The ordered matrix of the intensity of the failure and recovery flows of three autonomous systems is introduced in following asymmetric form

$R=\Vert \begin{array}{cccccccc}-{\lambda }_1-{\lambda }_2-{\lambda }_3& 0& {\mu }_1& 0& {\mu }_2& 0& {\mu }_3& 0\\ 0& -{\mu }_1-{\mu }_2-{\mu }_3& 0& {\lambda }_1& 0& {\lambda }_2& 0& {\lambda }_3\\ {\lambda }_1& 0& -{\mu }_1-{\lambda }_2-{\lambda }_3& 0& 0& {\mu }_3& 0& {\mu }_2\\ 0& {\mu }_1& 0& -{\lambda }_1-{\mu }_2-{\mu }_3& {\lambda }_3& 0& {\lambda }_2& 0\\ {\lambda }_2& 0& 0& {\mu }_3& -{\lambda }_1-{\mu }_2-{\lambda }_3& 0& 0& {\mu }_1\\ 0& {\mu }_2& {\lambda }_3& 0& 0& -{\mu }_1-{\lambda }_2-{\mu }_3& {\lambda }_1& 0\\ {\lambda }_3& 0& 0& {\mu }_2& 0& {\mu }_1& -{\lambda }_1-{\lambda }_2-{\mu }_3& 0\\ 0& {\mu }_3& {\lambda }_2& 0& {\lambda }_1& 0& 0& -{\mu }_1-{\mu }_2-{\lambda }_3\end{array}\Vert ,$(1)

The ordered determinant is associated with the matrix

$\Delta =\det \left[R\right].$ (2)

By reason of asymmetric structure of R matrix, it follows

$\Delta =0,$

i.e., R matrix is of the specific nature. The property serves as a verification criterion for mathematical models of the considered problem class.

Characteristic determinant is compiled for the ordered matrix R

$\begin{array}{l}\Delta \left(\nu \right)=\\ =\left|\begin{array}{cccccccc}-{\lambda }_1-{\lambda }_2-{\lambda }_3-v& 0& {\mu }_1& 0& {\mu }_2& 0& {\mu }_3& 0\\ 0& -{\mu }_1-{\mu }_2-{\mu }_3-v& 0& {\lambda }_1& 0& {\lambda }_2& 0& {\lambda }_3\\ {\lambda }_1& 0& -{\mu }_1-{\lambda }_2-{\lambda }_3-v& 0& 0& {\mu }_3& 0& {\mu }_2\\ 0& {\mu }_1& 0& -{\lambda }_1-{\mu }_2-{\mu }_3-v& {\lambda }_3& 0& {\lambda }_2& 0\\ {\lambda }_2& 0& 0& {\mu }_3& -{\lambda }_1-{\mu }_2-{\lambda }_3-v& 0& 0& {\mu }_1\\ 0& {\mu }_2& {\lambda }_3& 0& 0& -{\mu }_1-{\lambda }_2-{\mu }_3-v& {\lambda }_1& 0\\ {\lambda }_3& 0& 0& {\mu }_2& 0& {\mu }_1& -{\lambda }_1-{\lambda }_2-{\mu }_3-v& 0\\ 0& {\mu }_3& {\lambda }_2& 0& {\lambda }_1& 0& 0& -{\mu }_1-{\mu }_2-{\lambda }_3-v\end{array}\right|,\end{array}$(3)

Component heads identify the different components of your paper and are not topically subordinate to each other. Examples include Acknowledgements and References and, for these, the correct style to use is “Heading 5”. Use “figure caption” for your Figure captions, and “table head” for your table title. Run-in heads, such as “Abstract”, will require you to apply a style (in this case, non-italic) in addition to the style provided by the drop-down menu to differentiate the head from the text.

The characteristic equation is solved

$\Delta \left(\nu \right)=0.$

Real conjugated relative to a symmetry center characteristic equation roots are determined being expressed in the intensity of the failure and recovery flows of three autonomous subsystems

$\begin{array}{l}{\nu }_1=0,\\ {\nu }_3=-{\lambda }_1-{\mu }_1,\\ {\nu }_5=-{\lambda }_2-{\mu }_2,\\ {\nu }_7=-{\lambda }_3-{\mu }_3,\end{array}$ $\begin{array}{l}{\nu }_2=-{\lambda }_1-{\mu }_1-{\lambda }_2-{\mu }_2-{\lambda }_3-{\mu }_3,\\ {\nu }_4=-{\lambda }_2-{\mu }_2-{\lambda }_3-{\mu }_3,\\ {\nu }_6=-{\lambda }_1-{\mu }_1-{\lambda }_3-{\mu }_3,\\ {\nu }_8=-{\lambda }_1-{\mu }_1-{\lambda }_2+{\mu }_2.\end{array}$ (4)

Each of the eight roots are verified using the normalized determinants

$\Delta \left({\nu }_k\right)=0,\left(k=1,2,3,\cdots ,8\right).$(5)

Column matrix of the specified initial conditions for probabilities of the eight asymmetric states is introduced

$\left[P_i\left(0\right)\right]\text{,}\left(i=1,2,3,\cdots ,8\right).$ (6)

In the eight determinants $\Delta \left(v_k\right),k=1,2,3,\cdots ,8$, i^th column is replaced by $\left[-P_i\left(0\right)\right]$ column; the normalized determinants of a following type are formed

${\Delta }_i\left({\nu }_k\right),\left(i,k=1,2,3,\cdots ,8\right),$(7)

being components of octic square matrix $\left[{\Delta }_i\left(v_k\right)\right]$.

