Modified Hexagonal Split Ring Resonator Based on an Epsilon-Negative Metamaterial for Triple-Band Satellite Communication

A triple-band epsilon-negative (ENG) metamaterial based on a split ring resonator (SSR) with a modified hexagonal-shaped metal strip proposed in this study is a new combination of a single slit square resonator and a modified hexagonal-shaped metal strip. The desired unit cell FR-4 (lossy) that was selected as the substrate was 1.6 mm thick. Following the assessment of the unit cell, a high-frequency electromagnetic simulator like the computer simulation technology (CST) microwave studio was applied to assess the S-parameters. The proposed design exhibited resonance at 2.89, 9.42, and 15.16 GHz. The unit cell also demonstrated negative permittivity in the frequency ranges 2.912–3.728 GHz, 9.552–10.144 GHz, and 15.216–17.328 GHz, along with a negative refractive index. An effective medium ratio (EMR) of 11.53 is an indicator of the goodness of the metamaterial unit cell. It is deliberate at the lowermost resonance frequency of 2.89 GHz. Moreover, the simulated results that were validated using HFSS and equivalent circuit model indicated slight variations. The proposed design was finalised based on several parametric studies, including design optimisation, different unit cell sizes, various substrate materials, and different electromagnetic (EM) field propagations. The proposed triple band (S, X, and Ku bands) negative permittivity metamaterial unit cell can be utilised for various wireless applications, such as microwave communication, satellite communication, and long-distance radio communication.

Introduction
Materials found in natural surroundings can achieve positive permittivity and permeability. Contrarily, metamaterials yield both negative permittivity and negative permeability or either trait can be negative by itself. An engineered structure indicating both negative permittivity and permeability is called a double negative (DNG) metamaterial, while a structure manifesting either negative value is called a single negative (SNG) metamaterial [1,2]. DNG and SNG metamaterials are extensively used in different technological developments. The use of metamaterials in synthetic dielectrics of microwave application began just after the Second World War. The advancements in material sciences led to the industrialisation of metamaterials in radio frequency, microwave, and optical appliances. Meanwhile, the introduction of metamaterials with a negative refractive index was utilised to shield an object from view, termed cloaking [3]. Metamaterials are usually prepared by repeating the unit cell or proposed pattern, taking into consideration the wavelength of the desired phenomenon. Moreover, the properties of metamaterials depend on their design, size, forms, or assembly instead of the material ingredients [4]. In short, uncommon precise orientation, shapes, or sizes determine the unique or smart novel properties of metamaterials, particularly EM properties. Material researchers are passionate and interested in this field to harvest the benefits of metamaterials (absorbing, cloaking, blocking, bending or enhancing EM radiation) due to their novel EM properties [5,6].
The extant literature has indicated the uses of metamaterials in many applications. For instance, the L band stretchy metamaterial with an EMR of 4.8 using nickel aluminate substrate was designed for a sensing application [7]. Sekip Dalgac et al. [8] presented an S-shaped metamaterial comprising a rectangular split-ring resonator to forecast moisture in the air and temperature. In 1968, Victor Veselago, a Russian physicist, introduced a theoretical explanation of materials with negative permittivity and negative permeability, called the left-handed (LHM) characteristics [9]. Meanwhile, Pendry et al. revealed an SSR metamaterial based on synthetic magnetic materials with different shapes of SRR (circular, rectangular) on a dielectric substrate [10]. In another study, an S-shaped resonator was bordered by a ground ring [11]. A microstrip line was deployed in this metamaterial for feeding. This sort of metamaterial used an L-band with 1.8 GHz resonance frequency. An amalgamated metamaterial (12 × 12 mm 2 dimension) with an EMR of 7.44 recorded resonances at S, C, and X bands [12]. Meanwhile, a design of a Greek-key pattern operated at a frequency range from 1 to 5 GHz was fabricated on a substrate of Rogers RT5880 (10 × 10 mm 2 dimension). The design measured resonances at 2.4 GHz, 3.5 GHz, and 4 GHz [13]. After three decades, Smith et al. [14] described an experimental demonstration of functioning EM metamaterials using straightly mounding, intermittently, SRR, and thin-wire structures. In 2001, Shelby et al. [15] demonstrated a negative refractive index using a two-dimensional array of repeated unit cells that were also a combination of wire strips and split-ring resonators.
On the other hand, Winston E. Kock developed a material that has analogous features to the metamaterials in 1940 [16]. Flame Retardant-4 (FR-4) material is commonly used to design substrates. Modern technologies demand compounds that are cost-effective, compact, and lightweight, which led to the fabrication of low-cost and easily designed substrates. For example, an ENG metamaterial with an EMR value of 3.5 exhibited EM cloaking operation as it covered the C-band [17]. In another study [18], a dual SRR of hexagonal shape with a gap (GCHSRR) metamaterial (10 × 10 mm 2 dimension) exhibited microwave absorbance uses as it covered the C-and X-bands. In recent times, metamaterials have been aimed at supporting various frequency bands. Each frequency band has widespread use; the S-band covers a frequency range of 2-4 GHz; the X-band covers 8-12 GHz, and the Ku-bands covers 12-18 GHz. The three bands are extensively utilised in satellite communications [19]. Among them, the terrestrial signals can only slightly affect the Ku-band, which operates in a high-frequency range that allows the use of trivialsized dish antennas for VSAT uses. Meanwhile, a metamaterial with a harmonising SRR (5 × 5 mm 2 ) and EMR of 8 demonstrated a resonance frequency of 7.5 GHz [20].
This study proposed a new and novel metamaterial assembly whose unit cell consists of one SRR with a modified hexagonal-shaped metal stripe. It demonstrated resonance frequencies in the S-, X-, and Ku-bands (2-4 GHz, 8-12 GHz, and 12-18 GHz) in the microwave spectra, apart from indicating negative permittivity and negative refractive index at the same frequencies. Furthermore, the reliability of the performance of the proposed metamaterial was tested using different validation tests. Equivalent circuits were also designed in ADS, HFSS design, 2 × 1 and 2 × 2 arrays to simulate validation tests. The analyses were performed using different effective parameters like permittivity, permeability and the refractive index, together with transmission and reflection coefficients. The EMR was also used as a comparison of performances in the result section.

Design of the Unit Cell and Simulation
A vivid illustration of the projected metamaterial unit cell illustrated in Figure 1a is a combination of a square-shaped single slit resonator (SSR) and a modified hexagonal metal strip. Figure 1b demonstrates the simulation setup of the unit cell. A 0.035 mm thick copper (annealed) was used for SSR and modified hexagonal-shaped strip. Moreover, the dielectric substance FR-4 (lossy) with a thickness of 1.6 mm was used as a substrate with a dielectric constant of 2.2 and a loss tangent of 0.025. The dimension of the square shapedsubstrate was 9 × 9 × 1.6 mm 3 . The length and width of the unit cell SSR metal strip were defined as W1 and W2, respectively, while the length and width of the substrate were a and b, respectively. The slit width (g) and height were each 0.2 mm, with a 0.5 mm gap (G) between the two strips of the hexagon. However, the width (w) of all the metal strips in SSR was 0.5 mm, and the width of the hexagonal arm strips (y) was 1 mm. According to the simulation setup of the proposed unit cell (Figure 1b), an EM wave was applied along the z-axis. In addition, a perfect electrical and magnetic boundary were applied consecutively along the x-axis and y-axis. The dimensions of the unit cell, including all parametric values, are listed in Table 1.

