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Transient Behavior of a Two-Phase Queuing System with a Limitation on the Total Queue Size

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Abstract

The transient mode of a two-phase queuing system with a Poisson input flow, exponential distribution of service time in each phase, and a limitation on the total buffer size of the two phases is considered. Nonstationary probabilities of system states are found using the Laplace transform. A numerical calculation and analysis of the system performance characteristics in transient mode with parameters corresponding to new-generation optical networks were carried out.

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Funding

This work was supported by the Russian Science Foundation, project no. 23-29-00795. https://rscf.ru/en/project/23-29-00795/

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Correspondence to V. M. Vishnevsky, K. A. Vytovtov or E. A. Barabanova.

Additional information

This paper was recommended for publication by B.M. Miller, a member of the Editorial Board

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APPENDIX

APPENDIX

Formally, eliminating certain terms in Eqs. (4) and preserving the remaining ones can be done using the Heaviside function. However, this function is essentially logical, not analytical, and, therefore, does not allow one to write down an expression for the probabilities of system states in a general form. In particular, when using it in program code, it is necessary to organize additional loops. Therefore, to enable a compact analytical representation of the system of equations (4), the analytical function was introduced

$${{\sigma }_{1}}(x,{{x}_{0}}) = \frac{{\left| {x - {{x}_{0}}} \right| + x - {{x}_{0}}}}{{2\left| {x - {{x}_{0}}} \right|}}.$$
(A.1)

Thus, the function limiting from below the permissible states of the system has the form

$${{{v}}_{1}}(x,M) = \frac{{\left| {x - M + 0.5} \right| + x - M + 0.5}}{{2\left| {x - M + 0.5} \right|}}.$$
(A.2)

For example, for M = 0 the function \({{{v}}_{1}}\)(x, M) has the form shown in Fig. 6.

Fig. 6.
figure 6

Function \({{{v}}_{1}}\)(x, 0).

A shift of 0.5 along the time axis was chosen due to the fact that otherwise, in the state x = M of the system, this function would be indefinite, and its derivative would tend to infinity at this point. Similarly with (A.1), we introduce the function

$${{\sigma }_{2}}(x,{{x}_{0}}) = \frac{{\left| {{{x}_{0}} - x} \right| + {{x}_{0}} - x}}{{2\left| {{{x}_{0}} - x} \right|}}.$$
(A.3)

Thus, the function that limits from above the permissible states of the system can be written in the form

$${{{v}}_{2}}(x,K) = \frac{{\left| {K - x - 0.5} \right| + K - x - 0.5}}{{2\left| {K - x - 0.5} \right|}},$$
(A.4)

where K = \(\overline {0,N} \) is the state of the system. For K = 4, the function \({{{v}}_{2}}\)(x, M) has the form shown in Fig. 7.

Fig. 7.
figure 7

Function \({{{v}}_{2}}\)(x, 4).

Obviously, the function limiting the permissible range of values from the smallest M to the largest K takes the form

$$\begin{aligned} {v}(x,M,N) = {{{v}}_{1}}(x,M){{{v}}_{2}}(x,N) \\ = \frac{{\left( {\left| {x - M + 0.5} \right| + x - M + 0.5} \right)\left( {\left| {K - x - 0.5} \right| + K - x - 0.5} \right)}}{{4\left| {x - M + 0.5} \right|\left| {K - x - 0.5} \right|}}. \\ \end{aligned} $$
(A.5)

For example, with M = 0 and K = 4 it has the form shown in Fig. 8.

Fig. 8.
figure 8

Function \({v}\)(x, 0, 4).

In relation to the problem being solved, x can take the values n1, n2, n1 + n2, etc. The advantage of functions (A.2), (A.4), and (A.5) is the absence of conditions. However, it should be noted that such conditions still exist when the module is expanded. However, despite the fact that these functions do not speed up the calculation process, they allow for analytical study of the resulting expressions and simplify the program code.

To find the function ϑ(nk, nl) (see (6)), which transforms a pair of numbers nk, nl, characterizing the state of the system, into the column number of the matrix A, let us analyze the following pattern for N = 4: for nk = 0 the values of nl change from 0 to N, and the values of ϑ(nk, nl) change from 1 up to N + 1; for nk = 1 the values of nl change from 0 to N – 1, and the values of ϑ(nk, nl) change from N + 2 to 2N + 1; for nk = 2 the values of nl change from 0 to N – 2, and the values of ϑ(nk, nl) change from 2N + 2 to 3N; for nk = 3 the values of nl change from 0 to N – 3, and the values of ϑ(nk, nl) change from 3N + 1 to 4N – 2; for nk = 4 we have nl = 0 and ϑ(nk, nl) = 4N – 1. Therefore, the expression for ϑ(nk, nl) must contain the term nk(N + 1), as well as the term nl. Thus, for nk = 0:

$$\vartheta (0,{{n}_{l}}) = (N + 1){{n}_{k}} + {{n}_{l}} + 1 = (N + 1){{n}_{k}} + {{n}_{l}} + 0 \times ( - 0.5) + 1,$$
(A.6)

for nk = 1:

$$\vartheta (1,{{n}_{l}}) = (N + 1){{n}_{k}} + {{n}_{l}} + 1 = (N + 1){{n}_{k}} + {{n}_{l}} - 1 \times 0 + 1,$$
(A.7)

for nk = 2:

$$\vartheta (2,{{n}_{l}}) = (N + 1){{n}_{k}} + {{n}_{l}} + 0 = (N + 1){{n}_{k}} + {{n}_{l}} - 2 \times 0.5 + 1,$$
(A.8)

for nk = 3:

$$\vartheta (3,{{n}_{l}}) = (N + 1){{n}_{k}} + {{n}_{l}} - 2 = (N + 1){{n}_{k}} + {{n}_{l}} - 3 \times 1 + 1,$$
(A.9)

for nk =4:

$$\vartheta (4,{{n}_{l}}) = (N + 1){{n}_{k}} + {{n}_{l}} - 5 = (N + 1){{n}_{k}} + {{n}_{l}} - 4 \times 1.5 + 1,$$
(A.10)

for nk = m:

$$\vartheta (m,{{n}_{l}}) = (N + 1)m + {{n}_{l}} - (m + 1) = (N + 1)m + {{n}_{l}} - m\frac{{m - 1}}{2} + 1.$$
(A.11)

Thus, expressions (A.6)–(A.11) connect a pair of numbers to the corresponding ordinal number of the element in the row (column) of the coefficient matrix. It is easy to check that relation (6) is valid for any N.

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Vishnevsky, V.M., Vytovtov, K.A. & Barabanova, E.A. Transient Behavior of a Two-Phase Queuing System with a Limitation on the Total Queue Size. Autom Remote Control 85, 46–59 (2024). https://doi.org/10.1134/S0005117924010077

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