Products of a following type are formed

${\Pi }_k={\displaystyle \underset{\begin{array}{l}s=2\\ s\ne k\end{array}}{\overset{8}{\prod }}\left({\nu }_k-v_s\right)},$(8)

for each k^th root $k=1,2,3,\cdots ,8$; the column matrix is compiled

$\left[\frac{{\text{e}}^{v_k\cdot t}}{{\Pi }_k}\right],$(9)

where t is Markovian process time.

Matrix probability formula of temporal eight states of the system is applied for the purpose

$\left[P_i\left(t\right)\right]=\left[{\Delta }_i\left(v_k\right)\right]\cdot \left[\frac{{\text{e}}^{v_k\cdot t}}{{\Pi }_k}\right],\left(i,k=1,2,3,\cdots ,8\right).$

The result is verified if t = 0

$\left[P_i\left(0\right)\right]=\left[{\Delta }_i\left(v_k\right)\right]\cdot \left[\frac{1}{{\Pi }_k}\right],$ (11)

where $\left[P_i\left(0\right)\right]$ is a column matrix of the specified initial conditions.

The stable condition is used to verify probabilities of states for arbitrary time moment t

${\displaystyle \underset{i=1}{\overset{8}{\sum }}{P}_i\left(t\right)=1}.$ (12)

The stabilized (stationary) mode of random Markovian process is defined with the help of a limiting process $t\to \infty$ through a matrix formula in the form of the column matrix

$\left[P_i\left(\infty \right)\right]=\left[\frac{{\Delta }_i\left(v_1\right)}{{\Pi }_1}\right],$ (13)

In this context, marginal probabilities of states $P_i\left(\infty \right)$ meet the condition

${\displaystyle \underset{i=1}{\overset{8}{\sum }}{P}_i\left(\infty \right)=1}.$(14)

Moreover, following algebraic approach helps identify the marginal probabilities. Kolmogorov differential equations

$\frac{\text{d}}{\text{d}t}\left[P_i\left(t\right)\right]=R\cdot \left[P_i\left(t\right)\right]$(15)

are simplified in terms of the stabilized (stationary) mode of a continuous Markovian process owing to

$\frac{\text{d}}{\text{d}t}\left[P_i\left(\infty \right)\right]=\left[0_i\right],$(16)

where $\left[0_i\right]$ is a zero octic column matrix, i.e., Kolmogorov equations degenerate into the system of eight linear homogeneous algebraic equations

$R\cdot \left[P_i\left(\infty \right)\right]=\left[0_i\right].$(17)

where $\left[P_i\left(0\right)\right]$ is a column matrix of the specified initial conditions.

The stable condition is used to verify probabilities of states for arbitrary time moment t

${\displaystyle \underset{i=1}{\overset{8}{\sum }}{P}_i\left(\infty \right)=1}.$

The derived eight variants of equivalent systems of linear homogeneous algebraic equations has the only identically equal solutions being verification criterion for the results and the mathematical model on the whole.

The intensity of the failure and recovery flows of three autonomous subsystems are defined using statistical approaches within [0, T] time interval identified depending upon peculiarities of the applied problem being solved. Within the considered time interval, flow intensities are referred to as specified and constant. On the time interval T, continuous Markovian process has two different modes:

${Т}_{п}+{Т}_{у}=Т.$

Qualitatively, the nature of continuous Markovian process within time interval T is mainly defined by means of the specified initial conditions as well as the total of exponentials which indices are equal to characteristic equation roots. Transition process on the T time interval is defined through the total of the damped exponentials corresponding to negative roots of the characteristic equation. Stationary process within T_у time interval is identified asymptotically with the help of a zero root of the characteristic equation if $t\to \infty$. During the following time intervals, the intensity of the failure and recovery flows of three subsystems may vary discretely. Subsequently, dynamics of random Markovian process will be determined through new relevant roots of the characteristic equation; limit probabilities of states P_i(∞) at a previous time step are assumed as initial conditions.