Effective Medium Parameters Extraction Method
The EM simulator CST was applied for the mathematical and geometrical analyses based on a definite integration procedure. This method provided tractability in different composite symmetrical structure modelling and integrated the arbitrary distribution of materials with their properties. Simultaneously, a frequency-domain solver using a tetrahedral mesh in CST was used to simulate the proposed structure. The EM waves were fixed to promulgate to the z-axis. Perfect electric and magnetic fields were applied along the xand y-axes as a boundary condition to simulate 2.0-18.0 GHz of the resonance frequency. Additionally, transmission coefficient (S21), permittivity (ε r ), and refractive coefficient (n r ) were evaluated to determine the EM properties of the proposed structure using the Nicolson-Ross-Weir technique [21].
where, c, f, and d represent the speed of light, frequency, and the thickness of the substrate, respectively. S11 represents the reflection coefficient, and S21 is denoted the transmission coefficient. To obtain the refractive index (n r ), relative permittivity (ε r ), and permeability (µ r ), MATLAB codes were established based on Equations (1)-(3). The results of the parameters extracted through the NRW method was compared with the CST results.

Design Procedure
To maximise the benefit from a design, the best design was selected among the available trial designs. An iterative method was performed to assess the response of the unit cell. The length, width, the distance between the SSR, and the size of the slits were changed based on trial-and-error. Figure 2 represents some of the trial designs. Design-1 of Figure 2 contains a single slit square ring with a uniform width of 0.50 mm. This is an SSR with a dimension of 8 × 8 mm 2 and a thickness of 0.035 mm with a split gap of 0.40 mm. This SSR demonstrated two major responses at 3 GHz and 10 GHz. Design-2 illustrates a hexagonal loop with a width of 1 mm that was added to Design-1 (Figure 2b). Following the simulation, the resonance frequencies recorded were at 3 GHz, 10 GHz, and 17 GHz. Meanwhile, the modification to Design-2 gave rise to Design-3, which had an addition of two metal strips that were 6 mm long and a split gap of 1 mm. This modified design recorded resonances at 3 GHz and 10 GHz. Finally, the placement of two more metal strips with the same split gap in Design-3 in a horizontal orientation on the hexagon arms was represented as the proposed design ( Figure 2d). The proposed design measured three major frequencies at 2.89 GHz, 9.42 GHz, and 15.16 GHz. Figure 3 indicates the transmission coefficient (S21) for the four different designs.
The proposed design has a substrate with a dimension of 9 × 9 × 1.6 mm 3 ; the length and wide of the SSR unit cell metal strips were W1 = 8 mm and W2 = 8 mm, with a split width (g) of 0.20 mm. Moreover, G, the split gap between the placed metal bar strips of the hexagon was measured at 1 mm with every metal strip at 0.5 mm width. The proposed structure demonstrated a maximum number of resonance (frequency bands) with negative permittivity and negative index. The dimensions of the suggested metamaterial unit cell, including all designs tranmission coefficients results values, are listed in Table 2.

Effects of Changing of the Unit Cell Size
To optimise the size of the proposed metamaterial, the unit cell with different substrate sizes were simulated. Based on Figure 4, the substrate of sizes 11 mm and 13 mm with a thickness of 1.6 mm exhibited three resonances with low bandwidth. Contrarily, the 9 mm substrate with the same thickness demonstrated three major resonances at 2.89 GHz, 9.42 GHz, and 15.16 GHz. Hence, the 9 × 9 × 1.6 mm 3 dimension was fixed as the proposed unit cell size.

Effects of Changing Substrate Properties
According to a study, two types of Rogers (RT 5880 and RT 6002) and an FR-4 (lossy) with the same dimension (9 × 9 × 1.6 mm 3 ) were used to observe the properties of different substrates materials. The loss tangent and paramagnetic constant for substrate RT 5880 were 0.0004 and 2.2, as for RT 6002 were 0.0037 and 3.48, respectively. The S21 depicted in Figure 5 indicated three sharp resonances with narrow bandwidths for RT 5880 and RT 6002. Substrate FR-4 (Dk 2.2 and loss tangent 0.025) demonstrated three resonances with satisfactory bandwidth covering S-, X-, and Ku-bands. A high EMR of 11.53 was calculated for the first resonance frequency (2.89 GHz). Therefore, FR-4 (lossy) was selected as a substrate material for the fabrication of the proposed unit cell.

Effects of Changing the EM Field Propagation Direction
The electric field (Ey) was applied towards the y-axis, while the magnetic field (Hx) was applied towards the x-axis (Figure 6a). The resonances recorded at both axes were 6 GHz and 15 GHz, respectively. If the fields are interchanged with each other, an electric field (Ex) was applied in x-direction and the magnetic field (Hy) in y-direction ( Figure 6b); three major resonances were observed at 2.89 GHz, 9.42 GHz, and 15.16 GHz (Figure 7).

Surface Current, Electric Field, and Magnetic Field Analysis
The EM characteristics of metamaterials rely on different forces and fields, which are depended upon to produce charge. Maxwell's curl equations are the best tools to explain the generated electric and magnetic fields [22].
where vector operator, ∇ = [ ∂x ∂t , ∂y ∂t , ∂z ,∂t ]. Equations (4) and (5) are not sufficient to describe the relationship between matter, the electric field, and the magnetic field. However, these limitations were solved using the following equations.
The equations presented above were used to assess the surface current, magnetic, and electric fields functions. The current distribution at three different resonance frequencies was 2.89 GHz, 9.42 GHz, and 15.16 GHz (Figure 8). The current flow differs with the change of frequency. Based on Figure 9a-c, the outside square ring of the unit cell was determined to have been strongly dominated by the surface current. In other words, it meant that the outer ring belonged to the resonance at 2.89 GHz. Moreover, it was notable that the hexagonal ring was also subsidised as a major part of the current. The magnetic field distribution for the proposed unit cell at various resonance frequencies (2.89 GHz, 9.42 GHz, and 15.16 GHz) is depicted in Figure 10a-c.   According to the Biot-Savart and Ampere's law, a magnetic field is produced due to the motion of charged particles or electrical charges [23]. The flow of higher electrical charges increased the strength of the magnetic field that was observable from the H field ( Figure 10). The magnetic field centered in a wire at a distance (r) follows the equation (using the Biot-Savart law), B = µ 0 I 2πr , where µ 0 is the permeability of free space. A similar outcome was also yielded in the E field ( Figure 9). Compared to the electric field pattern presented in Figure 5 with the H field presented in Figure 6, it is evident that fewer changes in the H field at a particular point also measures low electric field intensity. Furthermore, the electric field intensity at the point of splits was higher since the split in the resonator ring gap formed a capacitor.

