QoE-Aware Cost-Minimizing Capacity Renting for Satellite-as-a-Service Enabled Multiple-Beam SatCom Systems

The advent of Satellite as a Service (SaaS) platforms has empowered satellite service providers (SPs) to rent portions of satellite capacity from infrastructure providers (IPs) to cater to the diverse demands of their users across multiple satellite services. To effectively manage costs and maintain a high Quality of Experience (QoE) for numerous concurrent connections, SPs should secure flexible capacity from IPs. However, the irregular and unpredictable nature of traffic demands from various applications complicates the capacity-renting framework. This study presents a dynamic capacity allocation framework that efficiently handles diverse traffic flows with varying arrival rates, aiming to minimize rental costs while meeting blocking probability and QoE requirements. Utilizing the $M_{t}/M_{t}/1$ queuing model and a continuous-time Markov chain, the technical designs are framed as a statistical optimization problem. In this context, the system waiting-queue lengths are estimated using the transient probabilities of Kolmogorov equations. Subsequently, cumulative distribution functions are employed to re-formulate this stochastic optimization problem into a convex form, which can be tackled through the Lagrangian duality method.Through extensive simulations and numerical assessments, we illustrate our method’s efficacy, with the proposed algorithm outperforming benchmarks by reducing costs by up to 9.85% and 3.1%.

Abstract-The advent of Satellite as a Service (SaaS) platforms has empowered satellite service providers (SPs) to rent portions of satellite capacity from infrastructure providers (IPs) to cater to the diverse demands of their users across multiple satellite services.To effectively manage costs and maintain a high Quality of Experience (QoE) for numerous concurrent connections, SPs should secure flexible capacity from IPs.However, the irregular and unpredictable nature of traffic demands from various applications complicates the capacity-renting framework.This study presents a dynamic capacity allocation framework that efficiently handles diverse traffic flows with varying arrival rates, aiming to minimize rental costs while meeting blocking probability and QoE requirements.Utilizing the   /  /1 queuing model and a continuous-time Markov chain, the technical designs are framed as a statistical optimization problem.In this context, the system waiting-queue lengths are estimated using the transient probabilities of Kolmogorov equations.Subsequently, cumulative distribution functions are employed to re-formulate this stochastic optimization problem into a convex form, which can be tackled through the Lagrangian duality method.Through extensive simulations and numerical assessments, we illustrate our method's efficacy, with the proposed algorithm outperforming benchmarks by reducing costs by up to 9.85% and 3.1%.
Index Terms-Time-varying queuing, capacity allocation, blocking probability, QoE-based optimization, satellite as a service.
Color versions of one or more figures in this article are available at https://doi.org/10.1109/TCOMM.2023.3334815.
Digital Object Identifier 10.1109/TCOMM.2023.3334815range of applications requiring high availability and resilience to critical areas that are unreachable by the current terrestrial networks [2], [3], [4].The growing attention on the dynamic SatCom systems supporting service-aware communications and seamless coverage, as well as its associated implementation costs, has inspired a number of big names in the industry to develop new "satellite-as-a-services" (SaaS) business platforms [5], [6].Such a platform comprises infrastructure providers (IPs) -the owners of the SatCom systems, and the service providers (SPs) which may rent certain satellite capacity from the IPs to provide different services to their end users [1], [5], [7], [8].This novel paradigm can enable the IPs to access new markets and it can release the SPs from investing a huge amount of money in order to build dedicated satellites [9].With the fluctuating demands from various applications and services, there's a growing interest in developing a flexible capacity renting mechanism for the SPs.This mechanism is expected to allow the SPs to obtain capacity as needed rather than committing to a fixed amount.Such flexibility can boost network utilization, enhance revenue for the IPs, and help the SPs save the costs while ensuring the user Quality of Experience (QoE) [10].However, to establish such an optimal capacity renting framework between the IPs and SPs, one should carefully manage and analyze a vast number of simultaneous data flows, each with distinct QoE requirements [11].Specifically, the SPs must swiftly make provisions on the required capacity while still guaranteeing the desired QoE for their users.On the SatCom IPs side, to realize the dynamic capacity allocation mechanism, the advanced digital transparent payloads (DTP) using "defacto" platform can be deployed [12], [13].Based on this, the satellites can assign capacity across beams effectively and route traffic of various types efficiently.
To ensure effective network resources renting from the IPs and optimal capacity allocation for customer communication services, the SPs have to cope with multiple challenges in maximizing their profit [9], [14].Herein, the business problems at the SPs include (but are not limited to) capacity estimation accommodating the irregular and unpredictable "time-varying" data traffic from heterogeneous apps/services, and efficient management of rented resources during both peak and off-peak periods to reduce costs [15], [16].Additionally, maintaining or expanding the customer base is a significant business challenge for both the IPs and SPs, and satisfying the diverse QoE requirements of customers is a key point of resource management that should be addressed carefully.Furthermore, the IPs and SPs should also establish an agreement detailing the speed of capacity adjustments and the maximum capacity SPs are allowed to use [7], [14].Consequently, the development of a dynamic, cost-effective, QoE-aware network resource renting mechanism in SatCom systems has emerged as an interesting and compelling research topic for both IPs and SPs.Facilitated by the challenging problem, this paper focuses on designing a new QoE-aware cost-minimizing capacity-renting framework in SaaS-enabled multi-beam SatCom systems.
In the realm of wireless communication, the QoE at users can be affected by a variety of factors, ranging from signal strength, data rate consistency, and connection reliability, to service interruptions [17].Among these aspects, a critical determinant of QoE in the SatCom schemes is the end-to-end latency where extended delay can significantly compromise a user's overall experience.While all QoE issues are essential, reducing the latent period is paramount [17].It is worth noting that the end-to-end data transmission latency encompasses both the signal propagation delay and the waiting time during which data resides in the system buffer before transmission [18], [19].However, in SatCom schemes, mitigating the propagation delay poses a formidable challenge due to the Line-of-Sight (LoS) connection and almost-fixed distances between the satellite and user terminals.This inherent physical limitation demands innovative solutions to ensure consistent and efficient communication, enhancing the overall user experience.
In light of this challenge, our study focuses on addressing the QoE issue by modeling it in terms of the waiting time of data packets in the system buffer.This approach underscores the significance of minimizing delays to improve overall user satisfaction.At a specific time instance, the waiting time of one beam is related to the stochastic queuing length which can be managed by dynamically allocating the capacity for the data transmission corresponding to this beam when the data arrival rates vary.In general, a longer queuing length returns a higher network congestion probability and a longer service time that the users may suffer because the data spends more time in the buffer before being processed and transmitted.When the queuing length approaches a limit, the waiting time may exceed users' tolerance; hence, the QoE rate can degrade.In addition, when the queue length violates the maximum buffer size of the system in some critical scenarios, the satellite system can be overloaded and the operating system can be blocked, resulting in packet loss.To cope with such an issue and conserve the required QoE, one demands a higher allocated capacity to increase the service rate.On another hand, letting the system operate with very short queuing lengths may imply that an over-needed amount of capacity is allocated, and the renting cost must increase.Then, our work aims to propose a novel dynamic capacity planning model that minimizes the total renting cost of SPs while maintaining a target blocking probability of the system and a target queuing delay requirement of customers.

