtain.

###  V-A Shapley Value

The concept of Shapley value [40] is applied to determine the deployment cost-
sharing among cooperative QKD service providers111In the remainder of this
paper, “QKD service provider” and “provider” are used interchangeably.. The
cooperative providers form a coalition to establish the shared QKD resource
pool. Given the characteristic function $v(\cdot)$ in (11) and (28), the
Shapley value of QKD service provider $s$ is expressed in as follows:

$\vartheta_{s}(v)={\displaystyle\sum_{\mathbb{D}\subseteq{\mathbb{C}}\setminus\\{s\\}}}\frac{\mathbb{|D|}!(|{\mathbb{C}}|-\mathbb{|D|}-1)!}{|{\mathbb{C}}|!}\big{(}v(\mathbb{D}\cup\\{s\\})-v(\mathbb{D})\big{)}.$
(45)

The cost management of the cooperative QKD service providers in coalition
${\mathbb{C}}$ based on the Shapley value can achieve four desirable
properties as follows:

  1. 1.

_Efficiency:_ It is
$\displaystyle\sum_{s\in{\mathbb{C}}}\vartheta_{s}(v)=v({\mathbb{C}})$ which
means that the sum of deployment cost of all cooperative QKD service providers
is minimum.

  2. 2.

_Symmetry:_ $\vartheta_{s}(v)$ is equal to $\vartheta_{l}(v)$ if
$v(\mathbb{D}\cup\\{s\\})=v(\mathbb{D}\cup\\{l\\})$ contains all QKD service
providers $s$ and $l$ given all coalitions $\mathbb{D}$ of other cooperative
QKD service providers. This means that the cost shared between QKD service
providers $s$ and $l$ is equal if they make an identical contribution to the
coalition.

  3. 3.

_Dummy:_ $\vartheta_{s}(v)$ is 0 if $v(\mathbb{D})=v(\mathbb{D}\cup\\{s\\})$
contains all coalitions $\mathbb{D}$ of all cooperative QKD service providers
except provider $s$. This means that the cost shared by QKD service provider
$s$ is zero if QKD service provider $s$ does not make a contribution to the
total cost of the coalition.

  4. 4.

_Additivity:_ $\vartheta(u+v)=\vartheta(v+u)=\vartheta(u)+\vartheta(v)$ if $u$
and $v$ are the characteristic functions.

With the Shapley value, individual fairness is achieved [40]. In particular,
the cost shared by the cooperative QKD service provider is not more than the
cost of the QKD service provider who does not cooperate with other QKD service
providers (i.e., $\vartheta_{s}(v)\leq v(\\{s\\})$). In addition, the
uniqueness of the Shapley value is ensured [40].

##  VI Cooperation Formation Among QKD Service Providers

In this section, we investigate the condition that QKD service providers aim
to minimize their own costs. Therefore, QKD service providers may form a
coalition and create a resource pool (i.e., QKD and KM wavelengths). We
analyze the cooperation formation of QKD service providers by considering
their individual costs.

###  VI-A Cooperation Model

We detail the model of cooperation formation of the QKD service providers with
the following information. Let a set of all providers (i.e., players) be
denoted by ${\mathbb{G}}$. There is an agreement among providers in
${\mathbb{C}}$ on cooperation (i.e., ${\mathbb{C}}\subseteq{\mathbb{G}}$).
${\mathbb{C}}$ is a set of cooperative providers such that the characteristic
function of cooperative providers is $v({\mathbb{C}})$, which is given by
(28). For example, given three providers, a set of providers is
${\mathbb{G}}=\\{1,2,3\\}$. The cooperation can be defined as ${\mathbb{C}}$
$\in\\{\\{\\{1\\},\\{2\\},\\{3\\}\\},\\{\\{1,2\\},\\{3\\}\\},\\{\\{1,3\\},\\{2\\}\\},\\{\\{2,3\\},\\{1\\}\\},$
$\\{\\{1,2,3\\}\\}\\}$. The coalition game with a set of players that are
providers can be defined as $({\mathbb{G}},\delta_{s}(\cdot))$. Therefore, we
can formulate a coalition game to model and obtain the equilibrium strategies.
The strategy of each provider is to cooperate with other providers. Let
$p_{s,l}$ denote the cooperation. If $p_{s,l}=1$ which means that providers
$s$ and $l$ cooperate, and $p_{s,l}=0$ otherwise. Let
$\mathbf{p}_{s}\in\mathcal{P}_{s}$ denote the strategy of provider $s$. Let
$\mathbf{p}^{*}_{s}\in\mathcal{P}_{s}$ denote the cooperation equilibrium
strategy. Let
$\mathbf{p}^{*}_{-s}\in\displaystyle{\prod_{l\in\mathbb{G}\setminus\\{s\\}}\mathcal{P}_{l}}$
denote the cooperation equilibrium strategies of all providers except the
strategy of provider $s$. Therefore, the strategy space of provider $s$
cooperating with other providers is expressed in (46). The cooperation
structure ${\mathbb{C}}$ is expressed in (49). Therefore, the cooperation
equilibrium of the cooperation formation of providers is expressed in (50).