Circuit Model of the Projected Metamaterial Unit Cell
The proposed metamaterial involved both inductive and capacitive components. All the metal bars and the metal strips of the rings (both SRR and hexagonal) acted as inductors, while the gaps within the metal bar or rings formed capacitors. Therefore, an LC resonance circuit was formed in this metamaterial unit cell. As such, the resonance frequency (f) can be derived using Equation (8) [17].
where L is inductance, and C is capacitance. The capacitance of the capacitor formed by the slits can be calculated using Equation (9).
where, ∈ 0 and ∈ r represent the permittivity in free space and the relative permittivity, respectively, and A is the cross-sectional area of the conducting strip, while d is the split gap.
Here, K g = Correction factor, w = width, l = length, and t = thickness of the strip Figure 11 represents an estimated equivalent circuit for the designed metamaterial unit cell. At the equivalent circuit, L1, L2, L3, L4, L5, and L6 refer to the inductors, while C1, C2, C3, C4, and C5 are capacitors. The circuit was optimised and simulated using the advanced design system (ADS). As such, inductor L1 and capacitor C1 were responsible for the first resonance (2.89 GHz), where L1 regulated the resonance frequency of S21, with C1 as the biasing factor that modifies the magnitude. Whereas C3 is the coupling capacitor, L3 and L4 refer to two series inductors, while L5 and L6 refer to two parallel inductors that contribute to the other two resonances. Figure 12 illustrates a comparative view of the simulated S21 derived from the CST and ADS describing the closest similarities between these two S21 values. However, the impedance was derived for equivalent circuits based on the Thevenin theorem.

Results and Discussion
The mathematical analyser CST was employed in this study to determine the Sparameters and different coefficients of the unit cell. Figure 13 represent the reflection coefficient (S11) and S21. The unit cell recorded resonance frequencies at 2.89 GHz, 9.42 GHz, and 15.16 GHz with −25 dB, −22.24 dB, and −28 dB amplitude, respectively. As for S11, negative peaks were observed at 3.78 GHz and 10.20 GHz with −23.72 dB and −8.98 dB amplitude, respectively. Bandwidths of 0.39 GHz at S-band, 0.94 GHz at X-band, and 2.64 GHz at Ku-band are also depicted in Figure 13. Moreover, the numerical data were extracted for permittivity, permeability, and refractive index using MATLAB code (Figures 14-16). According to Figure 13, it is found that every resonance of S21 was tracked by an S11 minimum. Since every minimum frequency of S21 was lower than that of the corresponding S11 minimum frequency, every resonance was treated as an electrical resonance in the unit cell. Meanwhile, Figure 14; Figure 15 represent the results of permittivity and permeability. It is observed that the value permittivity goes through from maximum to minimum while resonance takes place in S21. The figures also denoted that minimum values of µ were derived from minimum S11 points. The values of permittivity changed from positive to negative when µ positively changed the quantity. Since the refractive index was related to the frequency, the near-null property at the negative permittivity region is depicted in Figure 16. Due to the negative permittivity of the projected unit cell in a certain frequency region, the proposed design was termed a negative-permittivity (ENG) metamaterial. Applying the Drude function [24].
where Γ is the attenuation function due to energy disintegration in the plasmons, and ω p is the plasmon frequency which is determined by ω p = dq 2 0 m . Also, d represents effective density, q for electron charge, ∈ 0 for permittivity of free space, and m for the mass of the electron. If Γ ≈ 0 then ω < ω p and then permittivity (ε) will be negative. Thus, the EM wave will not propagate. This unique property (negative permittivity) of the metamaterial can widely be used in various types of communication systems, for instance, bandwidth improvement of the antenna [25] and microwave filter. Meanwhile, the negative refractive index property can be utilised for cloaking, sensing, and enhancing antennas. To validate the authentication of the result, the unit cell was simulated by CST and HFSS (S11 and S21) are illustrated in Figure 17. The extracted parameters and their respective frequencies values are listed in Table 3 and comparison of resonance frequencies between the unit cell and arrays are listed in Table 4.

Array Structure
The unit cell does not function satisfactorily as an independent unit. Hence, an array of unit cells is the best technique to obtain the expected EM properties from a metamaterial. The 2 × 1 and 2 × 2 arrays of the proposed unit cell with the same substrate material were simulated by CST (Figure 18a,b). A parametric study was performed using 2 × 1 and 2 × 2 arrays to investigate the coupling effect between the designed unit cells for different combinations of the array recorded nearly identical S21 responses to the unit cell. Therefore, the efficiency of the arrays was manifested. Figure 19 demonstrates the graphical presentation of S11 and S22 comparison by CST for array combination and the proposed unit cell. Although the resonance frequencies were a little distorted in the new array metamaterial, the triple band resonance properties were still functioning based on the simulated results of the array presented in Figure 19. The authentication between the previous configurations and proposed configuration values are listed in Table 5.   On the other hand, the symmetric pattern of the used SSR and hexagonal metal strip were important in achieving the abovementioned performance. This symmetric pattern of the unit cell helped to decrease the induction effects of the magnetic field. For this reason, the transmission coefficient (S21) is not affected by the harmonic and shows almost the same scattering behaviour and EM properties in the full frequency spectrum.

Conclusions
This study discussed a metamaterial with negative permittivity designed on an FR-4 substrate with resonance frequencies at 2.89 GHz (S-band), 9.42 GHz (X-band), and 15.16 GHz (Ku-band). Such metamaterials are nowadays designed and fabricated to influence EM radiation using new techniques. Many vital properties (optical and EM properties) of various natural materials have been converting through this chemistry. To investigate the performance of the proposed design, the dependency on the dielectric constant and an array or combination of the metamaterial characteristics (here three validations for a single unit cell and array) were incorporated to produce a comparative analysis. An extensive investigation was performed to analyse the surface current, electric field, and magnetic field distribution. The DPS-ENG media properties were also assessed using an EM wave interaction. The vital parameters like permittivity, permeability, and refractive index were determined using MATLAB coding, whereby the results exhibited negative permittivity with a negative refractive index. Whereas, 2 × 1 and 2 × 2 array metamaterials that were simulated using the same substrate material demonstrated minimal changes in resonance frequencies. On the other hand, the efficiencies of an equivalent circuit that was designed for the desired unit cell were tested via ADS. This result indicated S21 to be very close to the CST results. The proposed unit cell also yielded a resonance frequency of 2.89 GHz with an EMR of 11.53. Since S-, X-, and Ku-bands are widely utilised in satellite and RADAR communications, these tri-band spectra were utilised for microwave communication, long-distance radio communication, and satellite communication. Although this study utilised a simple and compact metamaterial structure, the proposed design manifested triple-band resonance frequencies with excellent EMR value (11.53). In conclusion, this unique metamaterial structure adapted several validation processes that led to its novelty.


Near Field RFID Tag for IoT in Sub-Six GHz Band

The present paper introduces the analysis and design of a near field RFID tag for IoT in sub-six GHz 5G frequency band. The proposed radio frequency identification technique is based on the near field interaction between the RFID tag and a wideband antenna reader. This near field interaction adjusts the resonances of the wideband antenna according to the used RFID tag. In addition, the far field RCS of the RFID tag is also investigated to study the relation between the near field and far field responses of the proposed RFID tag. The proposed RFID tag is characterized with adjustable six resonances based on concentric square rings printed on a dielectric slab. For manufacturing and experimental verification, the dielectric slab is assumed to be FR-4. However, the proposed structure can be generalized to other thin and flexible substrates like paper, plastic, and textile.