To interpret probabilities of the system states temporally, time integration takes place within [0, T] interval of an invariant condition

${\displaystyle \underset{0}{\overset{T}{\int }}{\displaystyle \underset{i=1}{\overset{8}{\sum }}{P}_i\left(t\right)}\cdot \text{d}t}={\displaystyle \underset{0}{\overset{T}{\int }}1\cdot \text{d}t}$ (18)

whence it follows

${\displaystyle \underset{i=1}{\overset{8}{\sum }}{\displaystyle \underset{0}{\overset{T}{\int }}{P}_i\left(t\right)\text{d}t}}=T$(19)

or

${\displaystyle \underset{i=1}{\overset{8}{\sum }}{\displaystyle \underset{0}{\overset{T_{п}}{\int }}{P}_i\left(t\right)\text{d}t}+{\displaystyle \underset{i=1}{\overset{8}{\sum }}{\displaystyle \underset{T_{п}}{\overset{T}{\int }}{P}_i\left(\infty \right)\text{d}t}}}=T,$ (20)

where

${\displaystyle \underset{0}{\overset{T_{п}}{\int }}{P}_i\left(t\right)\text{d}t}=T_{iп},{\displaystyle \underset{T_{п}}{\overset{T}{\int }}{P}_i\left(\infty \right)\text{d}t}=T_{iу}$ (21)

T_iп is the duration of stay at i^th state in the transitional mode; and

T_iу is the duration of stay at i^th state in the stationary mode,

i.e.,

$T_{iп}+T_{iу}=T_i$ (22)

T_i is the system duration stay at i^th state.

In this context

$T_{iу}=P_i\left(\infty \right){\displaystyle \underset{T_{п}}{\overset{T}{\int }}\text{d}t}=P_i\left(\infty \right)\cdot \left(T-T_{п}\right)=P_i\left(\infty \right)\cdot T_{у};$(23)

$T_{iп}={\displaystyle \underset{k=1}{\overset{8}{\sum }}\frac{{\Delta }_i\left(v_k\right)}{{\Pi }_k}}{\displaystyle \underset{0}{\overset{T_{п}}{\int }}{\text{e}}^{v_k\cdot t}\text{d}t}=\frac{{\Delta }_i\left(v_1\right)}{{\Pi }_1}{T}_{п}+{\displaystyle \underset{k=2}{\overset{8}{\sum }}\frac{{\Delta }_i\left(v_k\right)}{{\Pi }_k}\cdot \frac{1}{v_k}\left({\text{e}}^{v_k\cdot T_{п}}-1\right)}$(24)

In such a way, verification condition takes the form

${\displaystyle \underset{i=1}{\overset{8}{\sum }}{T}_i}=T.$(25)

Following indexing is introduced for the considered branches of the armed forces: land arms (1), air power (2), and marine troops (3). Probable states of every substructure are defined as combat-effective ⊕ or combat-ineffective ⊖. Hence, the military structure has eight probable states ( Figure 1 ).

Within the interval of military operation Т (being a week, i.e., seven days) specified by the problem, intensity of Poissonian flows of losses (λ₁, λ₂, λ₃) and recoveries (µ₁, µ₂, µ₃) of the combatant branches are assumed to be statistically defined on the average and constant

${\lambda }_1=2;$ ${\lambda }_2=1;$ ${\lambda }_3=0,5;$ ${\mu }_1=1,5;$ ${\mu }_2=0,5;$ ${\mu }_3=2.$ (26)

Initial state of the military structure (t = 0) is considered as the known; it corresponds to a probable state 1, i.e. the land arms, airpower, and marine troops are combat-effective. While using probabilities of the military structure states, we obtain

$P_1\left(0\right)=1;$ $P_3\left(0\right)=0;$ $P_5\left(0\right)=0;$ $P_7\left(0\right)=0;$

$P_2\left(0\right)=0;$ $P_4\left(0\right)=0;$ $P_6\left(0\right)=0;$ $P_8\left(0\right)=0.$ (27)

It is required to identify probable states of the military structure at the current time

$P_i\left(t\right)\left(i=1,2,3,4,5,6,7,8\right)$ (28)

and the anticipated future states ( $t\to \infty$), i.e. limit probability of the states

$P_i\left(\infty \right)\left(i=1,2,3,4,5,6,7,8\right).$ (29)

Relying upon the specified loss and recovery flows of the three combatant branches, analytical formulas are applied to calculate eight roots of the characteristic equation

$\begin{array}{l}{\nu }_1=0,\\ {\nu }_3=-3,5,\\ {\nu }_5=-1,5,\\ {\nu }_7=-2,5,\end{array}$ $\begin{array}{l}{\nu }_2=-7,5,\\ {\nu }_4=-4,\\ {\nu }_6=-6,\\ {\nu }_8=-5.\end{array}$ (30)

Each of the eight ${\nu }_k\left(k=1,2,3,\cdots ,8\right)$ roots is verified with the help of a characteristic determinant $\Delta \left(\nu \right)$ while k^th root substitution in the characteristic determinant, and its evaluation $\Delta \left({\nu }_k\right)$ if

$\Delta \left(v_k\right)=0\left(k=1,2,3,4,5,6,7,8\right).$ (31)