$\displaystyle\mathcal{P}_{s}=\Big{\\{}(p_{s,1},\dots,p_{s,l-1},p_{s,l},p_{s,l+1},\dots,p_{s,|{\mathbb{G}}|})$
$\displaystyle|p_{s,l}\in\\{0,1\\},l\in{\mathbb{G}}\setminus\\{s\\}\Big{\\}}$
(46) $\displaystyle
p_{s,l}=\left\\{\begin{array}[]{rcl}1~{}\mathrm{if}~{}s\in{\mathbb{C}}~{}~{}\mathrm{AND}~{}~{}l\in{\mathbb{C}}\\\
0~{}\mathrm{if}~{}s\notin{\mathbb{C}}~{}~{}\mathrm{OR}~{}~{}l\notin{\mathbb{C}}\\\
\end{array}\right.$ (49)
$\displaystyle\delta_{s}(\mathbf{p}^{*}_{s},\mathbf{p}^{*}_{-s})\leq\delta_{s}(\mathbf{p}_{s},\mathbf{p}^{*}_{-s}),\quad\forall
s$ (50)

$\delta_{s}(\cdot)$ is the cost of provider $s$ received from the QKD resource
allocation and cost management, i.e., $\delta_{s}(\cdot)=\vartheta_{s}$ in
(45).

###  VI-B Dynamics of Coalition Formation

The cooperation formation equilibrium among providers is determined based on
best-response dynamics. In particular, the provider iteratively makes a
decision to form a cooperation. The decision iteration is denoted by $\sigma$
(i.e., $\sigma=1,2,3,\dots,N$) where $N$ is a maximum number of iterations.
The strategy of provider $s$ in the decision iteration $\sigma$ is denoted by
$\mathbf{p}_{s}(\sigma)$. The strategies of all providers except the strategy
of provider $s$ in decision iteration $\sigma-1$ are denoted by
$\mathbf{p}_{-s}(\sigma-1)$. In each decision iteration, the provider assesses
the new strategy and then changes to the new strategy with the lowest cost.
The strategy of provider $s$ in decision iteration $\sigma$ (i.e.,
$\mathbf{p}_{s}(\sigma)$) can be determined as follows:

$\displaystyle\mathbf{p}_{s}(\sigma)\in\mathrm{arg}\;\min_{\mathbf{p}_{s}\in\mathcal{P}_{s}}\;\;\delta_{s}(\mathbf{p}_{s},\mathbf{p}_{-s}(\sigma-1)).$
(51)

In (51), given the knowledge of strategies of other providers in the prior
decision iteration (i.e., $\mathbf{p}_{-s}(\sigma-1)$), the best new strategy
is selected by the provider $s$. However, the provider may make a mistake with
a small probability (i.e., $\aleph$).

