Category: 3. Business

Analytical modelling of the edge-IoT environment as a single-channel queueing system with controlled shift in distributions

In modern edge-oriented IoT environments, there is an increasing need for adaptive load regulation that considers constraints related to latency, energy consumption, and traffic class. This is particularly relevant for systems such as URLLC or mMTC, where response speed or transmission stability is critically important. Under such conditions, the problem of formal control over waiting time and queue length becomes pressing, without disrupting the overall flow structure. One of the key solutions involves introducing a shift parameter into the arrival and service distributions. For this purpose, let us consider single-channel queueing systems of the A/B/1 type in Kendall’s classification, where the symbols A and B denote the probability distributions of inter-arrival and service intervals respectively, and 1 indicates the number of service channels. The single-channel model is the most appropriate abstraction of an edge node. It reflects the hardware constraints of NB-IoT, LoRa, or BLE devices, which process requests sequentially rather than in parallel. Moreover, this model preserves the mathematical clarity of the analytical apparatus, particularly when using the spectral method³¹ for the Lindley equation³².

In A/B/1 queueing systems, the probability density functions (alpha left( t right)) and (beta left( t right)), corresponding to the distributions of inter-arrival times and service durations, respectively, are defined as time-shifted functions (tau), where (tau>0) is a controllable parameter characterising the minimum delay in the system

Here, (tilde {alpha }left( t right)) and (tilde {beta }left( t right)) represent the original (non-shifted) density functions of inter-arrival and service intervals, respectively. The introduction of the shift (tau) enables continuous adjustment of the expected values of the corresponding stochastic variables without altering their functional form. As a result, a controlled reduction in the coefficient of variation occurs, which is one of the main factors influencing the mean waiting time. Consequently, the parameter (tau) becomes a controllable variable that can be used to shape delays as an optimisation tool, for instance, to balance between QoS classes or minimise buffer overflow. Henceforth, it is assumed that the base densities (tilde {alpha }left( t right)) and (tilde {beta }left( t right)) belong to the class of functions that admit Laplace transformation. This is a critical requirement for applying the spectral method, which serves as the principal analytical tool used to derive the numerical-analytical characteristics of queue waiting time. As a result of introducing the controllable shift (tau) into the density functions (1), not only the expected values of the corresponding intervals are altered, but also the shape of the system’s overall variation profile is affected. In particular, increasing the mean values of the intervals while keeping the variance fixed leads to a monotonic decrease in the coefficients of variation, which has a decisive impact on queue characteristics. Since the mean waiting time in a G/G/1 system is directly proportional to the squares of the coefficients of variation of inter-arrival and service intervals, managing the shift opens the way to analytically controlled delay optimisation.

From a mathematical standpoint, a system with a regulated shift does not preserve Markovian properties, and its dynamics are described within the general G/G/1 class. In this class, the arrival and service flows may follow arbitrary structures, provided they admit a Laplace transform. To describe the distribution law of the queue waiting time, the Lindley integral equation is used in the following interpretation:

where (Wleft( x right)) is the distribution function of the queue waiting time, and (Fleft( {{v_xi }} right)) is the distribution function of the stochastic variable (rho =beta – alpha), which describes the difference between the service time (beta) and the inter-arrival interval (alpha) of two successive requests. The variable ({v_rho }) in this context is the integration variable that spans all possible values of (rho), i.e., all scenarios of relative positioning of arrival and service completion events. If ({v_rho }>0), the current request is forced to wait; if ({v_rho } leqslant 0), the service begins immediately. The controllable shift parameter (tau), introduced at the level of the distributions (alpha left( t right)) and (beta left( t right)), indirectly shapes the behaviour of the function (Fleft( {{v_rho }} right)), and thus governs the entire dynamics of request accumulation in the system.

To obtain an analytical solution to the Lindley Eq. (2) under arbitrary (non-Markovian) arrival and service distributions, it is appropriate to apply the spectral method³². This approach is widely used in the analysis of queueing systems, as well as in applied problems of mathematical physics and signal processing. In our model, this method preserves analytical controllability even in the absence of simplifying assumptions about the form of the densities. The core idea of the method is to transition into the Laplace transform domain, where the densities (alpha left( t right)) and (beta left( t right)) are represented as functions ({{rm A}^ * }left( p right)) and ({{rm B}^ * }left( p right)). This transition allows the integral Eq. (2) to be rewritten in the form of an algebraic relation:

where (p in {text{F}}) is the complex Laplace transform parameter, and (aleft( p right)), (bleft( p right)) are analytical functions (typically polynomials) that approximate the integral structure in rational form. Such a transformation enables the analysis of the system’s spectral structure, in particular the identification of its zeros and poles, which directly influence the temporal characteristics of the service process and determine the asymptotic behaviour of the waiting time.

For the subsequent analysis, we select two of the most representative distributions that combine analytical transparency with practical relevance for IoT edge subsystems – the exponential and the second-order Erlang distributions. Their selection is motivated by the fact that these distributions, on the one hand, possess closed-form Laplace transforms, and on the other hand, allow for the modelling of both reactive and multi-phase behaviour of service or arrival processes.

The exponential distribution serves as a fundamental model for memoryless stochastic events, such as spontaneous request generation by sensors or short computational tasks. Its shifted distribution function is given in

where (lambda) denotes the intensity of the exponential process. This distribution is characterised by a zero coefficient of variation, which makes it convenient for analytical anchoring and spectral interpretation of systems with constant load.

The Erlang distribution of order two, in turn, enables the modelling of structured, staged processes, particularly in cases where a request undergoes several stages of preliminary processing (filtering, authorisation, encryption). Its distribution function with adjustable shift (tau) is given in

where (mu) denotes the intensity parameter of each phase. Unlike the exponential distribution, this one exhibits a lower coefficient of variation, which allows for more precise control over load fluctuations and more efficient management of waiting times in the system.

Both distributions form a unified parametric axis, enabling a smooth transition from a fully random (exponential) to a sequential-phase (Erlang) mode without losing analytical controllability. This makes it possible, within a single spectral scheme, to model a wide range of edge scenarios () from lightweight requests with immediate service to complex transactions with sequential processing.

Within the described analytical framework, we consider queueing systems in which interarrival and service intervals are modelled by continuous stochastic variables with shifted distribution functions. Specifically, let us assume that the system dynamics are defined by two functions of the form ({F^{left( i right)}}left( t right)=left{ begin{gathered} {{tilde {F}}^{left( i right)}}left( {t – tau } right)forall t geqslant tau , hfill \ 0forall 0 leqslant t leqslant tau , hfill \ end{gathered} right.), where ({tilde {F}^{left( 1 right)}}left( t right)) and ({tilde {F}^{left( 2 right)}}left( t right)) denote the base (unshifted) distributions for arrivals and service, respectively. This formalisation generalises the previously considered exponential and Erlang cases, allowing a more abstract representation of systems with controllable delay. Interpretatively, this means that each process in the system initiates no earlier than after a fixed time interval (tau), reflecting hardware, protocol, or energy constraints typical of real-time edge nodes. Such a shift enables the reproduction of internal buffering, adaptive delays, and minimum activity intervals without disrupting the overall structure of the model. Importantly, the shifted distributions retain all key properties of classical queueing models that underpin spectral and Laplace-based methods.

After formalising the shifted form of the distribution functions (4), (5) and analysing their properties in the time domain, a natural step is to transition to the spectral representation, which is implemented via the Laplace transform. In the classical formulation, it is defined as

where (fleft( t right)) denotes the probability density function of the corresponding random variable. This transition to the complex domain enables the replacement of integral operators with algebraic ones and reveals the structure of functional relationships between model components in the form of products, quotients, and poles.