Formulas of temporal probabilities of the structure states

$P_1\left(t\right)={\displaystyle \underset{k=1}{\overset{8}{\sum }}\frac{{\Delta }_1\left(v_k\right)}{{\Pi }_k}\cdot {\text{e}}^{v_k\cdot t}};P_2\left(t\right)={\displaystyle \underset{k=1}{\overset{8}{\sum }}\frac{{\Delta }_2\left(v_k\right)}{{\Pi }_k}\cdot {\text{e}}^{v_k\cdot t}};\cdots$ (32)

where

$\begin{array}{l}{\Pi }_1=\left(v_1-v_2\right)\left(v_1-v_3\right)\left(v_1-v_4\right)\left(v_1-v_5\right)\left(v_1-v_6\right)\left(v_1-v_7\right)\left(v_1-v_8\right))\\ {\Pi }_2=\left(v_2-v_1\right)\left(v_2-v_3\right)\left(v_2-v_4\right)\left(v_2-v_5\right)\left(v_2-v_6\right)\left(v_2-v_7\right)\left(v_2-v_8\right),\\ {\Pi }_3=\left(v_3-v_1\right)\left(v_3-v_2\right)\left(v_3-v_4\right)\left(v_3-v_5\right)\left(v_3-v_6\right)\left(v_3-v_7\right)\left(v_3-v_8\right),\\ ⋮\\ {\Pi }_8=\left(v_8-v_1\right)\left(v_8-v_2\right)\left(v_8-v_3\right)\left(v_8-v_4\right)\left(v_8-v_5\right)\left(v_8-v_6\right)\left(v_8-v_7\right).\end{array}$ (33)

${\Delta }_1\left(v_k\right)=\left|\begin{array}{cccccccc}-P_1\left(0\right)& 0& {\mu }_1& 0& {\mu }_2& 0& {\mu }_3& 0\\ -P_2\left(0\right)& -{\mu }_1-{\mu }_2-{\mu }_3-v_k& 0& {\lambda }_1& 0& {\lambda }_2& 0& {\lambda }_3\\ -P_3\left(0\right)& 0& -{\mu }_1-{\lambda }_2-{\lambda }_3-v_k& 0& 0& {\mu }_3& 0& {\mu }_2\\ -P_4\left(0\right)& {\mu }_1& 0& -{\lambda }_1-{\mu }_2-{\mu }_3-v_k& {\lambda }_3& 0& {\lambda }_2& 0\\ -P_5\left(0\right)& 0& 0& {\mu }_3& -{\lambda }_1-{\mu }_2-{\lambda }_3-v_k& 0& 0& {\mu }_1\\ -P_6\left(0\right)& {\mu }_2& {\lambda }_3& 0& 0& -{\mu }_1-{\lambda }_2-{\mu }_3-v_k& {\lambda }_1& 0\\ -P_7\left(0\right)& 0& 0& {\mu }_2& 0& {\mu }_1& -{\lambda }_1-{\lambda }_2-{\mu }_3-v_k& 0\\ -P_8\left(0\right)& {\mu }_3& {\lambda }_2& 0& {\lambda }_1& 0& 0& -{\mu }_1-{\mu }_2-{\lambda }_3-v_k\end{array}\right|,$

${\Delta }_2\left(v_k\right)=\left|\begin{array}{cccccccc}-{\lambda }_1-{\lambda }_2-{\lambda }_3-v_k& -P_1\left(0\right)& {\mu }_1& 0& {\mu }_2& 0& {\mu }_3& 0\\ 0& -P_2\left(0\right)& 0& {\lambda }_1& 0& {\lambda }_2& 0& {\lambda }_3\\ {\lambda }_1& -P_3\left(0\right)& -{\mu }_1-{\lambda }_2-{\lambda }_3-v_k& 0& 0& {\mu }_3& 0& {\mu }_2\\ 0& -P_4\left(0\right)& 0& -{\lambda }_1-{\mu }_2-{\mu }_3-v_k& {\lambda }_3& 0& {\lambda }_2& 0\\ {\lambda }_2& -P_5\left(0\right)& 0& {\mu }_3& -{\lambda }_1-{\mu }_2-{\lambda }_3-v_k& 0& 0& {\mu }_1\\ 0& -P_6\left(0\right)& {\lambda }_3& 0& 0& -{\mu }_1-{\lambda }_2-{\mu }_3-v_k& {\lambda }_1& 0\\ {\lambda }_3& -P_7\left(0\right)& 0& {\mu }_2& 0& {\mu }_1& -{\lambda }_1-{\lambda }_2-{\mu }_3-v_k& 0\\ 0& -P_8\left(0\right)& {\lambda }_2& 0& {\lambda }_1& 0& 0& -{\mu }_1-{\mu }_2-{\lambda }_3-v_k\end{array}\right|,$