A. Related Works
The majority of existing frameworks for SatCom resource allocation have primarily focused on maximizing overall power and spectrum utilization efficiency.Different techniques have been used, such as non-convex optimization for flexible power and capacity assignment [20], beam illumination and selective precoding [21], and joint beam selection and precoding [22].The work in [20] focused on satelliteuser association-oriented capacity allocation to minimize the total uplink transmit power for integrated satellite-terrestrial networks (ISTN).Research on QoE-aware dynamic capacity allocation to maximize user satisfaction in Orthogonal Frequency Division Multiple Access (OFDMA) terrestrial networks considering time-varying channels has been conducted [23], [24].These approaches aim to satisfy the overall demand by improving power, capacity, or both utilization efficiency.However, all previous works consider average beam demand which does not change over time.In addition, the cost of satisfying this demand, the QoE of users, and the amount of system capacity that remains unused is not well-documented in the literature.Other studies have discussed profit opportunities associated with 5G infrastructure dynamic leasing [14] and revenue management in SatCom systems [25].
Previous research has also applied different queue models for end-to-end latency estimation, capacity allocation, packet loss minimization, and buffer bloat prevention in various wireless communication systems.For example, the //1 queue model has been used to estimate transient queue length [26], latency estimation [27], [28], [29], the // queue model for dynamic buffer sizing [30], the //1 queuing model for Quality of Service (QoS) analysis [31], and the //∞ queue model to estimate the minimum required system capacity [32].While these studies use different queue models to maintain QoS in different wireless systems, they fail to consider the QoE of users and the time-varying nature of average arrival rates.
In other areas, time-varying queue model-based resource allocation has been explored [33].For example, in [34] and [35] the application of different time-varying queue models in large-scale service systems such as customer contact centers and hospital emergency departments is discussed.In these works, iterative staffing algorithms (ISA) are developed to optimize the staff levels at the customer center to satisfy the stochastic waiting time requirement of customers.Herein, the state-of-the-art   //() queue model is exploited where the   indicates time-varying Poisson arrivals, and the () indicates time-varying servers and the  indicates a constant serving capacity of the servers in every staffing interval.Then, an efficient time-varying human-resource-management framework is established by iteratively determining the staff level at a specific time.Similarly, the authors in [36] also employed   //() to develop a Deep-Reinforcement-Learning based framework to optimize the beam-hopping strategy adapting with time-varying data traffic flows coming to the multiplebeam SatCom systems.Next, [37] describes how time-varying Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
//∞ queue models can be applied to staffing and capacity planning of cloud services and protective equipment management in hospitals during the outbreak of a disease.All works given in [34], [35], and [37] focus on a fixed processing rate at one serve (staff).On a different approach, the changeable processing rate mechanism is studied in [38] where the   /  /1 queue model is used for efficient resource allocation in industrial production by estimating the stochastic probability of the system queue length.However, this work mainly focuses on developing the admission control strategy for network-slicing systems.Consequently, there is a very limited number of works considering multiple traffic flows accessing satellite systems as time-varying queue models and exploiting this to develop a QoE-aware dynamic capacity renting and allocation mechanism for the SPs to in multiplebeam SatCom systems.Therefore, the work in this paper aims to fill this gap in the literature.