From (51), discrete-time Markov chain can be applied to model the strategy
adaptation of the cooperation formation [41]. Under all cooperations of all
providers, the finite state space of the Markov chain is expressed as
$\Theta=\displaystyle{\prod_{s\in\mathbb{G}}\mathcal{P}_{s}=\mathcal{P}_{1}\times\dots\times\mathcal{P}_{|\mathbb{G}|}}$.
It is assumed that it has symmetric cooperation which is $p_{s,l}=p_{l,s}$.
The strategy $\mathbf{p}_{s}(\cdot)$ in (51) contains the cooperations of
provider $s$ and is part of the state $\tau$, i.e., $\tau\in\Theta$. For the
transition probability of Markov chain, $\tau$ and $\tau^{{}^{\prime}}$ are
the current state and the next state, respectively. Let
$\tau=(p_{s,1},\dots,p_{s,l},\dots,p_{s,|\mathbb{G}|}),\tau\in\Theta$ denote
the current state. Let
$\tau^{{}^{\prime}}=(p^{{}^{\prime}}_{s,1},\dots,p^{{}^{\prime}}_{s,l},\dots,p^{{}^{\prime}}_{s,|\mathbb{G}|}),\tau^{{}^{\prime}}\in\Theta$
denote the next state. The set of providers associated with the state change
from $\tau$ to $\tau^{{}^{\prime}}$ is defined in (52).

$\displaystyle\mathcal{Z}_{\tau,\tau^{{}^{\prime}}}=\\{s|p_{s,l}\neq
p^{{}^{\prime}}_{s,l},s\neq l,s,l\in\mathbb{G}\\}$ (52)

The transition probability from $\tau$ to $\tau^{{}^{\prime}}$ is expressed in
(53).

$\displaystyle\mathbf{T}_{\tau,\tau^{{}^{\prime}}}=\lambda^{|\mathcal{Z}_{\tau,\tau^{{}^{\prime}}}|}(1-\lambda)^{{|\mathbb{G}|}-{|\mathcal{Z}_{\tau,\tau^{{}^{\prime}}}|}}\prod_{s\in\mathcal{Z}_{\tau,\tau^{{}^{\prime}}}}\Xi_{s}(\tau,\tau^{{}^{\prime}}),$
(53)

where $\lambda$ is a probability of the provider updating the strategy in an
iteration of the decision. The probability of changing the strategy of
provider $s$ in an iteration of the decision is expressed in (56). Let
$\delta_{s}(\tau)$ denote the cost that is the function of the strategies of
all providers. In (56), the provider can change to the strategy that produces
lower cost (i.e., $\delta_{s}(\tau^{{}^{\prime}})<\delta_{s}(\tau)$). However,
the provider may irrationally change to the new strategy with probability
$\aleph$.

$\displaystyle\Xi_{s}(\tau,\tau^{{}^{\prime}})=\left\\{\begin{array}[]{rcl}1-\aleph~{}\mathrm{if}~{}\delta_{s}(\tau^{{}^{\prime}})<\delta_{s}(\tau)\\\
\aleph~{}\mathrm{otherwise}\end{array}\right.$ (56)

##  VII Performance Evaluation

In this section, we perform experiments with the following considerations.
First, we consider that the proposed SP model can obtain the efficient routing
to satisfy the QKD-SIC chain requests222In the remainder of this paper, “QKD-
SIC chain request” and “QKD chain request” are used interchangeably.. Each
routing consists of QKD and KM wavelengths in the reservation and the on-
demand phases of the SP model. The routing results are shown in Section
VII-B1. Second, we investigate the secret-key rates which have the effect on
increasing the deployment cost and consider that the deployment cost can be
decreased by optimally reserving the number of QKD and KM wavelengths in
advance. The results are presented in Section VII-B2.

Third, we show the upper and lower bounds of the SP model with the
consideration that if the gap between the solutions of the bound and the
solution of the SP model is small enough, we can apply the upper and lower
bounds to obtain the satisfied solutions instead of the SP model. This is due
to the fact that formulations of the upper and lower bounds are much lower
complexity than the formulation of SP model since the numbers of variables and
constraints of formulations of the upper and lower bounds are less than that
of formulation of SP model. In addition, we compare the SP model with the CO-
QBN algorithm [28] with different quantitative analyzes to show the
performance of the SP model. The results are shown in Section VII-B3. Fourth,
we show the SP model with a cooperative game and examine the impact of shared
QKD and KM wavelengths. In this experiment, we consider that, with a large
number of secret-key rates, the costs of cooperative providers are
significantly lower than that of noncooperative providers. The results are
shown in Section VII-B4.