INTRODUCTION
Radio frequency identification (RFID) technology is an automatic recognition system that uses radio waves to identify products and objects. RFID system consists of a tag, a reader, and a processing unit [1][2][3][4][5]. RFID tags can perform tasks such as retrieving, storing, and sending object information such as code, route, and coordinates. By reading the tag placed on the product or objects, the information can be automatically saved or changed. With this information, it is possible to easily determine which product belongs to which order, instantaneous changes can be made in the production schedule, and inventory tracking can be done more easily.
Modern digital solutions are attempting to complement the traditional security features through embedding RFID tags in official paper-work and banknotes [6][7][8]. On the other hand, wearable RFID tags on flexible substrates have importance in security applications, biomedical applications, structural health monitoring and other logistic applications [9][10][11][12][13][14][15][16][17][18][19][20]. Although traditional RFID systems based on the use of silicon RFID chips are very common and extensively used in practice, some of their limitations, such as cost and robustness, are driving many researches towards alternative solutions, namely chipless RFID. The most promising way for RFID is to directly print on a product or package like a bar code. The chipless RFID is quite similar in function to the optical bar code, and hence it is also known as RF bar code technology. Chipless RFID system is generally based on multiresonators which perform frequency signature encoding signed in [10][11][12][13][14][15][16][17][18][19][20]. In [11], multiple octagonal shaped resonators are used to present a nine-bit RFID tag on both flexible and non-flexible substrates for Internet of Things (IoT) applications. Depending on the type of the substrate, the range of frequencies was determined. On the other hand, in [12] different resonators of different shapes based on Arabic letters are used as different RFID tags. In this case, each letter corresponds to a unique RF signature which can be used for identification. These multi-resonance configurations can also be implemented by using slot resonators as in [13]. It is found that symmetric configurations of the multi-resonance structure would be more appropriate to make the response of the RFID tag is independent of the polarization of the reader antenna as discussed in [15]. In [19], an RFID tag is introduced consisting of crossed diagonal patches combined with L-shaped patches on a dielectric substrate to generate six resonances. It is used to detect cracks and corrosion in metallic structures. In this case, the resonance frequencies of this RFID tag are shifted due to the cracks or corrosion on the metallic structure behind the RFID tag.
In this paper, a simple passive RFID tag based on concentric square ring resonators on a dielectric slab is presented. Two techniques for radio frequency identification are studied. The first technique is based on a resulting radar cross section (RCS) of the designed RFID tag. The other technique is based on near field interaction of the proposed RFID tag with a planar wideband antenna. This near field interaction introduces sharp resonances on the reflection coefficient of the wideband antenna according to the used resonators on the RFID tag. By using a thin substrate, this configuration can be quite useful for paper currency and official papers like passports to increase their security level and facilitate their tracking process. A similar configuration based on a split ring resonator on a plastic film was discussed in [18] for the application of security ID cards.
In the following section, the basic configuration of the proposed RFID tag is presented. The RCS of this tag is studied. In Section 3, we discuss the applicability of using a wideband antenna as a near field reader for this RFID tag. Finally, Section 4 presents the conclusion.

CONCENTRIC SQUARE RINGS PASSIVE RFID TAG
The proposed RFID tag consists of concentric square rings printed on a dielectric slab as shown in Fig. 1. The width of each ring is 0.5 mm, and the spacing between successive rings is also 0.5 mm. The outer length of the greatest ring is 22 mm while the length of the substrate is 26 mm. Thus, the circumference of the outer ring is 88 mm. The difference between the circumferences of each successive two rings is 8 mm. The used substrate in the present analysis is FR-4 of thickness 0.8 mm, dielectric constant ε r = 4.3, and tan δ = 0.025. This square ring resonates when its circumference is nearly one guided wavelength. For a non-grounded dielectric slab, the effective dielectric constant can be approximated as: Thus, the guided wavelength would be λ g = λ 0 / √ ε e . Thus, according to the proposed dimensions and substrate parameters, the analytical six lowest resonances would be 2.094, 2.304, 2.88, 3.291, 3.84, and 4.61 GHz. Effectively, these values would be slightly changed due to the effect of the mutual coupling between the rings as shown in the full wave analysis. In addition, the most inner four concentric rings are added to adjust the effect of the mutual coupling while they resonate at much higher frequencies.
The required resonances can be controlled by adding or subtracting the corresponding ring resonator. Figure 2 shows the radar cross section of this RFID tag for six different configurations obtained by full wave analysis by using finite element analysis based on HFSS. The resonances of the tag with all rings are 2.7, 3, 3.4, 3.9, and 4.6 GHz. It can be noted that the resonance frequencies are slightly shifted from the corresponding analytical ones due to the coupling effect as mentioned above. However, this shift is decreased at higher resonances. This can be explained by that the approximation of the effective dielectric constant is more accurate at higher frequency. It can be noted that each ring corresponds to a resonance frequency in the corresponding RCS as shown in Figs. 2(a) to 2(f). Even though the resonances can be clearly discriminated between the different tags from their RCSs, the values of these RCSs for the different tags are very small. Thus, it is required to use high power transmitter combined with a sensitive receiver to discriminate between different tags at a distance more than or around 1 meter. This is the motivation here to introduce a near field reading system instead of using direct RCS measurement.

NEAR FIELD RFID SYSTEM WITH CONCENTRIC SQUARE RINGS PASSIVE RFID TAG
In this section, a near field identification is proposed by using a wideband antenna as a reader. The proposed antenna is shown in Fig. 3. It consists of a partially grounded patch with a tapered section to be matched with a 50 Ω microstrip line. The antenna is printed on a substrate RO3003 of thickness 0.25 mm with ε r = 3 and tan δ = 0.0013. The total size of the antenna is (x × y) = 38 × 41 mm 2 . The dimensions of the patch are xp = 21 mm, yp = 18 mm, x 1 = 11 mm, and y 1 = 5 mm. The width of the feeding 50 Ω microstrip line is w s = 0.6 mm. Fig. 4 shows the reflection coefficient of this antenna separately. It can be noted that this antenna has a reflection coefficient less than −10 dB in the frequency range from 2.2 to 4.5 GHz.  This wideband antenna is used as a near field reader for the designed RFID tag. The effect of the tag appears as resonance frequencies. Due to the presence of the conducting patch and the additional substrate of the antenna, the effective dielectric constant of the RFID tag is increased. Thus, the corresponding resonance frequencies are slightly shifted down from the free space RCS resonances of the RFID tag mention in the previous section. Fig. 5 shows the proposed RFID reader configuration. In this case, the proposed RFID tag is placed in the proximity of the reader antenna as shown in Fig. 5.    6 shows the simulated and measured reflection coefficients of the reader antenna for different RFID tags as discussed in Section 2. It can be noted that the RFID tag with all square rings has resonances at 2.12, 2.54, 2.92, 3.43, 4.09, and 4.39 GHz. It can be noted that these resonance frequencies are shifted toward lower frequencies due to the increase of the effective dielectric constant as mentioned before. Fig. 6 also shows the effect of removing rings on the resonances of the reader antenna. It is quite clear that each ring corresponds to a specific resonance frequency which can be classified as digital bit in the corresponding RFID tag. Based on these results, it can be concluded that this multi-ring resonator structure can be used as a six-bit RFID tag for a near field RFID system. The advantage of this simple configuration is that it has a significant difference more than 10 dB in the reflection coefficient between the presence and absence of the corresponding ring resonator. This significant difference can be easily detected by a simple wideband receiver system. Although the sample in the present case is printed on an FR-4 substrate, the basic theory of the proposed multi-ring resonator is valid for other dielectric substrates. Thus, it is expected that this configuration is also valid for other substrates like paper, plastic, and textile substrates.