B. Contributions
Our paper aims to propose a novel QoE-aware flexible capacity renting framework for SaaS-enabled multiple-beam SatCom systems to effectively manage the renting costs at the SPs.By employing the stochastic queuing theory, we first formulate the problem as a stochastic optimization problem.This allows us to examine the impact of system and user requirements on the rent cost and the trade-off between them, thereby assisting SPs in predicting the capacity they need to request IPs to maintain a satisfied customer base while minimizing rent costs.Our key technical contributions in this article can be summarized as follows: • First, we express the traffic flows using a time-varying (  /  /1) queuing model and estimate the stochastic queue length of packets waiting in the system by using the continuous-time Markov chain (CTMC).The analysis results are then employed to formulate a stochastic optimization problem for QoE-aware dynamic capacity allocation that includes queue status-based dynamic spectrum sharing among adjacent beams of the same cluster to minimize cost.The problem aims to help SPs to be able to efficiently allocate capacity and reduce un-utilized capacity as well as their rent costs.• Next, we estimate the stochastic blocking probability and the probability of violating the waiting-time requirement over the observation period, which allows SPs to predict the impact of capacity allocation decisions on user experience.We further analyze the trade-off between maintaining user satisfaction and capacity rent cost in Sat-Com systems will assist SPs in determining the optimal balance between these two objectives, based on which we provide a closed-form solution based on Lagrangian duality making use of the estimated blocking probability.• For comparison purposes, we introduce a greedy algorithm and modify the ISA frameworks in [34] and [35] to suit our design requirements.The proposed algorithms are validated through numerical results and Monte Carlo simulations using practical simulation parameters.The numerical and simulation outcomes have effectively confirmed the theoretical soundness of our proposed frameworks.In summary, our proposed approach is based on the stochastic queuing theory and aims to assist satellite service providers in predicting the capacity they need to maintain a satisfied customer base while minimizing rent costs by using a QoEaware dynamic capacity allocation model.Preliminary studies related to this objective were presented in [1].This current work further extends our previous results by regarding multi-beam settings with cross-beam interference avoidance constraints, presenting robust theorem analysis, and delivering more solid numerical and simulation demonstrations alongside added benchmark comparisons.The rest of the paper is organized as follows.In Section II, the system model and problem formulation are described.The queuing stochastic analysis and problem approximation are discussed in Section III.In Section V, the numerical results are discussed.Finally, Section VI concludes the paper.For notation, scalars are represented in a normal italic font such as  and , while the vectors and matrices are in bold, i.e., W and u.In addition, sans serif font is utilized to denote the "suffix" abbreviations, such as,  total ,  max .For ease of reference, a list of key notations used in this paper is provided in Table I.

II. SYSTEM MODEL AND PROBLEM FORMULATION
We examine a SaaS platform in a multi-beam geostationary satellite (GEO) communication system where the overall available capacity of a GEO satellite is owned by an IP that can be rented by SPs 1 The study focuses on a scenario where a specific SP2 rents a time-varying amount of capacity from the IP to provide broadband services to multiple users randomly distributed across  beams.Let  () (bps) represent the total capacity rented by the SP from IP to serve all traffic flows across all beams.Each beam is assigned a portion of this capacity, denoted by   (), which can range from 0 to  (), i.e. 0 ≤   () ≤  ().The network operates in a time-slot manner, with each time slot having a duration of  TS as the transmission time.Due to possible processing speed capability limitations of DTP reconfiguration combined with the consequent signaling through the tracking, telemetry, and command system (TT&C) links [12], it is assumed that   () remains constant for a cycle duration of  time slots.  () can only be reset at time-slot indices   ∈ {0, , 2, . . .,  , . ..} and   () =   ( ) if  ∈ (( − 1) TS ,   TS ] where  = 1, 2, . . ., .We refer to   ( ) as    , which represents the allocated capacity of beam  in cycle .The design framework considers a monitoring period of  cycles, equivalent to a total operation time of  =   TS (seconds).
Let  () stand for the cost function corresponding to  (bps) that the SP rents from IP. Normally,  () is a monotonic increasing function with respect to  which reflects the idea that renting more capacity incurs more rental cost.Then, the total renting cost that the SPs have to pay to the IP during  seconds can be expressed as,

A. Capacity Allocation and Frequency Reuse
To address the strong cross-beam interference between adjacent beams, a dynamic multi-color capacity allocation policy is employed in this multi-beam transmission system.This differs from traditional color-reuse schemes, where the spectrum is equally distributed.Here, the spectrum is freely allocated to beams, ensuring that different and non-overlapping frequency bands are assigned to two arbitrary adjacent beams.It follows that the sum of capacity assigned to any cluster of  adjacent beams must not exceed the maximum available spectrum band capacity.For instance, Figs.2a and 2b demonstrate settings of 3 and 4 adjacent-beam clusters with non-overlapping spectrum.
For a specific beam pattern, let  be the number of available -adjacent-beam clusters and U ∈ {0, 1}  × be the adjacency matrix.In the   ℎ row of U, only  elements corresponding to the indices of  adjacent beams of cluster  are set to one, while the others are set to zero.The dynamic multi-color reuse capacity allocation requirement can then be described by the following constraint: where W  =  1  , . . .,    , . . .,     ,  total (bps) indicates the maximum reusable capacity available per cluster, and ⊮  ×1 stands for a one-vector with size of  × 1.