Finally, we investigate and show how the coalition structure affects the costs
of QKD service providers. The stable coalition structure, which is a solution
to the cooperative game (i.e., the cooperation equilibrium), can be determined
by the Shapley value, a fair cost-sharing concept. In addition, we investigate
how the cost of shared QKD and KM wavelengths and cooperation affect the
stable cooperative structure. The reason for this is that we want to know
where the cooperation structure is stable when the cost increases. The results
are shown in Section VII-B5.

###  VII-A Parameter Setting

We consider the SIC over QKD networks as shown in Fig.1. Specifically, there
are three QKD service providers. Providers 1, 2, and 3 have 10, 15, and 20
available QKD wavelengths, respectively. Providers 1, 2, and 3 have 40, 55,
and 65 available KM wavelengths, respectively. We perform experiments using
the NSFNET and USNET topologies [28]. The distance $D$ between two QTxs is set
to 160 km [28]. We set $W^{\mathrm{qkd}}_{\mathrm{max}}=1,000$ and
$W^{\mathrm{kml}}_{\mathrm{max}}=300$ for the maximum number of wavelengths
for a QKD link between node $i$ and node $j$ and the maximum number of
wavelengths for a KM link between node $i$ and node $j$, respectively. We
consider the Shapley value for cost-sharing between providers. We implement
and solve the SP model using the GAMS/CPLEX solver [42].

For the SP model, we consider the random number of secret-key rates with
uniform distribution for ease of presentation. We consider the cost values of
five QKD network components which are composed of MDI-QTXs, MDI-QRXs, LKMs,
SIs, MUX/DEMUX components, and QKD and KM links. For the reservation phase, we
apply these cost values from [28]. The cost values of the components can be
presented in Table I.

TABLE I: Reservation, utilization, and on-demand cost values

Notations | Values($) | Notations | Values($) | Notations | Values($)
---|---|---|---|---|---
$\beta^{\mathrm{r}}_{\mathrm{tx}},\beta^{\mathrm{e}}_{\mathrm{tx}}$ | 1,500 | $\beta^{\mathrm{r}}_{\mathrm{rx}},\beta^{\mathrm{e}}_{\mathrm{rx}}$ | 2,250 | $\beta^{\mathrm{r}}_{\mathrm{km}},\beta^{\mathrm{e}}_{\mathrm{km}}$ | 1,200
$\beta^{\mathrm{r}}_{\mathrm{si}},\beta^{\mathrm{e}}_{\mathrm{si}}$ | 150 | $\beta^{\mathrm{r}}_{\mathrm{md}},\beta^{\mathrm{e}}_{\mathrm{md}}$ | 300 | $\beta^{\mathrm{r}}_{\mathrm{ch}},\beta^{\mathrm{e}}_{\mathrm{ch}}$ | 1
$\beta^{\mathrm{o}}_{\mathrm{tx}}$ | 6,000 | $\beta^{\mathrm{o}}_{\mathrm{rx}}$ | 9,000 | $\beta^{\mathrm{o}}_{\mathrm{km}}$ | 3,000
$\beta^{\mathrm{o}}_{\mathrm{si}}$ | 500 | $\beta^{\mathrm{o}}_{\mathrm{md}}$ | 900 | $\beta^{\mathrm{o}}_{\mathrm{ch}}$ | 4

###  VII-B Numerical Results

####  VII-B1 Routing

Figure 5: An example of three QKD-SIC chain requests in USNET topology.

Figure 5 illustrates the solutions of the SP model that satisfy three QKD-SIC
chain requests. In the solutions, the SP model can allocate QKD and KM
wavelengths (i.e., resources) in reservation and on-demand phases for the
requests in fiber optic networks. In Fig. 5, each request (i.e., a route)
utilizes QKD and KM wavelength within a fiber optic link. To obtain the
optimal deployment cost, it is interesting to mention that both QKD and KM
wavelengths in reservation and on-demand phases are utilized along the routes.
For example, with the QKD-SIC chain request from source node 1 to destination
node 23, the optimal deployment cost can be obtained from the route consisted
of nodes $1\rightarrow 6\rightarrow 9\rightarrow 12\rightarrow 16\rightarrow
22\rightarrow 23$. In particular, QKD and KM wavelengths in the on-demand
phase are utilized from node 1 to node 6 (i.e., $1\rightarrow 6$) and from
node 6 to node 9 (i.e., $6\rightarrow 9$) while the rest of the route utilize
the QKD and KM wavelengths in the reservation phase.