In the case of time-shifted functions (in particular, (fleft( {t – tau } right)), which equals zero for (left[ {0,tau } right))), the standard shift property ({text{L}}left[ {fleft( {t – tau } right)} right]={F^ * }left( p right)exp left( { – tau p} right)) is used, allowing the effect of controllable delay to be easily incorporated into the spectral image. This enables the previous relation (3) to be rewritten in the following form

where exponential factors associated with the shift parameter (tau) mutually cancel. As a result, the structural form of the spectral expression remains unchanged, which is a significant advantage: the shifted model does not require additional adjustment in the Laplace transform domain. This makes it possible to directly apply the spectral decomposition technique developed for classical systems, without any loss of generality or need to renormalise components.

Within the formulated model, we consider a queueing system in which both arrivals and service are described by two-phase Erlang densities with a symmetric delay structure. This approach reflects practical scenarios in which both incoming requests and their processing consist of sequential stages with a guaranteed minimum activation time, such as authentication and confirmation procedures. In the analytical representation, these densities take the form:

where (varphi ,phi>0) are the intensities of the phase components. Both densities are shifted to the right by (tau), ensuring consistency with the previously introduced logic of controllable delay.

After transitioning to the spectral domain, the corresponding Laplace transforms take the form:

where the factors (exp left( { – tau p} right)) arise as a consequence of the shift in the time domain. Since the exponential components in both transforms are synchronised, they cancel each other out within the product structure that appears in the spectral relation. After algebraic manipulation, we obtain:

where the coefficients ({k_0}), ({k_1}), ({k_2})​ depend solely on the model parameters and define the numerator as a third-degree polynomial. The pole structure of the fraction (10) is fully determined: singularities at the points (p=varphi) and (p=phi) define the dominant frequency behaviour of the system and determine the positions of the spectral peaks. This spectral localisation subsequently enables a precise analysis of the asymptotic characteristics of the waiting time.

Summarising the results of the spectral representation, we construct the Laplace transform of the waiting time function based on the rational structure of the fraction derived earlier. In the model with two-phase Erlang density distributions for arrivals and service, the corresponding transform ({W^ * }left( p right)) is given by

where ({Omega _+}left( p right)) is the regular part of the spectrum, and ({p_1}), ({p_2}) are the real positive roots of the denominator of the spectral decomposition, associated with the frequency characteristics of the system. Their presence determines the asymptotic behaviour of the waiting time function, including the dominant decay rates of the queue.

To complete the spectral construction, we refine the structure of the functions (aleft( p right)) and (bleft( p right)), which appear in relation (10) and define the spectral decomposition in the frequency domain:

where ({p_3}) is the pole of the function (bleft( p right)), located to the right on the complex axis and is the reciprocal of the characteristic time parameter of intensity (varphi). The rational form of both functions enables the efficient application of inverse transform methods and analytical approximation techniques.

The mean waiting time ({rm E}left[ W right]) is determined using the standard operator approach to the derivative of the spectral transform (11), or equivalently, through the analysis of partial fraction decomposition. We obtain:

which clearly illustrates the dependence of delay on the location of poles in the spectrum. According to formula (13), the value of ({rm E}left[ W right]) decreases with increasing ({p_1}), ({p_2})​, which, in turn, depend on the distribution parameters (primarily the mean intervals and coefficients of variation). Therefore, controlling these quantities opens the way to analytically formalised optimisation of delays in the system.

To proceed with the comparative analysis, we consider the generalised metric characteristics of the arrival and service flows. These quantities allow spectral results to be interpreted in terms of temporal scales and the dispersion properties of the system.

For the arrival flow, the corresponding values are calculated as:

where ({rm E}left[ {{T_varphi }} right]) is the mean interarrival time and ({c_varphi }) is the coefficient of variation, reflecting the degree of instability in the incoming traffic. Similarly, for the service flow we obtain:

In both cases, the coefficient of variation is a decreasing function of (tau), which confirms the earlier statement: increasing the shift stabilises the process by reducing relative dispersion and smoothing stochastic fluctuations.

In contrast to standard Erlang distributions without shift, where the coefficient of variation equals ({1 mathord{left/ {vphantom {1 {sqrt 2 }}} right. kern-0pt} {sqrt 2 }}), in the proposed model it is further reduced due to the presence of an unavailability phase. Consequently, both coefficients satisfy the inequality (0<{c_varphi },{c_phi }<0.5), indicating that the model belongs to the class of systems with limited variability, where the influence of random factors on the waiting time is significantly diminished. This creates the preconditions for predictable queue behaviour and effective real-time quality of service management.

Finally, let us consider the limiting case of the model with Erlang densities – Its transition to the exponential distribution with the same shift parameter (tau). This model corresponds to the classical M/M/1 system with activation delay, allowing an assessment of the impact of distribution order on the behaviour of the waiting time. In this case, the arrival and service densities take the form:

In contrast to the two-phase Erlang structure, this configuration exhibits memoryless behaviour with the highest possible variability (coefficient of variation (c=1)). In such a setup, the mean waiting time is determined by the classical formula for the M/M/1 model:

which, despite the presence of the shift (tau), retains its form due to the cancellation of exponential factors in the spectral domain, as demonstrated earlier.

The Laplace transforms of the densities (16) take the form:

and the product ({{rm A}^ * }left( { – p} right){{rm B}^ * }left( p right)) results in a rational fraction that describes the spectral structure of the model:

Unlike the previously considered cases, expression (19) has two simple poles, and the structure of the numerator is linear in (p), which simplifies inversion and facilitates the interpretation of queue dynamics. In this way, the shifted model remains fully compatible with the classical M/M/1 theory, while introducing a crucial element, controlled service unavailability over the interval (left[ {0,tau } right)), which is essential in realistic IoT scenarios.

In contrast to existing approaches that address task placement or energy-aware scheduling without formally modelling the internal queue structure, the proposed model introduces a parametrically controlled delay shift within a G/G/1 framework, enabling analytical control over key QoS metrics. For example, the study³³ formulates a stochastic game for distributed task coordination among UAVs, but does not explicitly model queue dynamics or device activation delay. Similarly³⁴, applies a vacation queue model to optimise application placement, yet its optimisation process is based on empirical heuristics and lacks an analytical linkage between service parameters and latency characteristics. The work³⁵ focuses on energy-efficient scheduling, but does not formalise the queue as a controllable element within the service process. The proposed model, by contrast, incorporates analytically derived expressions for mean waiting time (formula (13)) and coefficients of variation (formulas (14) and (15)), where the shift parameter (denoted (theta)) directly affects both temporal stability and service variability. The Laplace-domain representation (formulas (10) and (11)) enables spectral analysis of system behaviour under arbitrary input distributions. Furthermore, the shifted Erlang distributions defined in formula (8) allow precise modelling of activation delays typical for edge nodes. As a result, the proposed framework offers a mathematically grounded foundation for reinforcement learning that is explicitly sensitive to queue dynamics, structurally induced service delays, and decentralised real-time optimisation.

Intelligent control of the shift parameter in a queueing model for Edge-IoT environments using reinforcement learning

After formalising the analytical queueing model with a controllable shift, it is justified to proceed to the description of the mechanism for its intelligent control. The shift (tau), previously interpreted as a parameter defining the phase of system unavailability prior to processing, is hereinafter considered a controllable variable dynamically adjusted by the RL agent in response to the current system state. This is particularly relevant in edge-IoT environments, where load characteristics fluctuate unpredictably and the need to adapt to resource and timing constraints is critical.