B. Queuing Model
From user perspective, a single device can generate multiple data packets that correspond to various applications operating on it concurrently.It's assumed that packets corresponding to a specific application type have identical packet sizes.Consequently, the number of packets resulting from a particular application, which is run by several devices concurrently, can be consolidated into a traffic flow with a projected arrival rate.This number of packets adheres to a random process, with the estimated arrival rate serving as the mean value [39], [40], [41].This work deals with heterogeneous time-varying traffic rates generated by different applications such as voice, video streams, and web browsing.To handle this, the demands from end users are modeled as multiple queues accessing each beam.The data flows are classified based on their corresponding statistical parameters, such as arrival rates and packet lengths.In this model, the arrived packets are processed based on the basic first-come first-served strategy.
Consider  data-flows corresponding to  services tending to access each of  beams as shown in Figure 3.We further assume that the flow  carrying data packets of   -bit length comes to beam  at a time  following an independent Poison process3 [39], [40], [41], [42] with a time-varying arrival rate of   (), i.e.,   () is the number of packets that changes over the time.The total arrival rate in bits per second to a beam becomes Suppose that a packet of  bits is a normalized processing unit in this SatCom system.As a result, the arrived data can be divided into packets of  bits for transmission.Consequently, the total arrival rate in terms of the number of packets at beam  can be determined as Additionally, the service rate for all flows entering beam  at any time , which is estimated as the number of packets based on the corresponding allocated capacity, can be expressed as, Remark 1: The users' demands in terms of QoS can be captured by the traffic flow model.For instance, a higher transmission rate can be equated to an elevated arrival rate (   () is scaled-up) or an extended packet size (  is set to a larger value).Moreover, rapid time-varying requirements can be depicted by a more fluctuating function of the time-varying arrival rate setting,    ().

C. QoE Requirement and Problem Formulation
In this section, we aim to ensure that the allocated capacity for every beam does not become a bottleneck violating the required QoE.Here, we probabilistically model the QoE requirements in the manner of transmission delay (waiting time) for the users in each beam with the queuing length of packets available in the data buffer corresponding to that beam transmission.
Specifically, let   () denote the queuing length of data packets stored in the buffer of beam  at time .Then, one denotes  , () =  (  () = ) as the probability that there are  packets in the buffer of beam  at time .As discussed in [33], due to the time-varying arrival rate models,  , () can be expressed as a function of .Assuming that users in all beams have the same waiting time tolerance which corresponds to a QoE threshold of queuing length  QoE .The experience of network utilization is considered "acceptable" by the users if the probability of that such threshold is violated is less than a commitment factor PQoE , i.e., 0 < PQoE < 1.Therefore, the design in this work focuses on keeping the probability that the queue length surpasses  QoE packets over the window time of [0, ] less than PQoE for all the beams, which can be expressed as One further assumes that the length of every beam buffer is limited by  max , so-called the maximum buffer length.Herein, it also needs to ensure that the queue length at every beam does not surpass  max beams as much as possible, otherwise, the processing of all data flows to the particular beam will be blocked.Regarding the network admission requirements, our design needs to maintain the blocking probability below a predetermined threshold for every time slot [42].This requirement can be cast by the following constraint, where PBlk is the target blocking probability.Taking into account that    () ≥  = 1 −  =1  , (), our technical designs can be formulated into a statistical optimization problem as follows.
where W represents the matrix containing all    's.As observed, this presents a stochastic optimization problem Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.wherein the constraints correspond to a random process.The primary challenge in resolving this problem stems from the statistical formulas articulated in constraints (8b) and (8c).In this context, while the problem data remains uncertain, the queuing model incorporating the Poisson process, as discussed in Section II-B, serves as the foundation for our solution framework.
Remark 2: It is worth noting that  () is an increasing function so the SP has to pay more if it rents more capacity.Hence, problem (8)

III. QUEUING STOCHASTIC ANALYSIS AND PROBLEM APPROXIMATION A. Time-Varying Queuing Stochastic Brief Discussion
The total demand in each beam, which is the sum of arrivals of all data flows, varies over time and can be modeled as a continuously varying arrival rate.Consequently, the number of packets in the buffers in each time slot is a stochastic process.To model this behavior, we represent the number of packets at each time slot as a CTMC, as shown in Fig. 4.This stochastic process can be described using a system of ordinary differential equations (ODEs) known as the Kolmogorov equations [33], and the queue length follows a birth-death process.The transient solution of the Kolmogorov equation yields the stochastic queue length values as: where  0, () indicates the derivative of  , () at time .However, since the Kolmogorov equation does not yield explicit solutions for the transition probabilities, various more suitable methods have been developed to approximate the solutions, as described in [33], [38], and [43].Denote () as the queue length at time .Then, the transient probabilities can be approximated by a cumulative distribution function given by: The expected value of the queue distribution at the buffer of beam  provides, The beam utilization at any time slot t is calculated as, The continuous-model approximation is exact in steady-state [38].Therefore, the approximated probability of the availability of  packets in the buffer system of beam  at any time slot  can be expressed as

B. Problem Approximation
This section exploits the continuous-model approximation in (14) to express the stochastic queue length in terms of capacity and express the constraints as a function of capacity    .Expressing   () in ( 5) in terms of    for  ∈ (( − 1),  ) and  ∈ {1, ..,  }, one can rewrite (14) as, Since the beams can not serve beyond their maximum capacity, the system utilization constraint needs to satisfy   () ≤ 1 [38].Otherwise,   () > 1 indicates the system is over-congested and users are blocked from accessing the corresponding beam.Letting Ω  denote the set ( − 1),   , one can restate problem (9) as,

IV. DYNAMIC CAPACITY ALLOCATION DESIGN
In this section, we aim to explain the step-by-step approach to find the optimal dynamic capacity allocated across all beams.