####  VII-B2 Cost Structure Analysis

(a) The deployment cost.

(b) The solution under QKD wavelengths.

(c) The solution under KM wavelengths.

(d) The solution under QKD and KM wavelengths.

Figure 6:  (a) The deployment cost in the SP model, (b) The optimal solution
under different reserved QKD wavelengths, (c) The optimal solution under
different reserved KM wavelengths, and (d) The optimal solution under
different reserved QKD and KM wavelengths.

In Fig. 6(a), we investigate how the secret-key rate, which is an important
demand parameter from users and applications, affects the costs in different
stages. Here, the observation is that the second-stage cost increases much
faster than the first-stage cost. The reason is that the price of on-demand
resources in the second stage is much higher than that of reserved resources
in the first stage. In particular, the price of MDI-QTxs and MDI-QRxs in the
second stage is high than those in the first stage by four times. Therefore,
in the second stage, the price of MDI-QTxs and MDI-QRxs dominates the cost in
the second stage since not only the price of MDI-QTxs and MDI-QRxs in the
second stage (i.e., the on-demand phase) is much higher than those in the
first stage (i.e., the reservation phase) but also the price of MDI-QTxs and
MDI-QRxs directly depends on the secret-key rates.

In Fig. 6(b), we examine the performance of the SP model in obtaining an
optimal solution. In the first stage, we vary the number of reserved QKD
wavelengths and fix the number of reserved KM wavelengths. Then, we present
the optimal solution obtained by the SP model and the effect of reserved QKD
wavelengths on the solution. In Fig. 6(b), when the number of reserved QKD
wavelengths increases, the first-stage cost increases significantly. However,
the second-stage cost decreases dramatically when the secret-key rates check.
The reason is that the QKD wavelength in the on-demand phase (i.e., the second
stage) is forced to be minimum by utilizing the cheaper QKD wavelength in the
reservation phase (i.e., the first stage). As a result, at 150 QKD wavelengths
reserved, the optimal solution can be achieved, and the second-stage cost is
0. This is because the reserved QKD wavelengths meet demands (i.e., secret-key
rates), and therefore the on-demand QKD wavelengths in the second stage are
not utilized. After this point, the total cost and the first-stage cost
increase since it has a penalty cost for an excess of reserved QKD wavelength
to be charged. For Fig. 6(b), we can mention that over- and under-provision of
QKD wavelengths affect the high total cost significantly.

In a similar way to the investigation above, in Fig. 6(c), we vary the number
of KM wavelengths in the first stage and fix the number of QKD wavelengths to
show the optimal solution obtained by the SP model. In Fig. 6(c), clearly, the
first-stage cost increases steadily while the second-stage cost decreases.
However, the optimal solution is achieved when the number of reserved KM
wavelength is 50. The reason is that the reserved KM wavelengths of 50
successfully satisfy the demands. After this point, the total cost and the
first-stage cost increase since the penalty cost of an excess of reserved KM
wavelengths is charged. Therefore, we can explain that the high cost is
affected by not only the reserved QKD wavelengths but also the reserved KM
wavelengths. In addition, we vary the number of QKD and KM wavelengths to show
the optimal solution obtained by the SP model. In Fig. 6(d), apparently, the
optimal deployment cost can be achieved (i.e., the numbers of reserved QKD and
KM wavelengths being 180 and 60, respectively) when the number of QKD and KM
wavelengths increases.

According to the investigation of the secret-key rate, we present the number
of QKD and KM wavelengths utilized in reservation and on-demand phases and
explain the reason that the second-stage cost increases dramatically. Also,
this helps us to understand how the QKD and KM wavelengths affect the second-
stage cost.