The problem of optimal selection (tau) in this context is formalised as a Markov Decision Process (MDP), within which the RL agent observes the variation of queue parameters, selects actions from the set of admissible shifts, and receives a reward for reducing delay and improving system stability.

The state space (s) is defined by the key features of the current service configuration (S=leftlangle {q,rho ,{c_{ef}}} rightrangle), where (q) denotes the queue length, (rho ={varphi mathord{left/ {vphantom {varphi phi }} right. kern-0pt} phi }) represents the load intensity, and ({c_{ef}}) is the effective coefficient of variation. The latter can be specified as the average value between ({c_varphi }) and ({c_phi }), calculated according to formulas (14) and (15), which already incorporate the impact of the shift (tau) on the variability of incoming flows.

The action space ({rm T}) is a finite set of permitted shift values available for the agent to choose from: ({rm T}=left{ {{tau _1},{tau _2}, ldots ,{tau _n}, ldots ,{tau _N}} right}), ({tau _n} in left[ {0,{tau _{hbox{max} }}} right]). The boundaries of this set are determined by hardware, protocol, or energy constraints of edge devices, while its discrete nature allows for controlled complexity of the learning algorithms.

The reward function (Rleft( {s,{tau _n}} right)) integrates two key aspects of service performance: the average waiting time and the balance between the variability of service and arrivals. In its simplest form, it is expressed as

where ({rm E}left[ {Wleft( {{tau _n}} right)} right]) is defined by the spectral formula (13), which depends on the poles ({p_1}) and ({p_2}), indirectly influenced by the choice of shift (see expressions (10), (19)); (left| {{c_varphi } – {c_phi }} right|) serves as an indicator of variability imbalance; ({q mathord{left/ {vphantom {q {{q_{hbox{max} }}}}} right. kern-0pt} {{q_{hbox{max} }}}}) is the normalised queue length, directly reflecting the level of request accumulation; ({q_{hbox{max} }}) denotes the maximum permissible queue length; and the term (hbox{max} left( {0,rho – 1} right)) penalises situations where the arrival intensity exceeds the system’s computational capacity. The coefficients ({kappa _1},{kappa _2},{kappa _3} in {{mathbb{R}}^+}) define the relative importance of each criterion, taking into account architectural and service-level priorities.

The probabilistic transition function (Pleft( {s^{prime}left| {s,{tau _n}} right.} right)), which describes the change of state resulting from performing action ({tau _n} in {rm T}) in state (s in S), is empirically defined in most practical cases. The RL agent does not possess complete knowledge of the model; instead, it learns the queue dynamics through experience-based learning algorithms (off-policy).

The objective of the RL agent is to approximate the optimal policy ({pi ^ * }=arg mathop {hbox{max} }limits_{pi } {rm E}left[ {sumnolimits_{{t=0}}^{infty } {{gamma ^t}Rleft( {{s_t},{n_t}} right)} } right]), where (gamma in left( {0,1} right]) is the discount factor that determines the long-term significance of decisions.

The RL approach serves as a superstructure over the analytical framework outlined in subsection 2.1. It does not alter the structure of the Lindley equation or the spectrum (see expressions (3), (10), (19)), but rather uses them as a foundation for dynamic learning. Crucially, the RL agent operates not at the level of modifying the mathematical model itself, but at the level of managing its parameters, thus, enabling the system to adapt to load fluctuations and instability in incoming flows without sacrificing analytical predictability.

Function (20) formalises shift management (tau) as a MDP, in which the RL agent, interacting with the analytically grounded queuing system (see expressions (13)–(15)), develops a policy for dynamic action selection. However, in a practical edge-IoT environment, additional factors (such as buffer limitations, request losses, traffic class, and node energy capacity) play a decisive role alongside stability and service speed. Therefore, it is reasonable to introduce an extended reward function that complements function (20) with terms accounting for these application-specific requirements:

where (L={{{N_{drop}}} mathord{left/ {vphantom {{{N_{drop}}} {{N_{arrive}}}}} right. kern-0pt} {{N_{arrive}}}}) is the empirically estimated ratio of lost requests to total arrivals, (L in left[ {0,1} right]); ({L_{hbox{max} }}) is the permissible loss threshold defined by the QoS profile; (Eleft( {{tau _n},{c_{ef}}} right)) is the expected energy consumption, modelled in simplified linear form

where ({e_0},{e_1},{e_2} in {{mathbb{R}}^+}) are the parameters of the node’s energy profile corresponding to background consumption, delay cost, and processing of variable input flows; ({E_{hbox{max} }}) is the available energy consumption limit; and ({kappa _4},{kappa _5} in {{mathbb{R}}^+}) are the weighting coefficients. Each term in function (21) represents a measurable or predictable quantity calculated at the decision-making moment, ensuring a fully formalised agent policy without the need for heuristic tuning.

The extension of function (20) to the form (21) requires the construction of an agent-based architecture capable of making decisions regarding the value of the shift parameter (tau), based on observations of queue state, load characteristics, variability, losses, and energy consumption. Given that such key components of the reward as average waiting time and coefficients of variation are determined analytically (see expressions (13)–(15)), the RL agent does not approximate the service model, but rather operates as a strategic superstructure over an already adapted system.

The agent’s input is defined as a state vector (s=left( {q,rho ,{c_{ef}},L,E} right)), (s in S), where all variables are either available during execution (e.g. (q,rho)), (rho)) or computed using the mathematical framework defined in subsection 2.1. At the same time, reward components dependent on the selected action (in particular, ({rm E}left[ {Wleft( {{tau _n}} right)} right])) are not included in the state, as they are computed post hoc, after the action has been applied. As before, the RL agent’s action space is defined by the set of admissible shift values ({rm T}=left{ {{tau _i}} right}), (i=overline {{1,N}}). The discreteness of this set enables the use of tabular methods for policy learning. For such configurations, it is appropriate to apply the Q-learning algorithm, which updates the estimated utility of selecting ({tau _n} in {rm T}) in state s according to the rule:

where.(eta in left( {0,1} right]). is the learning rate, (left( {s,{tau _n}} right)) and (left( {s^{prime},{{tau ^{prime}}_n}} right)) denote the current and next states of the system, respectively; (Rleft( {s,{tau _n}} right)) is the reward function of the form (20) or (21), computed analytically based on the parameter ({tau _n}) and the observed state (s)s.

In cases where the dimensionality of the state space increases (for instance, due to the inclusion of additional QoS labels or changes in flow distributions), and the action set becomes broader, the RL agent can be implemented as a neural approximation of the Q-function, i.e. as a DQN. In this case, function (23) is modelled by a neural network with parameters (theta), which are updated by minimising the squared error between current and target estimates:

where ({theta ^ – }) denotes the parameters of the target network, updated with a delay. The DQN variant is appropriate in contexts where the management of (tau) is performed centrally using edge servers or gateway devices capable of real-time learning.

Thus, the optimal policy of the RL agent is defined as:

or, in the case of DQN:

The architecture generalised by expressions (23)–(26) implements a fully functional approach to system behaviour management without altering its internal structure. The RL agent, operating as a superstructure over the analytical core (13)–(15), performs adaptation to current load conditions, energy constraints, and service priorities (depending on the selected function (20) or (21)). This enables QoS-resilient, resource-aware control in practical edge-IoT scenarios, particularly in environments such as LoRaWAN, NB-IoT, or Smart Building Monitoring.

The construction of an effective policy for managing the shift parameter (tau) requires training the RL agent in a controlled environment that simultaneously reflects the analytical structure of the queuing model (see subsection 2.1) and allows flexible modelling of dynamic service conditions, losses, and energy consumption. Such simulation is a key instrument for validating the effectiveness of the chosen RL agent architecture and the reward function of the form (20), (21).