A. Problem Convexity Characterization
In order to solve problem (16), we first characterize its convexity.To begin with, we first define the lower bound capacity amount required per cycle in which   () is fixed by considering the following proposition.
Proposition 1: Constraints (16b) and (16d) in problem ( 16) can be merged into one constraint as Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
where  17) is a linear constraint, we state the convexity of problem (16) in the following theorem.
Theorem 1: Problem ( 16) can be transformed into the following optimization problem which is convex, (17), where   () =  TS / −    /   QoE +1 .Proof: The proof is given in Appendix B Thanks to Proposition 1 and Theorem 1, one can state that ( 16) is equivalent to convex problem (18).In the following section, a dynamic resource allocation algorithm is proposed by developing an optimization-based approach to obtain the optimal solution of this problem.

B. Duality-Based Dynamic Capacity Allocation Algorithm
where   and   are the Lagrangian multipliers; u  stands for the vector generated from the -th row of U; and W  = [ 1  ,  2  , . . .,    ]  .Then, the dual function of    can be defined as the minimum of the Lagrangian function as, g() = min W L (W, , ) s.t.(17).(20) To find the best lower bound that can be obtained from the Lagrange dual function, the dual problem can be written as, Since problem ( 18) is convex, the dual-gap between the primary and dual problem is zero [44].In the following, one will describe a searching approach to define the optimal solution.In particular, the dual problem is always convex, g(  ,   ) can be maximized by using the standard sub-gradient method where the dual variables   and   are first initialized to random values in the dual feasibility region of   > 0,   > 0, g(  ,   ) > −∞ [45].The dual variables can be iteratively updated as follows: and Given   [ℓ ] and   [ℓ ] , define (   ) ★ 's as in ( 25) and ( 26).
where the suffix [ℓ] represents the iteration index,  [ℓ ] is the step size, and [] + is defined as max(0, ).This sub-gradient method guarantees the convergence for any initial primary point of ( Proposition 2: The optimal solution of ( 24) is given as where, is as defined in (49) and  , is the element on row  and column  of matrix U.
Proof: The proof is given in Appendix C. 3) Proposed Duality-Based Algorithm: Thanks to the duality approach, the optimal solution of problem ( 18) can be obtained by alternatively solving problem (24) -the righthand-side of (20) -as presented in Proposition 2, and updating Lagrangian multipliers   and   as in (22) and (23) in each iteration until the convergence.The optimization-based approach is summarized in Algorithm 1 where the iteratively solving process can be stopped when the gaps

C. Greedy Algorithm
To mitigate the complexity of solving our problem given in the previous section, this section introduces a straightforward and efficient greedy algorithm.As evident, the primary challenge in addressing problem 16 arises from the coupling Algorithm 2 Greedy-based Dynamic Capacity Allocation Calculate  ,1 ,  ,2 ,  ,3 .

6:
end for 7: end for 8: Return W ★ . of all    throughout the entire time window  as illustrated in (16c) for each beam.In particular, directly handling this constraint needs to consider the average probability of QoE violation across the entire time window .To simplify this process, we disregard the average value, prompting the system to meet the QoE requirement at every given moment.As such, constraint (16c) is replaced by a more stringent version, which is presented as follows: This new constraint is similar to (16b).By employing the same approach handling (16b) given in Proposition 1, we first define the maximum arrival rate over one cycle as, Then, the new constraint in (27) can also be translated into, Thanks to Proposition 1, we are able to estimate the required capacity of each beam over every cycle by taking the maximum capacity that satisfies the problem constraints.That is, The greedy algorithm is summarized in Algorithm 2.