Figure 7(a) illustrates the number of QKD and KM wavelengths (i.e., resources)
in reservation and on-demand phases under different secret-key rates. In the
reservation phase, the numbers of QKD and KM wavelengths rise steeply until
the secret-key rates are 5 and 3 kbps, respectively. This is because the
reserved QKD and KM wavelengths are completely utilized at the secret-key
rates of 5 and 3 kbps, respectively. After these points, both QKD and KM
wavelengths are constant since the capacities of QKD and KM wavelengths are
limited. The capacities of QKD and KM wavelengths in the reservation phase are
constrained by the QKD service provider. As a result, to satisfy the high
secret-key rates, the QKD and KM wavelengths in the on-demand phase are more
utilized. In the on-demand phase, the numbers of QKD and KM wavelengths start
rising dramatically at 5 and 3 kbps, respectively, due to the limited
capacities of QKD and KM wavelengths in the reservation phase.

Figure 7(b) illustrates sub-path utilization according to the QKD-SIC chain
request (i.e., from a source to a destination) of QKD and KM wavelength in
reservation and on-demand phases under the different costs of QKD and KM
links. The full path in Fig. 7(b) is $1\rightarrow 6\rightarrow 9\rightarrow
12\rightarrow 16\rightarrow 22\rightarrow 23$, which is consisted of 6 sub-
paths. Source and destination nodes are 1 and 23, respectively. In Fig. 7(b),
it is straightforward to mention that when the cost of QKD and KM links
increases, the number of utilized sub-paths of both QKD and KM wavelengths in
the reservation phase changes to that of both QKD and KM wavelengths in the
on-demand phase. For example, with QKD and KM link cost $<50\$$, the sub-paths
of KM wavelengths in reservation and on-demand phases are utilized. However,
with QKD and KM link cost = 50$, the sub-paths of KM wavelength in the on-
demand phase are completely utilized (i.e., six sub-paths). To obtain the
optimal deployment cost, the sub-path utilization in the reservation phase is
completely changed to the sub-path utilization in the on-demand phase by the
SP model.

(a) QKD and KM wavelength utilization under different secret-key rates.

(b) QKD and KM wavelength utilization under different costs.

(c) Boundary.

(d) Cost comparison.

Figure 7: (a) The numbers of QKD and KM wavelengths in reservation and on-
demand phases under different secret-key rates, (b) The numbers of QKD and KM
wavelengths in reservation and on-demand phases under the different costs of
QKD and KM links, (c) Bounding of the objective value from the SP model, and
(d) Cost comparison between the SP model and the CO-QBN algorithm.

####  VII-B3 Performance Evaluation Under Various Parameters

We first evaluate the performance of the SP model by comparing it with the
_cooperative resource management (CRM) over the expected value deterministic_
and the _deterministic equivalent formulation_. In the case of CRM over the
expected value deterministic, the secret-key rates (i.e., demands) in the
first stage are considered to be the expected demands. On the other hand, the
secret-key rates are considered as the exact demands in the case of the
deterministic equivalent formulation. The former formulation is used to find
the values of the upper bound, while the latter formulation is used to find
the values of the lower bound.

In Fig. 7(c), the CRM over the expected value deterministic and deterministic
equivalent formulation yield the solutions that are the upper bound and lower
bound of the SP model, respectively. We observe that the gap to the upper
bound is sufficiently small. This result implies that using the expected value
of secret-key rates can achieve a satisfactory solution (i.e., close to the
optimal solution of the SP model). For example, when the QKD-SIC chain
requests are 10, which has the slight upper bound (i.e., 8.97%) shown in Fig.
7(c), we can obtain a good solution by using CRM over the expected value
deterministic instead of the SP model due to low computational complexity.

Then, we compare the SP model with the CO-QBN algorithm [28]. Figure 7(d)
illustrates a performance comparison of the SP model and CO-QBN algorithm by
varying the number of QKD-SIC chain requests. In Fig. 7(d), both the SP model
and CO-QBN algorithm can decrease the deployment cost when the QKD-SIC chain
requests increase. However, in comparison with the CO-QBN algorithm, the SP
model can significantly achieve a lower deployment cost under different QKD-
SIC chain requests. For example, with the QKD-SIC chain requests of 60, the SP
model can reduce the deployment cost compared with the CO-QBN algorithm by
8.15%. In addition, we observe that the performance of the SP model compared
with the CO-QBN algorithm under different QKD-SIC chain requests is quite
stable (i.e., 8.21%). Therefore, we can claim that the SP model significantly
outperforms the CO-QBN algorithm.