${\Delta }_3\left(v_k\right)=\left|\begin{array}{cccccccc}-{\lambda }_1-{\lambda }_2-{\lambda }_3-v_k& 0& -P_1\left(0\right)& 0& {\mu }_2& 0& {\mu }_3& 0\\ 0& -{\mu }_1-{\mu }_2-{\mu }_3-v_k& -P_2\left(0\right)& {\lambda }_1& 0& {\lambda }_2& 0& {\lambda }_3\\ {\lambda }_1& 0& -P_3\left(0\right)& 0& 0& {\mu }_3& 0& {\mu }_2\\ 0& {\mu }_1& -P_4\left(0\right)& -{\lambda }_1-{\mu }_2-{\mu }_3-v_k& {\lambda }_3& 0& {\lambda }_2& 0\\ {\lambda }_2& 0& -P_5\left(0\right)& {\mu }_3& -{\lambda }_1-{\mu }_2-{\lambda }_3-v_k& 0& 0& {\mu }_1\\ 0& {\mu }_2& -P_6\left(0\right)& 0& 0& -{\mu }_1-{\lambda }_2-{\mu }_3-v_k& {\lambda }_1& 0\\ {\lambda }_3& 0& -P_7\left(0\right)& {\mu }_2& 0& {\mu }_1& -{\lambda }_1-{\lambda }_2-{\mu }_3-v_k& 0\\ 0& {\mu }_3& -P_8\left(0\right)& 0& {\lambda }_1& 0& 0& -{\mu }_1-{\mu }_2-{\lambda }_3-v_k\end{array}\right|,$

${\Delta }_8\left(v_k\right)=\left|\begin{array}{cccccccc}-{\lambda }_1-{\lambda }_2-{\lambda }_3-v_k& 0& {\mu }_1& 0& {\mu }_2& 0& {\mu }_3& -P_1\left(0\right)\\ 0& -{\mu }_1-{\mu }_2-{\mu }_3-v_k& 0& {\lambda }_1& 0& {\lambda }_2& 0& -P_2\left(0\right)\\ {\lambda }_1& 0& -{\mu }_1-{\lambda }_2-{\lambda }_3-v_k& 0& 0& {\mu }_3& 0& -P_3\left(0\right)\\ 0& {\mu }_1& 0& -{\lambda }_1-{\mu }_2-{\mu }_3-v_k& {\lambda }_3& 0& {\lambda }_2& -P_4\left(0\right)\\ {\lambda }_2& 0& 0& {\mu }_3& -{\lambda }_1-{\mu }_2-{\lambda }_3-v_k& 0& 0& -P_5\left(0\right)\\ 0& {\mu }_2& {\lambda }_3& 0& 0& -{\mu }_1-{\lambda }_2-{\mu }_3-v_k& {\lambda }_1& -P_6\left(0\right)\\ {\lambda }_3& 0& 0& {\mu }_2& 0& {\mu }_1& -{\lambda }_1-{\lambda }_2-{\mu }_3-v_k& -P_7\left(0\right)\\ 0& {\mu }_3& {\lambda }_2& 0& {\lambda }_1& 0& 0& -P_8\left(0\right)\end{array}\right|.$

or

$\begin{array}{l}{\Delta }_1\left(v_1\right)=\left|\begin{array}{cccccccc}-P_1\left(0\right)& 0& {\mu }_1& 0& {\mu }_2& 0& {\mu }_3& 0\\ -P_2\left(0\right)& -{\mu }_1-{\mu }_2-{\mu }_3& 0& {\lambda }_1& 0& {\lambda }_2& 0& {\lambda }_3\\ -P_3\left(0\right)& 0& -{\mu }_1-{\lambda }_2-{\lambda }_3& 0& 0& {\mu }_3& 0& {\mu }_2\\ -P_4\left(0\right)& {\mu }_1& 0& -{\lambda }_1-{\mu }_2-{\mu }_3& {\lambda }_3& 0& {\lambda }_2& 0\\ -P_5\left(0\right)& 0& 0& {\mu }_3& -{\lambda }_1-{\mu }_2-{\lambda }_3& 0& 0& {\mu }_1\\ -P_6\left(0\right)& {\mu }_2& {\lambda }_3& 0& 0& -{\mu }_1-{\lambda }_2-{\mu }_3& {\lambda }_1& 0\\ -P_7\left(0\right)& 0& 0& {\mu }_2& 0& {\mu }_1& -{\lambda }_1-{\lambda }_2-{\mu }_3& 0\\ -P_8\left(0\right)& {\mu }_3& {\lambda }_2& 0& {\lambda }_1& 0& 0& -{\mu }_1-{\mu }_2-{\lambda }_3\end{array}\right|,\\ {\Pi }_1=\left({\lambda }_1+{\lambda }_2+{\lambda }_3+{\mu }_1+{\mu }_2+{\mu }_3\right)\left({\lambda }_1+{\mu }_1\right)\left({\lambda }_2+{\lambda }_3+{\mu }_2+{\mu }_3\right)\left({\lambda }_2+{\mu }_2\right)\left({\lambda }_1+{\lambda }_3+{\mu }_1+{\mu }_3\right)\left({\lambda }_3+{\mu }_3\right)\left({\lambda }_1+{\lambda }_2+{\mu }_1+{\mu }_2\right);\end{array}$