CONCLUSION
In this paper, we present a multi-ring resonator on a dielectric substrate without a ground plane. We studied the RCS of such multi-resonators for different numbers of rings. The RCS shows very small values which require high power transmitter and a sensitive receiver to discriminate between the different tags. We presented another more practical approach based on a near field reader. This near field reader is simply a wideband antenna. The RFID tags introduce different resonances in this wideband antenna which can be easily discriminated for the different tags. The present configuration in this paper is based on six resonances corresponding to six bits in the frequency range between 2 and 5 GHz. This configuration is quite suitable to be integrated with IoT and 5G applications in the sub-six GHz band. Although the present RFID tag is based on a rigid FR-4 substrate, the basic idea of this configuration is also applicable to flexible substrates like paper and plastic. This property makes this RFID tag suitable for other applications like wearable RFID and RFID for official papers and banknotes.


DEVISING MATRIX TECHNOLOGY FOR FORECASTING THE DYNAMICS IN THE OPERATION OF A CLOSED MILITARY LOGISTICS SYSTEM

Improvement of management of logistics processes for the training and combat use of military units and subunits occurs under the conditions of constant growth of the dynamics of this application and the shortage of material resources. This causes the need to develop scientific methods of prompt substantiation of management decisions made. This problem should be solved in theory and in practice by a new structure, which includes command, planning, and executive bodies and which is called the military logistics system. Logistics refers to the science of planning and optimizing material flows, service flows, as well as related flows, in a certain system or subsystem to achieve the goal due to these flows. Military logistics is a scientifically based, optimized comprehensive equipment of troops, units, and subunits of all types of the Armed Forces: – armament, transport, and means of management; – missiles and ammunition; – repair equipment and military property; – fuel and lubricants; – property, food, and medical supplies. The equipment takes place in peacetime, during preparation for combat operations and their implementation during the movement of troops, their activities in defense, and in the offensive. Optimization of processes in military logistics is carried out according to the criterion of minimum time spent on achieving the purpose of the functioning process and is organized in content, volume, time, and place. The novelty and complexity of the tasks facing the military logistics system require the development of new approaches to solve them. Therefore, it is necessary to use scientific methods in the analysis, planning, and optimization of the processes of the provision in the military logistics system. The following means are used to scientifically substantiate decisions on optimizing the processes of equipping troops, units, and subunits: – the scientific and methodological apparatus of inventory management; – software products based on predictive software models; How to Cite: Ugol’nikov, A., Diachenko, V., Kliat, Y., Kosenko, A., Shelukhin, S. (2021). Devising matrix technology for

Introduction
Improvement of management of logistics processes for the training and combat use of military units and subunits occurs under the conditions of constant growth of the dynamics of this application and the shortage of material resources. This causes the need to develop scientific methods of prompt substantiation of management decisions made. This problem should be solved in theory and in practice by a new structure, which includes command, planning, and executive bodies and which is called the military logistics system.
Logistics refers to the science of planning and optimizing material flows, service flows, as well as related flows, in a certain system or subsystem to achieve the goal due to these flows.
Military logistics is a scientifically based, optimized comprehensive equipment of troops, units, and subunits of all types of the Armed Forces: -armament, transport, and means of management; -missiles and ammunition; -repair equipment and military property; -fuel and lubricants; -property, food, and medical supplies. The equipment takes place in peacetime, during preparation for combat operations and their implementation during the movement of troops, their activities in defense, and in the offensive. Optimization of processes in military logistics is carried out according to the criterion of minimum time spent on achieving the purpose of the functioning process and is organized in content, volume, time, and place.
The novelty and complexity of the tasks facing the military logistics system require the development of new approaches to solve them. Therefore, it is necessary to use scientific methods in the analysis, planning, and optimization of the processes of the provision in the military logistics system. The following means are used to scientifically substantiate decisions on optimizing the processes of equipping troops, units, and subunits: -the scientific and methodological apparatus of inventory management; -software products based on predictive software models;

DEVISING MATRIX TECHNOLOGY FOR FORECASTING THE DYNAMICS IN THE OPERATION OF A CLOSED MILITARY LOGISTICS SYSTEM
A l e x a n d e r U g o l ' n i k o v -the methods of linear, dynamic programming to optimize the processes of comprehensive support; -an apparatus of computer calculations using MATLAB software; -staff dialog and information models implemented on a computer that works in real time.
The existence of a military logistics system requires the development of adequate models of the process of its functioning in the implementation of various practical tasks. This is due to the increasing complexity of the tasks that logistics must perform. The volume of information flows is increasing, the time for the implementation of various types of support is decreasing, the range of supplies is expanding, new vehicles are used for deliveries, for example, unmanned aerial vehicles, and so on. Modeling makes it possible to give an analytical description of the processes of functioning of the military logistics system in order to determine and improve its effectiveness. These factors determine the relevance of work in the area of forecasting the dynamics of the functioning of the closed military logistics system.