D. Other Benchmark Algorithm
This subsection introduces another benchmark solution for comparison purposes, which is developed by adapting the ISA given in [34].The ISA is well-established for time-varying human resource management to satisfy the stochastic waiting time requirement of customers.In [34], the   //() queue model is employed where the serving capacity of one server (or employees) is fixed () while the number of servers (employees) can be varied over the time.Herein, () represents the number of allocated employees at time , and the ISA is designed to determine () coping with the time-varying customer arrival rate efficiently.
As can be seen, the queuing model utilized in [34] is different from our   /  /1 scheme which is related to one server with variable serving capacity.Hence, in order to modify this work to address our problem, we assume that the capacity amount allocated to beam  in cycle  can be represented by a number of "fixed capacity packages".Let  0 (bps) be the capacity of one such package, and    denote the number of capacity packages assigned for beam  in cycle .Then, we have, In addition, the processing rate corresponding to one package can be estimated as Now, we can employ the ISA to optimize {   }'s by regarding the constraints (8b) and (8c) instead of the stochastic waiting time as designed in [34].Specifically, at the initialization, {   }'s are randomly selected and then adjusted over iterations.There are two components to define    in every iteration , the first is   bl, () -being updated according to the blocking probability requirement, and the latter is   QoE, () -being adjusted due to QoE-related demand.Here, constraint (8c) is considered for every cycle to update   QoE, ().To ensure compliance with the blocking probability requirement stated in (16b),   bl, () is updated as follows: where ⌈.⌉ and ⌊.⌋ indicate the ceil and floor operators and ψ  bl, () is the blocking-probability influence factor corresponding to    () and PBlk .In particular, ψ  bl, () can be given as where   bl, () indicates the maximum blocking probability during cycle  corresponding to    ().Specifically,   bl, is given as follows: Similarly, the   QoE, () in iteration  is updated as follows: where ψ  QoE, () is a QoE-related influence factor corresponding to    () and PQoE .Here, ψ  QoE, () is expressed as, where   QoE, () indicates the maximum probability of violating the target QoE requirement during cycle  with    () as Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
In addition, to meet the condition   () ≤ 1 of (16c), it is essential to carefully set    in a way  0    ≥   () for every  ∈ Ω  .Then,    can be updated as Accordingly, the adapted ISA is summarized in Algorithm 3.

E. Complexity Analysis
The complexity of Algorithm 1 arises from calculating    and also processing a number of loops in each of that   [ℓ ] , [ℓ ] , and (   ) ★ are estimated as given in ( 22), ( 23), (26), respectively.As given in (17), the complexity of estimating    is the order of O   2 max .Regarding the effort of estimating   (   ) and  =1   (   ), the complexity due to equation ( 22) can be given as O  2  QoE .Similarly, the complexity due to equation ( 23) is the order of O ( ).Next, the complexity due to equation ( 26) is associated with the summation of  elements and a power-of- QoE calculator.Hence, the computation effort for calculating Ŵ   's corresponding to  beams and  cycles can be the order of O ( ( +  QoE )).One assumes that implementing Algorithm 1 required ℓ (1)  iterations to get convergence and obtain the solution, the overall complexity of the algorithm taking the highest degree polynomial becomes This shows that our problem can be easily solved and converged in a polynomial amount of computation time.Considering the greedy algorithm, one can see that the outcome can be obtained by estimating  ,1 ,  ,2 ,  ,3 .According to (17) and ( 29), the required computation effort for implementing Algorithm 2 can be described as

TABLE II SIMULATION PARAMETERS
Consequently, we study the complexity of the ISA method.
As summarized in Algorithm 3, the process of ISA method encompasses multiple loops of estimating    through equations (33) and (36).In each loop, the heaviest task of determining    replies on calculating   bl, and   QoE, in ( 35) and (38), respectively.Therefore, the complexity of ISA can be given as This has shown a relatively low computation effort.While both the greedy and ISA approaches are simpler than the duality method, we favored the latter because of its superior efficiency, as showcased in Section V-B.Moreover, the duality method, unlike some algorithms which may require exponential time to converge, promises convergence in polynomial time.This not only ensures more predictable computational demands but also bolsters its viability as an optimal approach for real-world applications.

V. PERFORMANCE EVALUATION AND NUMERICAL RESULTS
In this section, we simulate and analyze a time-varying queuing model to estimate the stochastic blocking probability over time and to find the optimal capacity that can satisfy the defined QoE and blocking probability requirements.

A. Simulation Setup and Parameters
In this subsection, we conduct a Monte Carlo simulation consisting of 5000 independent data trials.The simulation includes generating random arrival rates based on a timevarying Poisson process [39], [40], [41] for various time slots, as well as assigning time-varying service rate values for different cycles.In each iteration, the arrival rate function is chosen to represent time-varying demand that varies between zero and the assumed system's maximum capacity, using a sinusoidal representation as described in [55].Three data flows are generated for every beam and the corresponding timevarying arrival-rate functions for beam  are given as where suffix  stands for data flow  and    is an influencing factor corresponding to the average number of arriving packets at time ,    is a positive number influencing the periodicity of the arrival time of packets, and    's are phase shift angles.Here,    is selected randomly in a range of [1,4] so that the average of the total demand (  =1    ()/ TS ) is not greater than the assumed beam capacity of 700 Mbps as presented in [49];    is selected randomly a range of (0, 1] as in [56] and [57], while    is chosen randomly within [0, 360] to ensure all flows have different peak and off-peak periods.The remaining parameters provided in Table II are adopted for all simulations unless specified otherwise. By utilizing the mean arrival rate function obtained from the Monte Carlo trials and the service rate values obtained from the optimal allocated capacity according to (5), we compute the queue length, blocking probability, and the probability of violating QoE requirements for each time slot .The simulation results as compared to the analytical results are explained in the next sub-section for different values of the considered parameters.Regarding the renting cost function  (), we exploit a linear form as  () =  [7], [58] where  represents the price per capacity unit, i.e., Euros/Mbits.Additionally, Fig. 6 illustrates the diagram of obtaining numerical, analysis, and simulation results in this section.As can be seen, Algorithms 1-3 are first employed to determine the rented capacity solutions based on which the costs can be calculated.They are so-called numerical results which are demonstrated in Figs.7-13.Furthermore, the capacity outcomes are utilized to obtain the analysis results by using (15), which are illustrated