$\begin{array}{l}{\Delta }_1\left(v_2\right)=\left|\begin{array}{cccccccc}-P_1\left(0\right)& 0& {\mu }_1& 0& {\mu }_2& 0& {\mu }_3& 0\\ -P_2\left(0\right)& {\lambda }_1+{\lambda }_2+{\lambda }_3& 0& {\lambda }_1& 0& {\lambda }_2& 0& {\lambda }_3\\ -P_3\left(0\right)& 0& {\lambda }_1+{\mu }_2+{\mu }_3& 0& 0& {\mu }_3& 0& {\mu }_2\\ -P_4\left(0\right)& {\mu }_1& 0& {\mu }_1+{\lambda }_2+{\lambda }_3& {\lambda }_3& 0& {\lambda }_2& 0\\ -P_5\left(0\right)& 0& 0& {\mu }_3& {\mu }_1+{\lambda }_2+{\mu }_3& 0& 0& {\mu }_1\\ -P_6\left(0\right)& {\mu }_2& {\lambda }_3& 0& 0& {\lambda }_1+{\mu }_2+{\lambda }_3& {\lambda }_1& 0\\ -P_7\left(0\right)& 0& 0& {\mu }_2& 0& {\mu }_1& {\mu }_1+{\mu }_2+{\lambda }_3& 0\\ -P_8\left(0\right)& {\mu }_3& {\lambda }_2& 0& {\lambda }_1& 0& 0& {\lambda }_1+{\lambda }_2+{\mu }_3\end{array}\right|,\\ {\Pi }_2=\left(-{\lambda }_1-{\lambda }_2-{\lambda }_3-{\mu }_1-{\mu }_2-{\mu }_3\right)\left(-{\lambda }_1-{\mu }_1\right)\left(-{\lambda }_2-{\lambda }_3-{\mu }_2-{\mu }_3\right)\left(-{\lambda }_2-{\mu }_2\right)\left(-{\lambda }_1-{\lambda }_3-{\mu }_1-{\mu }_3\right)\left(-{\lambda }_3-{\mu }_3\right)\left(-{\lambda }_1-{\lambda }_2-{\mu }_1-{\mu }_2\right);\end{array}$

$\begin{array}{l}{\Delta }_1\left(v_3\right)=\left|\begin{array}{cccccccc}-P_1\left(0\right)& 0& {\mu }_1& 0& {\mu }_2& 0& {\mu }_3& 0\\ -P_2\left(0\right)& {\lambda }_1-{\mu }_2-{\mu }_3& 0& {\lambda }_1& 0& {\lambda }_2& 0& {\lambda }_3\\ -P_3\left(0\right)& 0& {\lambda }_1-{\lambda }_2-{\lambda }_3& 0& 0& {\mu }_3& 0& {\mu }_2\\ -P_4\left(0\right)& {\mu }_1& 0& {\mu }_1-{\mu }_2-{\mu }_3& {\lambda }_3& 0& {\lambda }_2& 0\\ -P_5\left(0\right)& 0& 0& {\mu }_3& {\mu }_1-{\mu }_2-{\lambda }_3& 0& 0& {\mu }_1\\ -P_6\left(0\right)& {\mu }_2& {\lambda }_3& 0& 0& {\lambda }_1-{\lambda }_2-{\mu }_3& {\lambda }_1& 0\\ -P_7\left(0\right)& 0& 0& {\mu }_2& 0& {\mu }_1& {\mu }_1-{\lambda }_2-{\mu }_3& 0\\ -P_8\left(0\right)& {\mu }_3& {\lambda }_2& 0& {\lambda }_1& 0& 0& {\lambda }_1-{\mu }_2-{\lambda }_3\end{array}\right|,\\ {\Pi }_3=\left(-{\lambda }_1+{\lambda }_2+{\lambda }_3-{\mu }_1+{\mu }_2+{\mu }_3\right)\left(-{\lambda }_1-{\mu }_1\right)\left({\lambda }_2+{\lambda }_3+{\mu }_2+{\mu }_3\right)\left({\lambda }_2+{\mu }_2\right)\left(-{\lambda }_1+{\lambda }_3-{\mu }_1+{\mu }_3\right)\left({\lambda }_3+{\mu }_3\right)\left(-{\lambda }_1+{\lambda }_2-{\mu }_1+{\mu }_2\right);\end{array}$

Time interval in a day is considered

$0\le t\le 7.$

The computational results are summarized in Table 1 . Figure 2 illustrates them.

where [0, 5] is time interval of a transient process; and (5, 7] is time interval of stationary (stabilized) mode.