The simulator implements the integration of two components: the analytical core, which provides the computation of metrics (17)–(19), and the dynamic module, which updates the queue, overall costs, and energy expenditure. The current system state at step t is represented as ({s_t}=left( {{q_t},{rho _t},c_{{ef}}^{{left( t right)}},{L_t},{E_t}} right)). The RL agent selects an action ({tau _n} in {rm T}), corresponding to shift (tau _{n}^{{left( t right)}}), and the system transitions to a new state.

Within each simulation step of duration (Delta), the shift phase (tau _{n}^{{left( t right)}}) is implemented as a service delay. During this interval, arrivals continue, while processing is suspended. The new queue state is modelled according to the scheme:

where (Uleft( {tau _{n}^{{left( t right)}}} right)) is the number of new requests arriving during the shift, and (Dleft( {tau _{n}^{{left( t right)}}} right)) is the number of requests the system manages to process after the shift ends. The latter is computed as (Dleft( {tau _{n}^{{left( t right)}}} right)=hbox{min} left( {{q_t},phi left( {Delta – tau _{n}^{{left( t right)}}} right)} right)), which accounts for both queue limitations and the remaining service time. Losses are defined as the proportion of requests dropped due to buffer overflow: ({L_t}={{{N_{drop}}left( t right)} mathord{left/ {vphantom {{{N_{drop}}left( t right)} {Uleft( {tau _{n}^{{left( t right)}}} right)}}} right. kern-0pt} {Uleft( {tau _{n}^{{left( t right)}}} right)}}), and energy consumption is modelled as a linear function of the shift and flow variability: ({E_t}={e_0}+{e_1}tau _{n}^{{left( t right)}}+{e_2}c_{{ef}}^{{left( t right)}}). 

It is reasonable to train the RL agent under variable load conditions by following one of four typical scenarios:

• stationary (with constant (varphi), (phi));

• peak (with impulse load patterns);

• quasi-periodic (representing daily cycles in sensor networks);

• energy-constrained (with a variable energy budget).

To quantitatively assess the effectiveness of the strategy (pi left( s right)), the following metrics are accumulated:

where ({N_{rm T}}) denotes the number of iterations (simulation steps) during which the RL agent performs actions and the corresponding metric values are recorded.

The final stage of training the RL agent responsible for managing the shift parameter (tau) is the interpretation of the resulting policy (pi left( s right)) in terms of its stability, sensitivity to environmental changes, and generalisability beyond training scenarios. All actions of the RL agent are constrained within the discrete space ({rm T}), which ensures the preservation of the system’s spectral stability, particularly the invariance of the admissible pole placement in expression (10). The training procedure is formalised to ensure that the resulting policy (pi left( s right)) consistently reduces the average waiting time ({rm E}left[ W right]) while maintaining controlled losses ({rm E}left[ L right]) and balanced energy consumption ({rm E}left[ E right]). The sensitivity of the policy (pi left( s right)) to parametric changes was analysed through planned variation of (leftlangle {varphi ,{q_{hbox{max} }},{E_{hbox{max} }}} rightrangle) and the weighting coefficients ({kappa _i}).

To contextualise the proposed approach within the broader landscape of queueing and scheduling strategies for edge-IoT systems with strict QoS constraints, a comparative overview of relevant mathematical models is presented in the unnumbered table below. This summary outlines the structural and functional characteristics of classical stochastic queueing frameworks, threshold-based and protocol-imposed policies, task offloading schemes, and learning-driven delay management strategies. The models are compared in terms of their ability to regulate delay shifts, adapt to dynamic load conditions, and provide real-time responsiveness under decentralised operation. The final entry in the table summarises the distinctive contribution of this work, which formally integrates parameterised service delay control with reinforcement learning logic for locally autonomous decision-making.

Summary of Mathematical Models Considered in the Study.

Air Liquide announces today that it has signed a binding agreement with Macquarie Asia-Pacific Infrastructure Fund 2, for the acquisition of DIG Airgas, a leading national player in South Korea. It is expected to close in the first semester of 2026. The proposed transaction values DIG Airgas at an enterprise value of 2.85 billion euros / 4.6 trillion South Korean won. This major acquisition marks a significant strategic milestone for Air Liquide, substantially strengthening its market position in South Korea – the 6th largest manufacturing country by GDP, the 4th largest Industrial Gas market and the 2nd country in innovation spending ratio. It is recognized globally for its highly dynamic economy and rapidly expanding electronics, clean energy, mobility and Bio-Pharma sectors.

 

François Jackow, CEO of Air Liquide, commented on this announcement:

With this major acquisition, Air Liquide demonstrates once again its ability to strategically invest for profitable growth. 

First, this acquisition ideally positions Air Liquide on a major growth market. Driven by its outstanding innovation and manufacturing capabilities, Korea is leading the next waves of development in key sectors like the semiconductors industry, clean energy and mobility. Second, because it will enable us to build a reference player in the Republic of Korea. Not only are we perfectly complementary in our footprint and activities, but DIG Airgas also already boasts a backlog of nearly 20 secured projects. Finally, this opportunity, which we were able to seize thanks to our strong balance sheet, our ability to leverage synergies and our operational global footprint, will contribute to the net profit growth of the Group as soon as one year after Group integration.

We are thrilled to welcome DIG Airgas teams to the Air Liquide group and look forward to continuing the trustful relationships with our many key customers and partners. This significant step forward definitely confirms our long term commitment to the economic development of the Republic of Korea, and solidifies our  leading positions in Asia and Electronics globally.

With an annual turnover of 510 million euros / 752 billion South Korean won in 2024, DIG Airgas, founded in 1979, is a prominent industrial gas player in South Korea with close to 550 employees, 60 plants and 220 kilometers of pipeline networks. Its well-diversified business portfolio includes operations in all major industrial basins, supplying essential industrial gases to key customers across various sectors, including key Electronics players and Korean industrial leaders.

Under Macquarie Asset Management’s ownership over the last 6 years, DIG has become one of the largest industrial gas players in Korea and has successfully diversified its customer base and expanded into new growth industries such as semiconductor and secondary batteries. DIG is therefore well positioned to continue to secure new business opportunities to drive further growth, including securing mega projects in high growth industries.

Building on more than three decades of operations in South Korea, Air Liquide is an established local player and trusted business partner providing advanced gas solutions and technologies, as well as home healthcare services. Bringing perfect complementary with Air Liquide’s activities, this planned acquisition is poised to strengthen Air Liquide’s local presence and density. It also contributes to the broader Korean ecosystem, reaffirming Air Liquide’s role as a reliable, forward-looking partner with a long-term commitment to South Korea’s industrial, electronics and healthcare sectors as well as the coming energy transition.

Air Liquide’s management has a proven track record of executing and integrating acquisitions. This deal, driven by strong complementarities, will additionally deliver highly executable synergies, including secured growth from investment backlog. The proposed transaction values DIG Airgas at an enterprise value of 2.85 billion euros / 4.6 trillion South Korean won for 100% of the shares, representing an EBITDA 2024^[1] multiple of 20.2. The EBITDA multiple is reduced to 14.8 when taking into account DIG Airgas backlog of signed projects and the benefit of cost synergies. It decreases even further when considering other growth synergies that will be generated by this very complementary acquisition.

This acquisition is supported by a structured bridge loan. It will be (re)financed by bond issues. The closing of this acquisition is expected in the first semester of 2026, following the completion of Korean relevant regulatory clearances. 

---

Footnotes

1. including 49% of the 2024 EBITDA of the Green Air JV with Hyundai Steel ↑

The Directorate General of Customs Valuation Karachi has revised the customs values on the import of empty tin cans from all origins.