B. Numerical Results and Discussion
This section first shows the running time and convergence of the proposed algorithm, then we investigate the effect of varying parameters, namely  QoE , PBlk ,  max , PQoE , and , on the optimal allocated capacity and total renting cost to meet the time-varying demand.Assuming both polarizations are used in all beams, we can put a minimum of 3 adjacent beams per cluster to avoid cross-beam interference [50].Hence, our numerical results are based on 3-beam clustering framework arrangement with 10-beam footprints as described in Fig. 5.In particular, there are 10 clusters in this beam pattern setting which are (1,4,5); (1, 2, 5); (2,5,6); (2,3,6); (3,6,7); (4,5,8); (5,8,9); (5,6,9); (6,9,10) and (6,7,10).Accordingly, constraint (2) corresponding to cycle  is demonstrated as 1 0 0 1 1 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 1 1 0 0 1 Fig. 7 shows the running time of duality, greedy and ISA algorithms with varying values of , using Matlab (tic-toc method) on an Intel(R) Core(TM) i7 processor.The plot for the duality algorithm indicates that it is close to the expected polynomial time complexity which is proportional to power 2 of the size of cycles, which confirms the analysis results given in Section IV-E.Next, we present the convergence plot of our proposed algorithm alongside the benchmark ISA in Fig. 8.The figure in 8b illustrates the changes in capacities Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.assigned to the beams when Algorithm 1 is implemented.As shown, the assigned capacity for each beam decreases before stabilizing at its minimum value across iterations.Moreover, different beams necessitate various numbers of iterations to reach convergent capacity values.Similarly, the fluctuations in the total number of packages allocated to all beams in Fig. 8a are the result of employing Algorithm 3 across iterations.The consistent increment or decrement by 1 package in the plot Here, one assumes that the capacity of one package is set to 5 Mbps for implementing ISA algorithm, which is equivalent to the scenario of the transmission over a 1 MHz sub-channel with 32-QAM modulation at acceptable signal-to-noise ratio [59].The study results indicate that a system catering to users with a higher tolerance for waiting requires less capacity and thus incurs lower costs.Moreover, higher values of  QoE and  max can lead to lower required capacity, resulting in smaller costs.
Figs. 9b and 10b show the sum of optimal allocated capacity per cycle and total rental cost of all beams, respectively, when varying  max while keeping other parameters ( QoE = 15,  = 12, PBlk = 0.01, and PQoE = 0.05) constant.The results obtained through the Lagrangian duality and ISA method indicate that allowing more packets stored during congestion requires less allocated capacity.In contrast, the greedy algorithm assumes the maximum capacity required to meet the target  QoE requirement, neglecting the buffer size and its influence on capacity allocation.The figures also reveal that a larger buffer size and greater queuing delay tolerance result in lower allocated capacity and rental costs.
Fig. 9c demonstrates the allocated capacity per cycle at various PBlk values using the Lagrangian duality, ISA, and greedy algorithms at  max = 32,  QoE = 20, PQoE = 0.05, and  = 12.The Lagrangian duality and ISA approaches show that systems with lower blocking probability requirements have greater allocated capacities than those with higher blocking probabilities.However, the greedy algorithm provides the same optimal capacity for all PBlk values, as it only considers the maximum value associated with queuing delay violations and not with blocking probability.This makes our model more efficient in accounting for blocking probability.Fig. 10c shows the relationship between blocking probability and total rental costs at different  QoE values.As can be seen, the lower blocking probability requirements return higher allocated capacities and costs.For example, based on the obtained results, an increase in PBlk from 0.01 to 0.05 results in a cost Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.reduction of approximately 5%, while an increase from 0.01 to 0.1 leads to a reduction of 7.23%.
Next, Figs.9d and 10d depict the relationship between PQoE and total allocated capacity as well as rental costs obtained by implementing the three algorithms.As expected, the outcomes of all three algorithms imply that a smaller probability of violating the QoE requirement necessitates the SP to allocate a higher capacity and, conversely, less capacity for a higher probability of violation.For instance, the obtained result indicates, an increase in PQoE from 0.01 to 0.05 results in a reduction of the renting cost by 6.11%.
Figs. 11 and 12 present the total allocated capacities per cycle and the associated renting costs for the three algorithms for various values of .From the figures, it's evident that when  rises, there's a decrease in the optimal capacity allocation.This trend suggests that a more adaptive system can fulfill demand using less capacity, leading to reduced costs.For instance, results show that when  increases from 6 to 18, the renting cost drops by 11.38%, and a surge from 12 to 18 results in a decline of 6.1%.Parameters for this analysis, including  max = 32,  QoE = 20, PBlk = 0.01, and PQoE = 0.05, were consistently considered.
The results, as depicted in all the above figures, highlight the superiority of the duality method over both ISA and greedy algorithms in terms of flexibility and adaptiveness.For instance based on the obtained result and for the case where  max = 32,  QoE = 20, PBlk = 0.01, PQoE = 0.05 and  = 6, the proposed model can meet the requirement at a 9.85% and 3.1% lower cost compared to the greedy and ISA algorithms.However, as  increases, the greedy algorithm becomes as efficient as the proposed method, as larger values of  represent nearly immediate capacity changes, which are ideal conditions for greedy algorithms to perform well.
We need to examine if the capacity is shared among beams based on corresponding demand.We also need to assess whether the proposed model meets the requirements discussed in Section II-C.Fig. 13 displays the demand per beam as a function of the mean arrival rate (  ()/ TS ) and the optimal allocated capacity to different beams in a random cluster consisting of beams 2, 3, and 6.In every cycle, a higher capacity   The Figs. 14a, 14b and 15 depict the mean blocking probability over time for a randomly selected beam, the average of the mean blocking probability of all beams, and the mean QoE requirement violation probability respectively.The target blocking probability of PBlk = 0.01 was set with the parameters  max = 32,  QoE = 20, PQoE = 0.05, and  = 6.The results show that all techniques satisfy the blocking probability requirement.Figures 14a, 14b, 15, and Table III clearly illustrate the close alignment between our analytical and simulation methods.Furthermore, the minor differences in   average blocking probability and queue length across various beams, as depicted in Figs.16 and 17, affirm the accuracy and validity of our proposed method.This shows an effective integration of the inherent randomness and stochastic nature of user traffic demands for capacity management.Additionally, Table III shows that the queuing delay requirement is duly met across all instances of  surpassing the threshold  QoE .Interestingly, the greedy and ISA algorithms result in lower blocking and QoE violation probabilities than the proposed duality method.Similarly, the greedy and ISA algorithms result in lower queuing length than the proposed duality method as shown in Fig. 17.This is because of that these benchmarks return the higher assigned capacity for beams.