####  VII-B4 Impact of Available QKD and KM wavelengths

(a) Total deployment cost under different QKD and KM wavelengths shared by
provider 2.

(b) Cost of providers 1 and 2 under different QKD wavelengths.

(c) Cost of providers 1 and 2 under different KM wavelengths.

Figure 8: (a) The deployment cost under different number of shared QKD and KM
wavelengths of provider 2, (b) Costs of providers 1 and 2 with and without
cooperation under different available QKD wavelengths of provider 2, and (c)
Costs of providers 1 and 2 with and without cooperation under different
available KM wavelengths of provider 2.

We consider a situation that the QKD service providers 1 and 2 cooperate to
share the QKD and KM wavelengths. Figure 8(a) illustrates the total deployment
cost when provider 2 shares the number of QKD and KM wavelengths. It is clear
to explain that, with the high demands, the total deployment cost increases if
provider 2 performs over-sharing wavelengths and under-sharing wavelengths.
Figures 8(b) and (c) illustrate the optimal deployment cost obtained from the
SP model when provider 2 increases the number of shared QKD and KM
wavelengths. The main observation is that the providers experience lower costs
when they are in cooperation. This is due to the fact that the providers can
utilize the available resources of each other. As illustrated in Figs. 8(b)
and 8(c), when the QKD and KM wavelengths of provider 2 increase, provider 1
receives a benefit from using the extra QKD and KM wavelengths of provider 2
and provider 2 can utilize QKD and KM wavelengths shared in the resource pool
by provider 1. On the other hand, without cooperation, the QKD and KM
wavelength resources cannot be shared. Hence, increasing the available QKD and
KM wavelengths of provider 2 does not have an effect on the deployment cost of
provider 1. Another observation is that, with cooperation, the cost of
provider 1 can slightly increase when the QKD and KM wavelengths of provider 2
increase. This is because provider 1 does not contribute more to the QKD and
KM wavelength resource pool. Therefore, the cost-sharing of provider 1
increases when provider 2 contributes more resources.

####  VII-B5 Cooperation Formulation and Impact of Cooperation Cost

TABLE II: Payoff matrix for three coalition game

The Shapley value
---
| QKD wavelength pool | KM wavelength pool
Cooperation | Provider | Provider | Provider | Provider | Provider | Provider
structure | 1 | 2 | 3 | 1 | 2 | 3
$\mathbb{C}_{1}=\\{\\{1\\},\\{2\\},\\{3\\}\\}$ | 3,271,643.12 | 2,998,812.40 | 2,725,981.68 | 35,647,210.00 | 35,260,720.00 | 35,131,890.00
$\mathbb{C}_{2}=\\{\\{1,2\\},\\{3\\}\\}$ |  2,562,990.84* |  2,890,160.12* |  2,725,981.68* | 32,199,190.00 | 37,227,700.00 | 35,131,890.00
$\mathbb{C}_{3}=\\{\\{1,3\\},\\{2\\}\\}$ | 2,562,990.84 | 2,998,812.40 | 3,217,329.40 | 32,263,605.00 | 35,260,720.00 | 40,773,285.00
$\mathbb{C}_{4}=\\{\\{2,3\\},\\{1\\}\\}$ | 3,271,643.12 | 3,026,575.48 | 3,353,744.76 | 35,647,210.00 | 37,485,360.00 | 40,966,530.00
$\mathbb{C}_{5}=\\{\\{1,2,3\\}\\}$ | 2,108,660.56 | 2,572,245.20 | 2,899,414.48 |  26,300,931.67* |  31,522,686.67* |  35,068,271.67*