The directorate issued Valuation Ruling 2022 of 2025 on Thursday.

Details reveal that the customs values of empty tin cans were previously determined under Valuation Ruling No. 1962-2025. M/s. Fine Daily (Pvt) Ltd. filed a review appeal under Section 25D of the Customs Act, 1969, before the Director General, Directorate General of Customs Valuation. The Director General, through Order-in-Revision No. 281/2025, remanded the case back with directives for the Directorate to re-examine the issue, providing ample opportunity for the petitioners to be heard and considering relevant documentary evidence.

The Directorate subsequently initiated the exercise for the re-determination of customs values for the aforementioned items under Section 25A of the Customs Act, 1969.

M/s. Fine Daily (Pvt) Ltd. submitted documents indicating higher values for the goods and claimed to hold a significant market share of the imported items. However, other stakeholders argued that various challenges, particularly the higher exchange rate of the dollar and rising inflation, have adversely impacted their businesses and market sales.

They requested the Directorate to reconsider the valuation ruling in light of these prevailing dynamics and proposed aligning the valuation ruling with international raw material prices. Relevant documents were also submitted by the stakeholders for review.

The viewpoints of all stakeholders were thoroughly examined during the proceedings, ensuring that each party’s concerns and recommendations were duly considered. To ensure objectivity and accuracy, a comprehensive analysis was conducted, incorporating multiple dimensions of evaluation.

M/s. Fine Daily (Pvt) Ltd. imported four consignments in May 2025 under the Export Facilitation Scheme, with declared customs values of US 2.72/𝑘𝑔 𝑎𝑛𝑑 𝑈𝑆 2.72/kg and US 2.04/kg. Despite their claim of holding a significant market share, the limited number of consignments did not provide a reliable basis for determining customs values under Section 25(1) of the Customs Act, 1969. Data revealed that the company’s market share during the said period was approximately 2.8% of total imports, which does not substantiate its assertion of a dominant market position.

Additionally, prevailing international price trends for raw materials were analyzed, along with the value addition arising from the processing and manufacturing stages of the goods. This assessment ensured that the impact of such processing on the final customs value was accurately determined. The valuation exercise was conducted holistically, based on empirical data, market intelligence, and stakeholder input, in accordance with the provisions of Section 25 of the Customs Act, 1969.

“After gathering, analyzing, and utilizing market information from surveys, credible online sources, and raw material prices, the customs values of the subject goods were determined under sub-section (9), read with sub-section (8), of Section 25 of the Customs Act, 1969,” the Directorate added.

Wafi Energy Pakistan Limited has announced its intention to explore potential investment and acquisition opportunities in the oil marketing sector.

The decision was made during a Board of Directors meeting held on August 21, 2025.

In a disclosure to the Pakistan Stock Exchange (PSX), the company stated that the Board has granted in-principle authorization to the management to assess the feasibility and viability of such opportunities.

This move signals Wafi Energy’s strategic interest in expanding its footprint within the oil marketing sector.

(Bloomberg) — US stock futures paused this week’s string of declines in uneasy trading ahead of Federal Reserve Chair Jerome Powell’s Jackson Hole address, with markets scaling back bets on imminent rate cuts.

S&P 500 contracts erased early losses to rise 0.1% after the US benchmark slipped for five straight sessions. Futures for the Nasdaq 100 were little changed. European stocks advanced 0.2%, nudging closer toward an all-time high. US Treasuries held steady after Thursday’s pullback, with the 10-year rate at 4.33%. The dollar traded flat.

Listen to the Stock Movers podcast on Apple, Spotify or anywhere you listen.

A selloff in big tech this week has halted US stocks’ record-breaking rally. Investors are awaiting Powell’s latest policy blueprint, weighing whether the Fed will stay cautious on inflation, which is showing signs of stickiness, or tilt toward supporting a softer labor market.

Swaps have sharply reduced the odds of aggressive near-term easing, now pricing about a 70% chance of a cut next month and fewer than two moves this year. Little more than a week ago, markets were betting on a full quarter-point reduction in September, with some traders even positioning for a half-point move.

The stakes are heightened by pressure from the Trump administration to cut rates and growing divisions within the Fed’s rate-setting committee. To keep his options open, Powell may emphasize that the Fed’s September move will be guided by employment and inflation figures set for release early next month.

He is due to speak at 10 a.m. New York time.

“If the Fed doesn’t cut in September, markets will drop because they are expecting the Fed to do something. If they cut too much, markets may take it as a sign that the Fed is losing its independence, which may trigger much higher inflation,” said Joachim Klement, a strategist at Panmure Liberum. “It’s like Goldilocks with two bears and a bull.”

In premarket trading, Alphabet Inc. rose more than 1%, leading gains among the Magnificent Seven heavyweights. Nvidia Corp., meanwhile, was the sole laggard, slipping about 1%. The firm instructed component suppliers to halt production of its H20 AI chip, The Information reported.

A Bloomberg equal-weighted index of the tech titans has dropped 3.4% since Monday, setting it on track for its steepest weekly decline since April’s market rout.

Corporate News:

Meta Platforms Inc. is hiring another key Apple Inc. artificial intelligence executive, even as the social networking company prepares to slow its recruitment, according to people familiar with the matter. Nvidia Corp. has instructed component suppliers including Samsung Electronics Co. and Amkor Technology Inc. to stop production related to the H20 AI chip, The Information reported, citing unidentified sources. Air Liquide SA agreed to buy South Korea’s DIG Airgas for an enterprise value of €2.85 billion ($3.3 billion) as it seeks to expand in Asia. Fonterra Co-operative Group has agreed to sell its global Consumer and related businesses to French dairy giant Lactalis for NZ$3.85 billion ($2.2 billion). The Trump administration will not seek equity stakes in chipmakers like Taiwan Semiconductor Manufacturing Co. and Micron Technology Inc. that are boosting their US investments, as talks continue on taking a stake in Intel Corp., a US official said. The US Department of Justice has rejected claims by two whistleblowers that it failed to properly investigate allegations of sanctions violations by Standard Chartered Plc, the bank said. Some of the main moves in markets:

Stocks

The Stoxx Europe 600 rose 0.2% as of 10:26 a.m. London time S&P 500 futures rose 0.2% Nasdaq 100 futures were little changed Futures on the Dow Jones Industrial Average rose 0.3% The MSCI Asia Pacific Index rose 0.1% The MSCI Emerging Markets Index rose 0.2% Currencies

The Bloomberg Dollar Spot Index was little changed The euro was little changed at $1.1597 The Japanese yen fell 0.2% to 148.69 per dollar The offshore yuan was little changed at 7.1833 per dollar The British pound was little changed at $1.3416 Cryptocurrencies

Bitcoin rose 0.4% to $112,856.14 Ether rose 1.6% to $4,309.99 Bonds

The yield on 10-year Treasuries advanced one basis point to 4.34% Germany’s 10-year yield was little changed at 2.75% Britain’s 10-year yield advanced three basis points to 4.76% Commodities

Brent crude was little changed Spot gold fell 0.3% to $3,329.60 an ounce This story was produced with the assistance of Bloomberg Automation.

–With assistance from James Hirai and Subrat Patnaik.

©2025 Bloomberg L.P.

Elon Musk’s company, Tesla, should have its application to supply energy to UK homes blocked on national security grounds, Ed Davey has told ministers.

The Liberal Democrat leader argued that giving the electric car manufacturer a foothold in the British energy market would be “a gravely concerning move considering Elon Musk’s repeated interference in UK politics”.

Tesla has a clean energy arm and applied in July for a licence to supply power to British homes.