C. Discussion on Feasibility of Practical Implementation
The simulation results underscore the computational efficiency of the proposed method, demonstrating its capability to adeptly manage the dynamic and fluctuating traffic demands inherent to SatCom systems.Specifically, as illustrated in Fig. 7, the method takes mere minutes (under 100 seconds when  ≤ 12) to determine the optimal amount of rented BW over a one-hour time window.It's pertinent to highlight that this execution time can be further trimmed when the proposed algorithm runs on a more powerful industrial-grade computer.Such a run-time is practically viable, allowing the SPs to ascertain the necessary capacity prior to entering rental agreements with the IPs.Another pivotal factor for the successful implementation of our proposed algorithm in the practical systems is a deep understanding and accurate estimation of the time-varying arrival rate function, specifically,    () 's.However, within the scope of this study, Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
one does not delve into traffic model estimation.Several existing studies, including those by [60], [61], and [62], have dedicated efforts to unpack this intricate domain.Their insights suggest that a machine-learning-based model, which adjusts based on real-time data and historical patterns, could be the most efficient way to determine the stochastic information of the network traffic.The prospect of capacity management informed by traffic flow predictions is intriguing and is a topic we intend to address in future studies.

VI. CONCLUSION
In conclusion, this paper has proposed a novel and efficient dynamic capacity allocation model for multi-beam GEO satellite systems.The method aims to minimize the renting cost while ensuring the target blocking probability and QoE violation requirements.Traffic arrivals are modeled using the   /  /1 queueing model, and the stochastic queue length was estimated using the CTMC.The optimization problem has been solved using the Lagrangian duality method and the obtained results demonstrate its effectiveness and superiority over the benchmark ISA and greedy algorithms.Future work can include the extension of this model to a more complex network architecture, for example, the integration of the proposed model with terrestrial networks for a more comprehensive solution.Additionally, exploring the potential of machine learning techniques to further optimize the capacity allocation process can also be of great interest.Furthermore, considering the impact of flow prioritization and different traffic characteristics on the proposed model can provide deeper insights into the system's behavior and enhance its practicality.

Manuscript received 6
April 2023; revised 4 September 2023 and 10 November 2023; accepted 10 November 2023.Date of publication 20 November 2023; date of current version 19 March 2024.This research was funded in whole by the Luxembourg National Research Fund (FNR), with two granted projects corresponding to two grant references IPBG19/ 14016225/INSTRUCT (INSTRUCT project) and C23/IS/18073708/SENTRY (SENTRY project).An earlier version of this paper was presented at the IEEE 96th Vehicular Technology Conference (VTC-Fall 2022) [DOI: 10.1109/VTC2022 Fall57202.2022.10013084].The associate editor coordinating the review of this article and approving it for publication was N. Miridakis.(Corresponding author: Teweldebrhan Mezgebo Kebedew.)

]. 2 )
Solving the Optimization Problem Related to Dual Function: This section focuses on minimizing the Lagrangian function, min W L (W, , ) s.t.constraint (17). .

Fig. 10 .
Fig. 10.Total renting cost of the optimal allocated capacity to all beams.

Fig. 11 .
Fig. 11.Sum of the optimal allocated capacity of all beams at different values of .

Fig. 13 .
Fig.13.Optimal capacity allocation to beams in the same cluster for different arrival rates.

Fig. 17 .
Fig. 17.Mean value of queue length over time-window  and 5000 trials due to beams 1 − 5.

TABLE III PQoE
(2)UES AT  ≥  QoE = 20 THRESHOLD USING OPTIMAL CAPACITY is assigned to the beam with the highest arrival rate, which corresponds to the highest demand, as demonstrated in the figure.This allocation meets the requirements in equation(2).For this demonstration, the parameters used are  max = 32,  QoE = 20, PBlk = 0.01, PQoE = 0.05, and  = 12.