Literature review and problem statement
In work [1], the tasks of provision are considered in terms of minimizing the cost of storage and supply of industrial products. From a mathematical point of view, the problem is reduced to finding a minimum of the objective function, depending on many variables.
Article [2] investigates a complicated situation where the objective function is not continuous and has non-sliding gaps of the first kind. In [3], to optimize the operation of the motor transport enterprise of the military logistics base, linear programming methods are used in the multi-criteria scorecard. The problem of optimal placement of goods (the problem of three-dimensional packaging) is solved in work [4] provided that cars are additionally loaded at specialized points.
Work [5] is methodological. It examines the existing approaches to modeling logistics systems -deterministic-optimal, probabilistic, and knowledge-oriented, assessing the possibility of their use to describe processes in the auto-technical units of military units. The analysis of the advantages and disadvantages of each of these approaches and models based on them leads to the conclusion that the introduction of a knowledge-oriented approach is the most effective in modern military logistics. There are no calculated algorithms in the cited work, which reduces its applied value.
Paper [6] reports the development of a methodology for building a model of the system of ensuring the work of the enterprise engaged in production and sales activities. The substantiation of the importance of the development of general theoretic research methods is given. A universal method of developing a support system, starting with the simplest tank systems, is suggested. The model is used to solve optimization problems of building systems for ensuring the operation of an enterprise. Optimization is carried out using numerical methods. Study [7] considers the task of choosing a safe cargo supply route in war zones. Unlike the classic setting of the task about the salesman, a parameter is introduced into consideration, which characterizes the safety of the selected route (the probability of delivery of goods to its intended destination). The criterion for the optimality of the route is the maximum probability of delivery of goods to all desti-nations. It is noted that the formulated problem is the task of integer programming and it is proposed to solve it by the method of branches and bounds.
Article [8] proposes solving a narrow problem of building a statistical model for assessing and repairing combat damage to weapons and equipment. On this basis, it is proposed to deploy the forces of the military logistics system and the formation of its resources.
Work [9] builds on the research started in [8], and considers the development of a simulation model to study the factors influencing combat damage to equipment. Modeling results are planned to be used as input information to build a more complex model of logistics of the military industry.
Another work by the same team of authors [10] proposes the construction of a model of optimal solutions based on statistical material and simulation modeling. The result of the integrated system is a built model of optimal solutions for the elimination of combat damage to equipment.
Work [11] considers the issue of improving the reliability of logistics by the forces of the military logistics system. Two deception strategies are considered: the inclusion of empty convoys on routes and the spread of misinformation about the route. It is assumed that convoys can be controlled on routes by the enemy. A two-step integer model of linear programming is proposed, which determines the quantitative parameters of both options for deceiving opponents. In addition, the model takes into consideration several other tasks of the military logistics system. Among them is the presence of restrictions on the range at points of supply, compliance with time restrictions, the choice of the best route option.
Article [12] describes the peculiarities of logistical support in the implementation of military expeditionary operations and humanitarian missions. The model of a non-stationary mass service network is used, which makes it possible to take into consideration uncertainties and risk assessment. To verify the model, the data of logistical support of combat operations during the operation in Iraq were used.
A common feature of those studies is the focus on the interaction of the military logistics system with consumers of its services. At the same time, not enough attention is paid to the internal problems of the functioning of the military logistics system itself.
The problem of increasing (in publication) the adequacy of the model, as well as reliable risk assessment, is not solved. The functioning of the military logistics system is always carried out under conditions of uncertainty of both random and antagonistic nature. Under these conditions, the application of the specified mass service model requires the use of pre-known parameters of the law of the flow of random orders of consumers, which during practice are difficult to determine. In this situation, predicting the dynamics of the functioning of the military logistics system can make unacceptable mistakes in determining risks.
Under these conditions, it seems more reliable to use models that are not based on the laws of a probability distribution of the flow of applications and refusals.
The use of more adequate (in such a situation) models of the dynamics of mean, that is, the apparatus of discrete Markov processes, seems more acceptable. And overcoming the uncertainty of the antagonistic type, for example, with the help of the minimax principle, that is, taking into consideration the expected options for the enemy's actions and options for counteracting the logistics system, does not create technical difficulties.
That is why the use of more adequate models, which are based on the widespread use of matrix calculation technology in real time, is promising. It should be noted that the input data, in this case, are the data of experience surveillance and use under the conditions of high dynamics of changes in the situation of the investigated support processes.

The aim and objectives of the study
The purpose of this work is to devise an informative and convenient (in use) method for describing and predicting the dynamics of the functioning of the military logistics system. This technique makes it possible to find the probabilities of the process of functioning of the system in its main states, which contributes to the adoption of informed decisions when ensuring the implementation of combat missions.
To accomplish the aim, the following tasks have been set: -to build a system of differential equations describing within the model the process of functioning of the military logistics system when performing various tasks; -to derive a solution to the system of differential equations in the form of expressions to calculate the probability of the functioning of the logistics system in its main states; -to illustrate the performance of the model, using it to describe the process of functioning of the military logistics system in the performance of specific support tasks, for example, forecasting the survivability of the transport logistics subsystem during the march of the military unit.

The study materials and methods
One of the effective methods of researching complex phenomena and processes is the method of mathematical modeling. The functioning of the military logistics system is influenced by a significant number of poorly defined factors of random and antagonistic nature. Simplification of solving the problem can be achieved by introducing into consideration some generalized performance indicator [13]. In work [14], as such a generalized indicator, it is proposed to use the probability of the military logistics system being in its working condition when performing specific tasks. This way of assessing the effectiveness of the system makes it possible to take into consideration the uncertainties that quantify the change in time, that is, the dynamics of the functioning of the logistics system in real conditions. Therefore, it was chosen during the construction of the model in this work.
Analysis of the dynamics of the functioning of the military logistics system makes it possible to present its functioning as a discrete Markov process [14]. This process is characterized by a set of typical states that the system can enter, as well as state transitions. The intensity and probability of these transitions are determined by specific conditions of application of the logistics system for its intended purpose. The possibility of using the Markov model is based on the fact that the next state of the military logistics system depends only on its current state and does not depend on how the system entered that state. Transitions of the Markov system between states occur many times, and this reflects the real transitions of the military logistics system between states in the process of its functioning.
The dynamic model of the functioning of the military logistics system, which can be in n+1 different states S i (i=0, 1, 2, …, n) with the probabilities P i (t) and make transitions from any state to any other, is considered. The system can be depicted using the state and transition graph. Figure 1 shows an example of such a graph for a system that can be in four states and execute all possible transitions between them.
The transition from S j state to S i state is characterized by the intensity h ij and the probability H ij . Then the coefficients a ij =h ij •H ij (i≠j) characterizes the impact of the probabilities P j (t) of the system staying in the S j state on the probability

5.
Results of studying the model of the functioning of the military logistics system

1. A system of differential equations describing the dynamics of the process model
These are some practical examples of building a system of differential equations that are inherent only in the military logistics system and worthy in solving the typical problems of this system.
We have proposed an effective matrix technology for solving applied problems of military logistics. Unlike known ones, this computation technique is based on the use of the well-known apparatus of discrete Marking processes and Laplace transformations. This could contribute to a significant acceleration of the calculation of forecast estimates of the probability levels of the system being in a state of effective functioning, which is important information for the command staff when planning the processes of comprehensive provision of actions and planning maneuvers by material means.
The probability Р � of the system being in the S i state increases due to the transition of the entire system from other states S j and decreases due to transitions in the opposite direction. Then the differential equations describing the time dependence of the probabilities P i (t) of the presence of the studied system in states S i , take the form:

t a P t a P t a P t P t a P t a P t a P t P t a P t a P t a P t P t a P t a P t a P t
is the probability derivative by time. The properties of the system of equations (1) are fully determined by the matrix A of the coefficients of the system: Unknown P i (t) functions are subject to additional restrictions. These are, first, the initial conditions that specify the probability values at the initial point in time (t=0): and, second, the following normalization condition that is, a set of all possible system states forms a complete group of events. Obviously, the presence of the system in one of these states is a reliable event, the probability of which is equal to one.
When applying the proposed model to the description and forecasting of the functioning of the military logistics system in the performance of typical provision tasks, it is necessary to build a graph of states and transitions and determine the quantitative characteristics of all transitions. This can be done using regulatory documents and an expert evaluation method.