If the licence is granted by the regulator, Ofgem, the US company could be competing with big UK domestic energy suppliers such as British Gas and Octopus as soon as next year.

In a letter to the energy secretary, Ed Miliband, Sir Ed cited comments by Musk on social media appearing to encourage violent riots in the UK last summer, and accused the Tesla CEO of “peddling misinformation to millions”.

Musk, in a series of posts after last summer’s riots that followed the deadly mass stabbing in Southport, wrote that “civil war is inevitable” in the UK, which he labelled a “tyrannical police state”.

He has expressed sympathy with the far-right activist Tommy Robinson, whose real name is Stephen Yaxley-Lennon.

The Daily Mirror reported in January that the Home Office’s homeland security group was monitoring social media posts by Musk as a possible security risk.

Davey cited reports that Musk had been in regular contact with Vladimir Putin and wrote that the billionaire had “repeatedly demonstrated his recklessness when in control of critical infrastructure” through his management of the Starlink satellite service.

Reuters reported earlier this summer that Musk had ordered Starlink to cut internet coverage in parts of Ukraine during a crucial counteroffensive during the early stages of the Russian invasion. The Wall Street Journal reported in 2024 that Musk had been in regular contact with Putin over the previous two years.

Davey said that allowing Tesla into the British energy market would be “a security, as well as a diplomatic, disaster” and urged ministers to examine the application using national security laws.

Friday 22 August is the deadline for Ofgem to consider feedback on the application before it decides whether to grant the licence. More than 8,000 people have written to the regulator to object after a campaign by Best for Britain.

Sales of Tesla cars have plummeted in Europe amid a backlash over Musk’s support for hard-right politicians across the continent and growing competition from rival electric vehicle makers, including China’s BYD.

Tesla Europe and Ofgem were contacted for comment. The Department for Energy Security and Net Zero declined to comment.

Davey said in a statement: “Elon Musk has shown time and time again he’s no friend of the UK – whether it’s claiming that civil war here is inevitable or peddling misinformation to millions. I am seriously concerned that allowing Musk to supply energy to British homes would pose a risk to our national security.

“The energy secretary should put the national interest first and use his powers to block this licence. Failure to do so would suggest the government is more worried about upsetting Elon Musk than standing up for Britain.”

Millions of people in the UK face travel disruption on busy roads and due to a rail strike and engineering weeks this bank holiday weekend.

There is likely to be significant disruption on the rail network on the CrossCountry route from Aberdeen to Cornwall because of a strike by the RMT union over pay, safety and staffing.

The RAC has also warned roads will be busy on Friday, with three million getaway journeys planned, and particularly heavy traffic on the M5 between Bristol and Devon.

Monday is a bank holiday in England, Wales and Northern Ireland.

Network Rail has advised all passengers to check their journeys before travelling due to the strikes and rail works.

There will be no CrossCountry services on Saturday, meaning there will be no direct trains connecting Birmingham to Leicester, Derby or Nottingham.

On Sunday, although industrial action is not taking place, alterations and cancellations are to be expected.

On Monday, there will be a strike, but there will be limited services on all Cross Country lines between 08:00 BST and 18:00.

Trains between Birmingham, Reading and the South Coast will not run, as well as services between Leicester, Cambridge and Stansted airport.

There will only be a very limited service to the south west and north of York.

Meanwhile, on the East Coast Main Line, LNER will have no direct trains to London King’s Cross on Sunday, with engineering projects taking place on other routes too.

On Saturday, western England, Wales, Northern Ireland and Scotland will see sunny spells in the afternoon but also a chance of showers. Scotland will also have some sunny spells.

Sunday will be mainly dry with spells of sunshine and a few areas of cloud, along with the chance of a light shower in the north.

Central and Southern England will be the warmest spots over the weekend, with temperatures predicted to be between 23 and 25C.

Bank Holiday Monday will be a largely dry and warm day with lots of sunshine. Rain will move into Northern Ireland in the evening.

Analysis of the impact of inflammatory and nutritional markers on prognostic factors in lung cancer patients

Univariate analysis

This study ultimately included 500 patients with lung cancer, comprising 178 patients in the survival group and 322 patients in the deceased group. Patients in the survival group had significantly higher levels of lymphocyte count, serum albumin, hemoglobin, prognostic nutritional index (PNI), lymphocyte-to-monocyte ratio (LMR), hemoglobin-to-red cell distribution width ratio (HRR), hemoglobin-albumin-lymphocyte-platelet (HALP) score, albumin-to-globulin ratio (ALB/GLB), as well as a greater proportion of stage I cases and patients with high-to-moderate tumor differentiation. In contrast, patients in the deceased group exhibited significantly higher levels of serum globulin, age, systemic immune-inflammation index (SII Log), platelet-to-lymphocyte ratio (PLR Log), neutrophil-to-lymphocyte ratio (NLR), and a higher proportion of cases with distant metastasis, stage IV disease, Eastern Cooperative Oncology Group performance status (ECOG PS) ≥ 2, and poorly differentiated tumors. Table 1.

LASSO regression analysis and risk prediction formula

In this study, mortality in lung cancer patients was used as the dependent variable. Independent variables included clinical stage (coded as 1 = Stage I, 0 = Stage II/III/IV), differentiation grade (coded as 1 = low differentiation, 0 = moderate/high differentiation), ECOG PS (coded as 1 = ECOG PS 0–1, 0 = ECOG PS ≥ 2), serum albumin (measured value), LMR (measured value), HRR (measured value), ALB/GLB (measured value), and age (measured value).LASSO regression analysis identified ECOG PS 0–1, ALB/GLB, and age as independent prognostic factors for lung cancer. ECOG PS 0–1 and higher ALB/GLB levels were protective factors, while age was a significant risk factor. Specifically, the analysis indicated that each additional year of age increased the risk of mortality by approximately 7%.Table 2. The following formula was provided to calculate the mortality risk prediction score for individual patients: Exp(x) = [−0.94909] + [−0.47464 × 1(Stage I)] + [0.54761 × 1(Low differentiation)] + [−0.85073 × 1(ECOG PS 0–1)] + [−0.00427 × Serum albumin] + [−0.04974 × LMR] + [−0.28638 × HRR] + [−0.98366 × ALB/GLB] + [0.06838 × Age].The probability of mortality is calculated as: Probability = Exp(x)/[1 + Exp(x)].

Development of a risk prediction model

Feature selection using LASSO regression for dimensionality reduction

In this study, LASSO regression analysis with 10-fold cross-validation was used to select the optimal log(λ) values. When log(λ) was − 2.8203 and − 3.5645, the data demonstrated stability and statistical significance. Figure 2 shows the coefficient distribution curve for different log(λ) values. Figure 3 illustrates the stepwise feature selection process, reducing 46 variables to 1. Each curve represents the trajectory of different predictor coefficients as the parameter changes.The optimal eight variables selected for constructing the lung cancer risk prediction model based on inflammation and nutrition markers were: age, Stage I, low differentiation, ECOG PS 0–1, serum albumin, LMR, HRR, and ALB/GLB.Age was identified as an independent risk factor for poor prognosis, with increasing age associated with worse outcomes. Patients with Stage I disease and ECOG PS 0–1 had better prognoses. In contrast, low differentiation, decreased serum albumin levels, and lower ALB/GLB were associated with worse outcomes. Elevated LMR and HRR were linked to improved prognosis.Table 3. Figure 2. Figure 3.

Near-Zero variance test of training data

To further refine variable selection and understand their local characteristics and distribution patterns, a near-zero variance test was conducted on multiple variables. The results showed that patient age, ALB/GLB, and ECOG PS scores (≥ 2 and 0–1) were evenly distributed, suggesting their relevance to prognosis. The proportions of LMR, HRR, and ALB/GLB were 93.79%, 94.99%, and 98.20%, respectively, indicating high individual variability among these variables.Table 4.