At every time moment t, the computations are verified in terms of the invariant condition

${\displaystyle \underset{i=1}{\overset{8}{\sum }}{P}_i\left(t\right)=1}.$

It is easy to show with the help of the Table that the total of each column elements is equl to 1. Specifically, it should be mentioned that under t = 0 and t→∞ the general formulas of state probabilities are simplified and take the form

$\begin{array}{l}{P}_1\left(0\right)={\displaystyle \underset{k=1}{\overset{8}{\sum }}\frac{{\Delta }_1\left(v_k\right)}{{\Pi }_k}};P_2\left(0\right)={\displaystyle \underset{k=1}{\overset{8}{\sum }}\frac{{\Delta }_2\left(v_k\right)}{{\Pi }_k}};\cdots \\ P_1\left(\infty \right)=\frac{{\Delta }_1\left(v_1\right)}{{\Pi }_1};P_2\left(\infty \right)=\frac{{\Delta }_2\left(v_1\right)}{{\Pi }_1};\cdots \end{array}$ (34)

where the specified initial conditions $P_1\left(0\right),P_2\left(0\right),\cdots$ as well as limit state probabilities $P_1\left(\infty \right),P_2\left(\infty \right),\cdots$ comply with the equations

${\displaystyle \underset{i=1}{\overset{8}{\sum }}{P}_i\left(0\right)=1};{\displaystyle \underset{i=1}{\overset{8}{\sum }}{P}_i\left(\infty \right)=1}.$

Eight variants of equivalent systems of algebraic levels relative to the limit state probabilities are compiled in expanded form. Limit state probabilities are calculated using Cramer’s formulas.

Variant 1.

$\Vert \begin{array}{cccccccc}1& 1& 1& 1& 1& 1& 1& 1\\ 0& -{\mu }_1-{\mu }_2-{\mu }_3& 0& {\lambda }_1& 0& {\lambda }_2& 0& {\lambda }_3\\ {\lambda }_1& 0& -{\mu }_1-{\lambda }_2-{\lambda }_3& 0& 0& {\mu }_3& 0& {\mu }_2\\ 0& {\mu }_1& 0& -{\lambda }_1-{\mu }_2-{\mu }_3& {\lambda }_3& 0& {\lambda }_2& 0\\ {\lambda }_2& 0& 0& {\mu }_3& -{\lambda }_1-{\mu }_2-{\lambda }_3& 0& 0& {\mu }_1\\ 0& {\mu }_2& {\lambda }_3& 0& 0& -{\mu }_1-{\lambda }_2-{\mu }_3& {\lambda }_1& 0\\ {\lambda }_3& 0& 0& {\mu }_2& 0& {\mu }_1& -{\lambda }_1-{\lambda }_2-{\mu }_3& 0\\ 0& {\mu }_3& {\lambda }_2& 0& {\lambda }_1& 0& 0& -{\mu }_1-{\mu }_2-{\lambda }_3\end{array}\Vert \cdot \Vert \begin{array}{c}{P}_1\left(\infty \right)\\ P_2\left(\infty \right)\\ P_3\left(\infty \right)\\ P_4\left(\infty \right)\\ P_5\left(\infty \right)\\ P_6\left(\infty \right)\\ P_7\left(\infty \right)\\ P_8\left(\infty \right)\end{array}\Vert =\Vert \begin{array}{c}1\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\Vert .$(35)

Solution is

$P_i\left(\infty \right)=\frac{{\Delta }_i}{\Delta },\left(i=1,2,3,\cdots ,8\right),$ (36)

where

$\Delta =\left|\begin{array}{cccccccc}1& 1& 1& 1& 1& 1& 1& 1\\ 0& -{\mu }_1-{\mu }_2-{\mu }_3& 0& {\lambda }_1& 0& {\lambda }_2& 0& {\lambda }_3\\ {\lambda }_1& 0& -{\mu }_1-{\lambda }_2-{\lambda }_3& 0& 0& {\mu }_3& 0& {\mu }_2\\ 0& {\mu }_1& 0& -{\lambda }_1-{\mu }_2-{\mu }_3& {\lambda }_3& 0& {\lambda }_2& 0\\ {\lambda }_2& 0& 0& {\mu }_3& -{\lambda }_1-{\mu }_2-{\lambda }_3& 0& 0& {\mu }_1\\ 0& {\mu }_2& {\lambda }_3& 0& 0& -{\mu }_1-{\lambda }_2-{\mu }_3& {\lambda }_1& 0\\ {\lambda }_3& 0& 0& {\mu }_2& 0& {\mu }_1& -{\lambda }_1-{\lambda }_2-{\mu }_3& 0\\ 0& {\mu }_3& {\lambda }_2& 0& {\lambda }_1& 0& 0& -{\mu }_1-{\mu }_2-{\lambda }_3\end{array}\right|,$(37)