2. Solving a system of model equations that describes the process of functioning
Solving the system of linear differential equations with constant coefficients is advisable to perform using the method of operating calculus, which is based on the Laplace transformation. Next, for a more compact notation, the argument t in parentheses after the sign of the function P i (t) or its time derivative Ṕ i (t) is omitted.
Since the system equations (1) are bound by a condition of normalization, they are not independent. This makes it possible to exclude one of the equations and consider a system containing n independent equations. The exclusion of the first equation (with index "0") and the substitution of the corresponding expression to all other equations produce the following system The first equation is removed, the terms of other equations are regrouped, and the designation is entered to bring the so-called shortened system: The next step is to apply the Laplace transformation to system (2) using the conversion table and the original differentiation formula: - is the mapping of the derivative; -a i0 →a i0 /p is the mapping of the constant value. As a result, the system of linear differential equations relative to probabilities P i (t) is replaced by a system of linear algebraic equations relative to their mapping X i (p): , , , n n n n n n n n n nn n a pX p P a X a X a X p a pX p P a X a X a X p a pX p P a X a X a X p Introducing into consideration the matrix of the shortened system A', the matrix-column of free terms B(p), and the matrix-column of unknown X(p): 11 allows the system (3) to be written as a single matrix equation where E is the unit matrix of the n-th order. The methods of solving systems of linear algebraic equations are well known.
A solution to the matrix equation is convenient to present in the form where det(A'-pE) is the main determinant of system (3) (characteristic polynomial of the matrix A'), and the elements R i (p) of the matrix-column R(p) with accuracy to the multiplier p are the auxiliary determinants of this system: To obtain the solution P=P(t), one must inversely convert the mapping of X(p) by Laplace. X i (p) elements of the matrix X(p) are correct rational fractions, so they decompose into the sum of elementary fractions by difference (p-p j ) -1 (j=0, 1, …, n), where p 0 =0, p j ≠ 0 are the proper numbers of matrix A' (the roots of its characteristic equation det(A'-pE)=0). From the content of the problem, it follows that the roots must be valid, and, in real conditions, they are all simple. As a result, the decomposition of X i (p) (i=1, 2, …, n) elements is: In matrix form, we obtain the equation X(p)=C•D(p). The symbols C and D indicate the matrix of decomposition coefficients and the matrix of decomposition parameters, respectively: Calculation of unknown elements of matrix C, in the case of simple real roots, is performed by substitution of zeros of the denominator p j ( j=0, 1, 2, …, n). The algorithm is that fractions in the right parts of expressions in each line of system (6) lead to a common denominator, and then equate the numerator of the resulting fraction to the corresponding numerator R i (p) on the left side. These equalities must hold at all values of the complex variable p, including at the values p=p j ( j = 0, 1, 2, …, n), which are zeros of the denominator. The substitution of these values gives the following expressions for calculating the C ij elements of matrix C: Determining C ij coefficients and using a tabular formula

In matrix form, we obtain P(t)=C•F(t), where F(t) is the matrix of the inverse transformation of Laplace of the matrix D(p):
The solution (8) is derived without any additional assumptions, that is, it is precise and fully analytical. This makes it possible to use this result to solve specific tasks of ensuring the actions of the military logistics system, that is, to evaluate the quality of its functioning.

3. Illustrating the feasibility of the forecasting model for the dynamics of a military logistics system
The calculation algorithm developed using operational and matrix calculus is quite generalized.
We are talking about a set of consistent actions, rules that make it possible to achieve the goal of the operation to determine the effectiveness of the functioning of military logistics systems under typical difficult conditions for the implementation of military tasks.
It can be applied to the description of the processes of functioning of the military logistics system in different conditions. The specificity of particular tasks is determined by the list of the main possible states of the system, the list of possible transitions, and the numerical values of input parameters of transitions. To determine them, it is advisable to use normative documentation and practical experience (in the form of an expert evaluation method). Features of the model are illustrated in several examples.
The subsystem of technical support of the military logistics system, which ensures the functioning of the transport flow, is considered. Analysis of the dynamics of the process shows that the system can be in three main states and make transitions between them. A graph of the main states and transitions of the military logistics system, which provides the transport flow, is shown in Fig. 2.
In this practical example, the content and typical nature of each state are defined. The following state notations are used: -S 0 -the system is in a working, non-busy state, that is, it is not involved before the march and at rest, if there is no damaged equipment that needs to be restored; -S 1 -the system is in a working, busy state, that is, the movement of the column occurs, and there is no damaged equipment that needs to be restored; -S 2 -the system is involved in the elimination of damages caused by equipment during movement or at rest. The commander makes the decision on the march personally based on clarification of the task received, assessment of the situation, the route of nomination and tactical calculations carried out by the headquarters. The march purpose includes: -determining the structure of marching order and the distribution of forces and means in columns; -composition, tasks, and remote protection; -routes and average speed; -output frontier (point) and frontiers (points) of regulation; -the number and duration of halts over the first daily transition; -organization of air defense on the march. When organizing a long-distance march, the number and size of daily transitions is also determined; the number of routes at each transition areas of the day (night) daily rest of units and time of stay in them.
It is planned to carry out a daily march. After moving for six hours, it is assumed to rest for an hour and a half, four hours is the time of restoration of weapons and military equipment that failed due to operational reasons, as well as due to damage by the enemy. According to the established standards, the daily operational failures of weapons and mil-itary equipment (WME) are equal to 5 %, daily damage to these means at rest -6 %, daily damage during movement -7 % [15]. Recovery rates during the march of WME column for vehicles that failed due to operational reasons are 95 %, and the means that failed as a result of combat losses -65 %. From this data, one can calculate values for the input parameters of the model (the intensity and probability of each state transition).
The intensity of transition from any state to another is defined as the value inverse to the time of the system's stay in this state. The WME provision system is during the march in a non-busy state S 0 only at rest. Therefore, the intensity ν of the system transitions from this state to states S 1 or S 2 is equal to 1 1 0.667 hour .
Accordingly, the intensity μ of transition of the supply system from state S 1 to states S 0 and S 2 is equal to The intensity λ of transition of the support system from state S 2 (restoration of WME failed during the march or at rest) to the state S 0 , is equal to 1 1 0.25 hour . 4 − λ = = According to the above-mentioned standards, the probability P F of WME failure at rest should be accepted equal to P F =0.06.
The probability P D of WME damage during movement consists of the amount of probabilities of operational losses and losses as a result of the enemy's actions and is equal to P D =0.05+0.07=0.12.
Finally, the probability Р В of WME recovery over the planned time consists of three additions. This is the probability of restoration of equipment that failed during operation, and the probability of restoration of equipment that was damaged due to the enemy's actions during movement or at rest. Hence 0.05 0.95 0.06 0.65 0.07 0.65 0.132.
It is advisable to assume that at the time t=0 the march support system was in state S 0 (the system is working, nonbusy). This assumption corresponds to the initial conditions The plot in Fig. 3 demonstrates that during the march of the column, the technical support system is highly likely (~0.70) in the state S 1 . In this case, the movement of the column occurs and there is no damaged equipment that needs to be restored. This fully corresponds to a real situation and indicates the adequacy of the proposed model and the realism of the selected values of the input parameters.
To assess the quality of the technical support system, an efficiency indicator is used. It is introduced, for example, as the ratio of the total probability of the system staying in a state of working capacity to the probability of its stay in a state of disability: The plot of dependence of this indicator on time for the case in question is shown in Fig. 4.
The proposed method of modeling the technical support system of military logistics can be applied to more complex cases, with a greater number of possible system states and transitions between them.
As a second example, a generalized model of the process of technical support for the restoration of military equipment that is damaged in battle is considered. This system, according to our analysis, can enter one of the four main states: -S 0 is the state of combat use of equipment; -S 1 is the state of preparation of equipment for use; -S 2 is the state of restoration of equipment after its damage; -S 3 is the state of maintenance of equipment before or after the end of hostilities.
The graph of states and transitions of the system is shown in Fig. 5.
System transitions between states are characterized by the following parameters: -a, A is the intensity and probability of transitions of the military-logistic support system of the weapons and military equipment from the state of preparation of equipment to the state of its maintenance; -b, B is the intensity and probability of transitions from the state of maintenance to the state of combat use of automotive equipment; -c, C is the intensity and probability of transitions from the state of combat use to the state of maintenance of equipment; -d, D is the intensity and probability of transitions from the maintenance state to the state of recovery of equipment after damage; -e, E is the intensity of the probability of transitions from the state of preparation to the state of recovery of equipment after damage; -f, F is the intensity and probability of transitions from the state of preparation to the state of combat use of equipment; -g, G is the intensity and probability of transitions from the state of combat use to the state of training of equipment for the purpose of its use; -h, H is the intensity and probability of transitions from the state of recovery of equipment after damage to the state of its combat use; -i, I is the intensity and probability of transitions from the state of combat use of equipment to the state of recovery after its damage.
The a ij elements of matrix Therefore, we obtain the matrix of the system of differential equations in the form The probability values of the system states at the initial point in time (initial conditions) are accepted equal to The curves of dependence of probabilities of staying of the system of ensuring hostilities in its possible states on time are shown in Fig. 6.
The indicator of the effectiveness of the combat operations support system is defined as the ratio of the probability of its stay in a state of combat use to the probability of its stay in the state of recovery, takes the form