Collinearity test on training data (Stepwise VIF Selection)

To improve the predictive performance of the model and address irrelevant variables, a collinearity test (VIF) was performed to further refine the feature selection. Variables such as Stage II, moderate/high differentiation, and ECOG PS ≥ 2 were stepwise excluded. The retained variables included Stage I, Stage III, Stage IV, low differentiation, ECOG PS 0–1, serum albumin (g/L), LMR, HRR, ALB/GLB, and age.The analysis identified age, clinical stage, low differentiation, ECOG PS 0–1, serum albumin levels, LMR, HRR, and ALB/GLB as independent prognostic factors.Table 5.

Recursive feature elimination

To identify the most relevant feature variables, recursive feature elimination (RFE) was performed. The selected variables included Stage I, low differentiation, ECOG PS 0–1, serum albumin (g/L), LMR, HRR, ALB/GLB, and age.The analysis revealed that LMR had the most significant predictive performance. Serum albumin, ALB/GLB, and HRR also demonstrated strong predictive capabilities. Age, Stage I, low differentiation, and ECOG PS 0–1 showed moderate predictive value.Table 6.

Variable importance

Finally, this study systematically evaluated the factors influencing prognosis and identified age as the most critical determinant. The factors ranked in descending order of impact were ECOG PS 0–1, ALB/GLB, poor differentiation, stage I, HRR, LMR, and serum albumin. Table 7.

Model evaluation

ROC curve (Internal validation performed 500 Times)

The performance of the model was evaluated using the ROC curve, with an area under the curve (AUC) of 0.7652 (95% CI: 0.7246–0.8029) and an accuracy of 0.711 (95% CI: 0.669–0.751). The model demonstrated high sensitivity (0.847) and moderate specificity (0.466), indicating good discriminatory power, making it suitable for preliminary disease screening. The positive predictive value (0.741) exceeded the negative predictive value (0.629), suggesting the model is more reliable in identifying positive cases.Internal validation was conducted using 500 bootstrap resamples. Calibration was assessed with isotonic regression fitting. The results showed an F1-score of 0.791 (range: 0–1, with higher values indicating better balance), confirming that the model achieved a good trade-off between precision and recall. These findings highlight the model’s strong predictive accuracy, good sensitivity, and reliable calibration, demonstrating its overall predictive and fitting performance.Table 8. Figure 4.

Optimal cutoff point

The ROC cut-off points, sensitivity, and specificity for eight inflammation-nutritional markers, including NLR, PLR, SII, LMR, PNI, HALP, HRR, and ALB/GLB, are presented in Supplementary Table S1. Users can select appropriate cut-off points based on their clinical context to predict future outcomes (see Supplementary_Table_S1.pdf).

Model calibration

The predictive performance of the model was evaluated using a visual calibration curve. The calibration curve closely aligned with the ideal curve, with a slope of 1. This indicates that the risk prediction model has good calibration performance and can provide reliable risk estimates. Figure 5.

Clinical utility of the model

The decision curve analysis demonstrated that the predictive model (PRED.MODEL1) maintained stable performance across various risk thresholds. In the low-risk range (0.2–0.3), the model effectively identified high-risk patients requiring intervention while reducing overtreatment. In the moderate-risk range (0.3–0.6), it performed optimally, balancing treatment benefits with potential risks. Even in the high-risk range (0.6–0.8), the model retained good discriminatory ability, aiding in the identification of patients needing aggressive intervention.The model’s curve consistently remained above the reference line and was most prominent in the moderate-risk range (0.3–0.6). These findings indicate that the predictive model can effectively guide clinical decision-making. Figure 6.

Web-Based calculator for prognostic risk prediction in patients with lung cancer

We have developed a web-based calculator for predicting the prognostic risk of lung cancer patients. Instructions for Use: After accessing Risk Prediction Model for Overall Survival in Lung Cancer Patients, input the patient’s information as follows: Select “Yes” or “No” for “Stage I.“Select “Yes” or “No” for “Poor Differentiation.“Select “Yes” or “No” for “ECOG PS 0–1.“Enter the serum albumin level (g/L) in the designated field.Input the LMR, HRR, and ALB/GLB values in their respective fields.Enter the patient’s age in years.Once all fields are completed, click “Calculate Risk.” The probability of mortality for lung cancer patients will be automatically displayed at the bottom of the interface. For reference, see the webpage calculator interface in Fig. 7.

The Directorate General of Customs Valuation Karachi has revised customs values on the import of a wide range of medical items and equipment from China.

The directorate issued Valuation Ruling 2020 of 2025 on Thursday.

After a long period of eight years, the directorate has updated the values for the import of such equipment from China.

According to the ruling, the directorate had previously issued Valuation Ruling No. 1202 of 2017 for medical items and equipment under Section 25A of the Customs Act. As the valuation ruling was eight years old, an exercise for the re-determination of customs values of the subject goods was initiated, based on an analysis of import data, current market trends, and the difference in market prices and customs values.

Meetings were convened to re-evaluate the customs values of the subject goods. During these meetings, stakeholders submitted relevant documents along with their proposed values and samples, which were duly recorded.

The customs values of the subject goods have been determined under Section 25(9), read with Section 25(7), and Customs Rule 2001.

The rule provides that the methods of valuation employed under sub-section (9) of Section 25 of the Customs Act, 1969, may include those laid down in the sub-sections of the said section. However, reasonable flexibility in the application of such methods is in conformity with the aims and provisions of sub-section (9) of that section, it added.

In the current trial, the effectiveness of a single dose of the low-level laser before each stage of the orthodontic leveling and alignment treatment and paracetamol– caffeine combination drug in reliving the orthodontic pain were evaluated.

The current research work was a three-arm, parallel-design study. Some researchers have claimed that the split-mouth design is superior to the parallel design in the trials that evaluate patient-centered outcomes to overcome interpersonal differences such as sex, age, and psychiatric changes¹⁷. However, an inherent shortcoming of this design was the possibility of carrying the perception of pain from one side to another, resulting in difficulty in accurately determining pain by patients, especially after archwire replacement^2,32. On the other hand, some studies did not find any relation between orthodontic pain perception and age or gender³³.

Adult patients with mild to moderate upper and lower crowded arches were included in the current study. Adults can express their feelings more accurately than younger people³⁴. The applied orthodontic force depends on the position of the teeth and the crowding level, especially after orthodontic separator placement or initial archwire engagement, as the greater force may cause greater pain^2,35. Many previous studies did not consider the effect of crowding degree and the selected age group in their study designs^21,22,25.

Similar to many previous studies^{2,4,7,18,21,22}, a diode Ga-Al-As laser was used with a wavelength of 810 nm (near-infrared electromagnetic spectrum). Infrared radiation has a low absorption coefficient in hemoglobin and water; thus, more depth of penetration in the irradiated tissue, possibly reaching cortical and alveolar bone tissues⁴. To achieve the best results, it is important to use an appropriate amount of laser energy. In the current study, a power of 3.5 joules per point was applied, similar to the studies of Almallah et al., Owayda et al. after separators placement^2,22 and studies of Qamruddin et al. and Domínguez et al. after archwires replacement^4,28.