${\Delta }_1=\left|\begin{array}{cccccccc}1& 1& 1& 1& 1& 1& 1& 1\\ 0& -{\mu }_1-{\mu }_2-{\mu }_3& 0& {\lambda }_1& 0& {\lambda }_2& 0& {\lambda }_3\\ 0& 0& -{\mu }_1-{\lambda }_2-{\lambda }_3& 0& 0& {\mu }_3& 0& {\mu }_2\\ 0& {\mu }_1& 0& -{\lambda }_1-{\mu }_2-{\mu }_3& {\lambda }_3& 0& {\lambda }_2& 0\\ 0& 0& 0& {\mu }_3& -{\lambda }_1-{\mu }_2-{\lambda }_3& 0& 0& {\mu }_1\\ 0& {\mu }_2& {\lambda }_3& 0& 0& -{\mu }_1-{\lambda }_2-{\mu }_3& {\lambda }_1& 0\\ 0& 0& 0& {\mu }_2& 0& {\mu }_1& -{\lambda }_1-{\lambda }_2-{\mu }_3& 0\\ 0& {\mu }_3& {\lambda }_2& 0& {\lambda }_1& 0& 0& -{\mu }_1-{\mu }_2-{\lambda }_3\end{array}\right|,$(38)

${\Delta }_2=\left|\begin{array}{cccccccc}1& 1& 1& 1& 1& 1& 1& 1\\ 0& 0& 0& {\lambda }_1& 0& {\lambda }_2& 0& {\lambda }_3\\ {\lambda }_1& 0& -{\mu }_1-{\lambda }_2-{\lambda }_3& 0& 0& {\mu }_3& 0& {\mu }_2\\ 0& 0& 0& -{\lambda }_1-{\mu }_2-{\mu }_3& {\lambda }_3& 0& {\lambda }_2& 0\\ {\lambda }_2& 0& 0& {\mu }_3& -{\lambda }_1-{\mu }_2-{\lambda }_3& 0& 0& {\mu }_1\\ 0& 0& {\lambda }_3& 0& 0& -{\mu }_1-{\lambda }_2-{\mu }_3& {\lambda }_1& 0\\ {\lambda }_3& 0& 0& {\mu }_2& 0& {\mu }_1& -{\lambda }_1-{\lambda }_2-{\mu }_3& 0\\ 0& 0& {\lambda }_2& 0& {\lambda }_1& 0& 0& -{\mu }_1-{\mu }_2-{\lambda }_3\end{array}\right|,$(39)

${\Delta }_3=\left|\begin{array}{cccccccc}1& 1& 1& 1& 1& 1& 1& 1\\ 0& -{\mu }_1-{\mu }_2-{\mu }_3& 0& {\lambda }_1& 0& {\lambda }_2& 0& {\lambda }_3\\ {\lambda }_1& 0& 0& 0& 0& {\mu }_3& 0& {\mu }_2\\ 0& {\mu }_1& 0& -{\lambda }_1-{\mu }_2-{\mu }_3& {\lambda }_3& 0& {\lambda }_2& 0\\ {\lambda }_2& 0& 0& {\mu }_3& -{\lambda }_1-{\mu }_2-{\lambda }_3& 0& 0& {\mu }_1\\ 0& {\mu }_2& 0& 0& 0& -{\mu }_1-{\lambda }_2-{\mu }_3& {\lambda }_1& 0\\ {\lambda }_3& 0& 0& {\mu }_2& 0& {\mu }_1& -{\lambda }_1-{\lambda }_2-{\mu }_3& 0\\ 0& {\mu }_3& 0& 0& {\lambda }_1& 0& 0& -{\mu }_1-{\mu }_2-{\lambda }_3\end{array}\right|,$(40)

${\Delta }_8=\left|\begin{array}{cccccccc}1& 1& 1& 1& 1& 1& 1& 1\\ 0& -{\mu }_1-{\mu }_2-{\mu }_3& 0& {\lambda }_1& 0& {\lambda }_2& 0& 0\\ {\lambda }_1& 0& -{\mu }_1-{\lambda }_2-{\lambda }_3& 0& 0& {\mu }_3& 0& 0\\ 0& {\mu }_1& 0& -{\lambda }_1-{\mu }_2-{\mu }_3& {\lambda }_3& 0& {\lambda }_2& 0\\ {\lambda }_2& 0& 0& {\mu }_3& -{\lambda }_1-{\mu }_2-{\lambda }_3& 0& 0& 0\\ 0& {\mu }_2& {\lambda }_3& 0& 0& -{\mu }_1-{\lambda }_2-{\mu }_3& {\lambda }_1& 0\\ {\lambda }_3& 0& 0& {\mu }_2& 0& {\mu }_1& -{\lambda }_1-{\lambda }_2-{\mu }_3& 0\\ 0& {\mu }_3& {\lambda }_2& 0& {\lambda }_1& 0& 0& 0\end{array}\right|.$(41)