P t E t P t =
As an example of a system that can be in five different states, the system of military logistics of comprehensive support of combat operations is considered with the dominance of the importance of the functioning of the military logistics system of comprehensive support of troops' actions in full readiness. According to the analysis, the system can be in the following main states: -S 0 is the state of stay of the functioning of the military logistics system of comprehensive support of troops' actions in full readiness; -S 1 is the state of stay of functioning of the combat (operational) subsystem in unpreparedness; Obviously, the main task is to maintain the state of operation of the comprehensive readiness of the military logistics system for use. This is clearly demonstrated by the graph of states and transitions of this system, shown in Fig. 7, where the S 0 state occupies a special place.
The numeric intensity and probability values of all possible state transitions have been selected as arbitrary but realistic. The result is the following matrix A for this system: The initial conditions for this system are as follows: The dependences of probabilities of staying by a subsystem of comprehensive provision of combat operations in its possible states and performance indicator on time are shown in Fig. 8.
An indicator of the effectiveness of the system of comprehensive provision is the ratio of the probability of its stay in a state of comprehensive readiness to the total probability of being in a state of unpreparedness . P t E t P t P t P t P t = + + + The efficiency indicator of this support system is always reduced during the operation and its limit value, as a rule, decreases with an increase in the complexity of the system, that is, with an increase in the number of types of provision.

Discussion of results of studying the model of functioning of a military logistics system
In the Markov model, the process of functioning a military logistics system is depicted as a graph of states and transitions (Fig. 1). This graph corresponds to the system of linear differential equations of the first order (1), which describes the probability of the functioning of the logistics system in its main states. The use of such a simple mathematical apparatus favorably distinguishes the proposed model from those reported in [1][2][3][4][5][6][7] (methods of minimizing the functions of many variables or methods of integer programming).
The application of the Laplace transformation and matrix calculus when solving the system of equations (1) led to obtaining a precise analytical solution in the form of ratios (8). As a result, a simple and easy-to-use tool for predicting the dynamics of the functioning of the military logistics system in ensuring the actions of troops was con- structed. All calculations can easily be implemented on personal computers or tablets using effective numerical algorithms of linear algebra. The proposed model and calculation technique, developed on its basis, were applied to the forecasting of the dynamics of the functioning of the military logistics system in the execution of various typical tasks for ensuring the actions of troops. For the first example (Fig. 2), the algorithm for determining the input parameters of the model has been described in detail. In all cases, qualitative coordination with the real situation was obtained (Fig. 6, 8), and in the first example -quantitative (Fig. 4, 5).
All this allows us to assert that a reliable and convenient technique for predicting the dynamics of the functioning of the military logistics system has been devised.
Both the model under consideration and its mathematical implementation are not free from restrictions, some of which are purely technical in nature, and some are fundamental. The first of the technical limitations is that a fully analytical solution can be obtained only with a number of possible system states of no more than five. With a larger number of states to find their own numbers of a shortened matrix, it is necessary to use numerical algorithms. The second constraint irelated to the fact that only the case when all the proper numbers of the shortened matrix are different was considered. If some of them coincide, then, to determine the coefficients in probability expressions, it is necessary to use a different method than the method used in this work. It should be noted that this restriction is not significant since such a situation is somewhat artificial and unlikely under real conditions.
The fundamental limitation of the model is that the coefficients of the initial system of differential equations themselves may depend on time. The nature of such dependence is currently unknown. In addition, even in the case of the simplest linear dependence, the system would be described by nonlinear differential equations, solving which is a complex mathematical problem. Obviously, the development and improvement of the proposed model should tackle these areas.
The toolset proposed to predict the dynamics of the process and the effectiveness of the functioning of the military logistics system does not require, unlike models reported, for example, in [12], the use of differential probability distribution laws for input parameters during calculations. In the study, the use of experience data on input parameters is enough. At the same time, due to errors in the case of non-compliance of the parameters of these laws with real parameters in practice, this could cause [12] a repeated increase in errors in the results of solving an actual task.
This statement does not contradict the recommendations of E. S. Wenzel about the expediency to pay more attention not to sophisticated calculations but to the search for reliable input data.
This study may be advanced by improving the toolset offered for its application under conditions where the parameters of the state and transition graph are the functions of time.

Conclusions
1. To describe and predict the functioning of a military logistics system, a model of discrete Markov processes was used since this system can be considered a system without memory. For a process with an arbitrary number of possible states, a system of differential equations has been built, describing a change in the probability of the process of functioning in its possible states. This result makes it possible to adequately take into consideration the influence of the factor of the uncertainty of operating conditions on the nature of the process.
2. The exact solution to the system of differential equations was derived by using the method of operational calculus. The application of matrix calculus has made it possible to obtain ultimate results in the form of simple analytical expressions for the probability of the functioning of the military logistics system in its main states. This also allows the use of effective numerical algorithms of linear algebra during calculations.
3. The performance of the proposed model has been illustrated by several examples. They consider the processes of functioning of the military logistics system and its subsystems in solving various tasks of ensuring the operation of troops. The sequence of calculations describes in detail the procedure for determining the input parameters of the model. The results reported in this work can be used to plan the processes of functioning of a military logistics system and its operational adjustment in real time.