We evaluate the cooperation formation behavior of the three providers and then
scrutinize the cooperation costs which comprise shared QKD wavelength (i.e.,
$C^{\mathrm{qkd}}(s)$), shared KM wavelength (i.e., $C^{\mathrm{kmw}}(s)$),
and cooperation (i.e., $C^{\mathrm{coc}}_{s}$). Table II illustrates the
Shaplye values (i.e., the cost of each provider), which is $\vartheta_{s}(v)$
= $\vartheta_{s}(v)$ \+ $C^{\mathrm{qkd}}(s)$ \+ $C^{\mathrm{kmw}}(s)$ \+
$C^{\mathrm{coc}}_{s}$, for the QKD and KM wavelength resource allocation.
With cost management and QKD and KM wavelength allocation based on the SP
model, the stable coalition structure of the QKD wavelength pool is obtained
to be $\mathbb{C}_{2}^{*}$ (i.e., providers 1 and 2 cooperate).
$\mathbb{C}_{2}^{*}$ is a stable coalition structure since providers 1 and 2
achieve the lowest cost, while provider 3 does not have a better choice to
decrease its cost. In addition, the stable coalition structure of KM
wavelength pool is obtained to be $\mathbb{C}_{5}^{*}$ (i.e., providers 1, 2,
and 3 cooperate). $\mathbb{C}_{5}^{*}$ is a stable coalition structure since
providers 1, 2, and 3 achieve a lower cost compared with other coalition
structures.

(a) Transition cooperation structure under different cost of QKD wavelength.

(b) Transition cooperation structure under different cost of KM wavelength.

(c) Transition cooperation structure under different cooperation cost of each
provider.

Figure 9: (a) The transition cooperation structure under different costs of
QKD wavelength, (b) KM wavelength, and (c) cooperation of each provider.

We investigate how the costs of parameters (i.e., shared QKD and KM
wavelengths, and cooperation) affect the stable cooperation structure. In
Figs. 9(a), 9(b), and 9(c), the major observation is that the cooperation
structure will move towards $\\{\\{1\\},\\{2\\},\\{3\\}\\}$ which means that
providers work separately when the costs of shared QKD wavelength (i.e.,
$C^{\mathrm{qkd}}(s)$), shared KM wavelength (i.e., $C^{\mathrm{kmw}}(s)$),
and cooperation (i.e., $C^{\mathrm{coc}}_{s}$) increase. The reason is that,
without cooperation, the providers mainly utilize the resources (i.e., QKD and
KM wavelengths) in reservation and on-demand phases to obtain the minimum
cost. When the costs increase, it is not worth for the provider to cooperate
and increase their resources to accommodate demands. Alternatively, it is more
beneficial for the provider to work separately.

Figure 10: Stable coalitions under different sharing costs of QKD and KM
wavelengths.

Figure 10 illustrates the adaptation of the stable coalitions under different
sharing costs of QKD and KM wavelengths. With a large number of secret-key
rates (i.e., demands), each provider tends to move to the coalition structure
$\mathbb{C}_{5}=\\{\\{1,2,3\\}\\}$ when the sharing cost of QKD and KM
wavelengths is low. For example, with the sharing cost of QKD and KM
wavelengths being 600,000$ and 10,000$, respectively, the stable coalition is
$\mathbb{C}_{5}=\\{\\{1,2,3\\}\\}$ which is indicated by “1” in Fig. 10. This
is because each provider obtains the lowest cost, and the resource can be
better utilized with more providers in the coalition. However, when the
sharing cost of QKD and/or KM wavelengths increases, the stable coalition
tends to move to the coalition $\mathbb{C}_{1}=\\{\\{1\\},\\{2\\},\\{3\\}\\}$
which is indicated by “3” in Fig. 10. This is due to the fact that each
provider obtains a high cost when they cooperate to share their QKD and/or KM
wavelengths. As a result, there are no QKD and KM wavelength pools, and each
provider separately supports the demands.

##  VIII Conclusions

In the proposed QKD-SIC system, edge devices will need security requirements
in QKD nodes. Therefore, the QKD service providers must provide QKD resources
to achieve the minimum deployment cost. We have proposed a decision-making
scheme for QKD service providers in QKD-based SIC. The proposed scheme
comprises methods for resource allocation to secure semantic information
transmission of edge devices, cost management, and cooperation among QKD
service providers. In particular, the QKD service providers can cooperate and
establish the QKD resource pool. For the resource allocation, we have applied
the two-stage stochastic programming to resolve the solutions of the QKD
service providers to support edge devices to transmit the semantic information
that utilizes resources in the resource pool. For cost management, we have
incorporated Shapley values to share the deployment cost that is received from
the resource pool with cooperative QKD service providers in a fair and
interpretable manner. For the coalition formation among QKD service providers,
we have achieved the cooperation equilibrium solutions for the cooperative QKD
service providers who contribute to the resource pool.

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