Due to the superiority of paracetamol-caffeine (500–65 mg) over acetaminophen alone¹², and a greater level of safety when compared to other locally acting NSAIDs, it was chosen in the current trial as opposed to other investigations^8,9 that solely used paracetamol for orthodontic pain management. Sandue et al. demonstrated that the effectiveness of analgesics depends on the drug’s pharmacokinetics in terms of dose and timing of administration³⁶. To maintain a healthy drug plasma level in the current study, all patients were instructed to take their prescription an hour before any replacement and to continue taking a pill every 8 h for the first couple of days. Then, as needed, in the following days. This instruction was given as the orthodontic pain peaks after 24–48 h of force application^2,21.

A small amount of spontaneous and chewing pain was perceived directly after each archwire replacement (a mean of 1 to 2 units on the NRS). The compression of the periodontal ligament fibers after orthodontic force application might be responsible for this immediate pain sensation^2,23. This study showed that most patients experienced the most pain 24 to 48 h after undergoing orthodontic replacement, consistent with the results of many previous reports^2,4,8,16,28. The time course of orthodontic pain may be related to biological changes after an orthodontic force is applied, specifically, the concentration of interleukin (IL) 1-beta that reaches its highest levels after 24 h³⁷.

Intergroup findings

Differences less than 2 degrees on the Numeric Rating Scale (NRS) were considered not clinically significant. Understanding the minimally clinically important difference (MCID) is essential for evaluating the effectiveness of pain management modalities³⁸.

The spontaneous and chewing pain levels induced by orthodontic separators were significantly lower than those from the drug and control groups, just at the peak of pain after 48 h of insertion. These results were similar to Almallah and Qamruddin et al.^21,22. Qamruddin et al. used 940 nm of GaAlAs laser with 200 mW for 20 s and irradiated at three points: mesial, distal, and the middle of the first molar from the buccal side only. Almallah et al. compared single versus double laser irradiation doses with 4 J/cm² energy applied for 28 s on each of the eight selected points (4 buccal and four palatal). Neither the LLLT nor the medication could relieve the orthodontic pain induced by orthodontic separators in the other assessment times. The current results contrasted what was reported by Mirhashemi et al. who evaluated double laser irradiations with 15 J/cm² energy applied for 11 s on each point of cervical points for mesial and distal roots of the maxillary 1 st molar²⁰. The difference in efficacy of LLLT may be due to variations in overall laser energy, application protocol, or study methodology, as suggested by Farzan et al.¹⁷. Similarly, the above results match the outcomes of Patel et al., who used only paracetamol one hour before and after 3 and 7 h of separator placement⁹. In contrast, Najafi et al. suggested that preemptive paracetamol alone could be an effective option for managing orthodontic pain; however, their study lacked a control group, and the results should be viewed with care⁸.

The pain levels experienced by the participants in the laser, drug, and control groups when the initial archwire (0.014 NITI) was installed were similar to those found in previous studies^7,15. The 0.014 and 0.016 -NiTi-induced Pain mean scores in LLLT and paracetamol-caffeine groups were similar to those of the control group without significant differences. Conversely, Qamruddin et al. demonstrated that a single dose of Al-Ga-As laser irradiation with a power of 75 J per tooth can reduce the pain associated with the initial archwire⁴. The specific ability of LLLT to reduce the peak pain experienced after the separator placement, rather than after the installation of initial archwires, may be because separators induce significantly greater levels of pain than round archwires, leading to a greater concentration of inflammatory mediators, such as PGE2, IL-1β, and TNF. These mediators create a more suitable environment for laser irradiation to work effectively in reducing pain¹. The study’s results indicate that the combination of paracetamol and caffeine did not provide significant relief for spontaneous and chewing pain caused by initial archwires, similar to what was observed with pain caused by separators. In a previous study by Alshammari et al., it was reported that using paracetamol or chewing gum after initial archwire installation had similar effects²⁵. However, their study was limited by the absence of a control group, making it unclear whether the modalities used effectively controlled pain. This is particularly relevant to the current study, as pain mean scores in their study were compared to those of a control group in the current study.

The insertion of rectangular archwires caused mild levels of pain that peaked 24 h after insertion and subsided by the end of the week in all three study groups. The laser in the current study reduced pain levels during its peak at 24 and 48 h after rectangular wire insertion, but the amount of reduction was not clinically important. These results were similar to those reported by Dominguez and Velasquez, who evaluated the effect of a low-energy GaAlAs laser with a wavelength of 830 nm and an energy of 100 J/cm2 for 22 s on the vestibular and palatal/lingual sides after 0.019*0.025 SS archwire insertion in crowded cases that were resolved without extraction²⁸. This effect may be attributed to the cumulative effect of successive laser doses applied from the first to the final stage in this study. In the drug group, the mean values of the spontaneous and chewing pain scores after 24 and 48 h of placing the 0.016*0.022 NiTi and 0.019*0.025 NiTi archwires were the lowest among the tested groups. However, the multiple comparisons did not reveal any statistically significant differences between the drug and control groups. The medication used was not effective in alleviating mild pain levels in stages from 0.016*0.022 NITI to 19*25 SS, and this may be attributed to the fact that the pain levels induced in these stages were mild; there was no need to relieve these mild pain levels, and this psychological factor may have affected the drug’s efficacy.

The drug intake did not significantly reduce pain levels compared to receiving no treatment, although there was some decrease. Additionally, the addition of caffeine did not appear to enhance the pain-relieving effect of paracetamol on induced pain.

Within group findings

Most previous studies in this area have only examined one stage, either separators^2,8,21,22 or the initial archwire^4,7,15,25, as the source of force. We believe that studies focusing on a single component as the source of force may draw inaccurate conclusions because the nature of pain caused by each stage may have different characteristics compared to the pain caused by other stages. Therefore, to the limit of our knowledge, this study is the first to assess all stages of leveling and alignment treatment with a fixed appliance, providing a comprehensive understanding of orthodontic pain from A to Z.

In the current study, pain levels (spontaneous and chewing) were significantly greater in the separation and initial archwire stages compared to other stages after 24 and 48 h in the laser, drug, and control groups. However, there was no significant difference between these two stages. It is commonly believed that orthodontic pain increases proportionately to the force applied, which may explain why there was more intense pain during initial archwire alignment but decreased with the 0.016-in wire and became almost non-existent later on⁴. This also could be attributed to the patient’s exposure to new procedures that could increase anxiety and pain levels³⁹. Additionally, patients may become accustomed to the fixed appliance and the pain it causes over time, although there is currently no research to support this theory.

Specifically, the pain levels during separation were slightly greater than those induced by the 0.014 NiTi archwire. This could be interrupted by the fact that the pain caused by the separator differs from that caused by the archwire, as the separator exerts only lateral forces for teeth separation. In contrast, fixed orthodontic treatment using an archwire involves more complex movements, such as extrusion, intrusion, and uncontrolled tipping, which can contribute to a different type of pain⁷.

Additionally, no significant differences were detected between the pain mean scores recorded for the initial archwire and 0.016*0.022 NiTi or 0.019*0.025 NiTi. One possible explanation is that the 0.016*0.022 NiTi archwire is the first rectangular archwire that transfers torque data, and the 0.019*0.025 NiTi wire is the largest wire that conveys the majority of torque data through the brackets⁴⁰. Orthodontic pain can be a significant concern following the initial stages of orthodontic treatment, particularly after the placement of separators and the initial archwire installation. However, using buccal tubes can help avoid the need for separators.

Limitations

One of the limitations of the current study was that masking was not applicable due to the long-term follow-up period and patients in the control group did not receive any placebo procedure, and the placebo procedures could play an intervention role in pain studies. Additionally, the current study did not assess participants’ pain threshold before enrollment. no drop-outs were planned during sample calculation of this trial. Another limitation of the current study is that the use of medication in the drug group after the first couple of days of treatment was not tracked or analyzed through subgroup or sensitivity analysis.