=Paper=
{{Paper
|id=Vol-1995/paper-15-970
|storemode=property
|title=
Sensitivity Analysis of Steady State Reliability Characteristics of a Repairable Cold Standby Data Transmission System to
the Shapes of Lifetime and Repair Time Distributions of its Elements
|pdfUrl=https://ceur-ws.org/Vol-1995/paper-15-970.pdf
|volume=Vol-1995
|authors=Hector G. K. Houankpo,Dmitry V. Kozyrev
}}
==
Sensitivity Analysis of Steady State Reliability Characteristics of a Repairable Cold Standby Data Transmission System to
the Shapes of Lifetime and Repair Time Distributions of its Elements
==
https://ceur-ws.org/Vol-1995/paper-15-970.pdf
107
Sensitivity Analysis of Steady State Reliability Characteristics
of a Repairable Cold Standby Data Transmission System to the
Shapes of Lifetime and Repair Time Distributions
of its Elements
Hector G. K. Houankpo, Dmitry V. Kozyrev
Department of Applied Probability and Informatics
Peoples’ Friendship University of Russia (RUDN University)
6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation
Email: gibsonhouankpo@yahoo.fr, kozyrev_dv@rudn.university
Continuous development of computer networks and data transmission systems emphasizes
the increasing need for adequate mathematical models and tools that allow for the study of
their performance and reliability.
We consider the problem of sensitivity analysis of reliability characteristics of a repairable
cold standby data transmission system with exponential life time and general repair time dis-
tributions to shapes of the input distributions. The simulation model based on the discrete-
event approach is introduced to obtain results in case of a general (non-exponential) distri-
bution of repair time of the elements.
There are several approaches to model the reliability of systems with general life- and
repair times distributions. Anyhow all of them are reduced to markovization of the process
that describes the system behavior [1].
The proposed analytical methodology allows to assess system-level reliability in case of
failures of system elements. Explicit analytical expressions were obtained for the stationary
probability distribution of system states, which enable to analyze other operational variables
of the system with respect to the redundant element’s performance.
The demonstrated analytical and simulation results show excellent asymptotic insensitivity
of the stationary reliability of the system under “fast” recovery of its elements to the type
of repair time distribution. Comparison of numerical and graphical results obtained using
both analytical and simulation approaches, shows that they have close agreement, so the
elaborated simulation model can be used in cases when explicit analytical solution can not
be achieved, or as a part of a more complex simulation model.
Key words and phrases: system reliability, steady state probabilities, sensitivity, math-
ematical modeling and simulation.
1. Introduction
Continuous development of computer networks and data transmission systems em-
phasizes the increasing need for adequate mathematical models, and tools that allow
for the study of their functioning. You must have the help of both the design stage (for
comparison-making) and exploitation (Service Quality Management) network systems.
Indeed, the development of complex technical system requires not only high-quality
simulation to check how well it is constructed logically, but also required a priori verifi-
cation of system performance during the design phase. The aim is to conduct analytical
and simulation of the system hM3 |GI|1i.
2. The Model and Analytical Results
Consider a random process v(t) — the number of failed elements at time t, the set
of states of the system E = 0, 1, 2, 3, 4. To describe the behavior of the system using the
Markov process, we introduce an additional variable x(t) ∈ R+ 2 — time spent at time
Copyright © 2017 for the individual papers by the papers’ authors. Copying permitted for private and
academic purposes. This volume is published and copyrighted by its editors.
In: K. E. Samouilov, L. A. Sevastianov, D. S. Kulyabov (eds.): Selected Papers of the VII Conference
“Information and Telecommunication Technologies and Mathematical Modeling of High-Tech Systems”,
Moscow, Russia, 24-Apr-2017, published at http://ceur-ws.org
108 ITTMM—2017
t, for the repair of the failed element. We obtain a two-dimensional process (v(t), x(t)),
with an extended state space = (0), (1, x), (2, x), (3, x).
We denote p0 (t) — the probability that at time t the system is in the state i = 0,
pi (t; x) — density distribution (in continuous component) the probability that at time
t the system is in state i(i = 1, 2, 3), and the time taken to repair the failed element is
in the range (x, x + dx).
p0 (t) = p{v(t) = 0},
p1 (t, x)dx = p{v(t) = 1, x < x(t) < x + dx},
p2 (t, x)dx = p{v(t) = 2, x < x(t) < x + dx},
p3 (t, x)dx = p{v(t) = 3, x < x(t) < x + dx}.
With the help of the formula of total probability we move to a system of Kolmogorov
differential equations
Zt
p0 (t + ∆) = p 0 (t) · (1 − α∆) + p1 (t, x)δ(x)∆dx,
0
p1 (t + ∆, x + ∆) = p1 (t, x) · (1 − α∆) · (1 − δ(x)∆),
p2 (t + ∆, x + ∆) = p2 (t, x) · (1 − α∆) · (1 − δ(x)∆),
p3 (t + ∆, x + ∆) = p3 (t, x) · (1 − δ(x)∆) + p2 (t, x) · α∆,
Zt
p (t + ∆, 0)dx = p (t) · α∆ + p2 (t, x)δ(x)∆dx,
1 0
0
Zt Zt
p2 (t + ∆, 0)dx =
p1 (t, x)α∆dx + p3 (t, x)δ(x)∆dx,
0 0
and a passage to the limit ∆ → 0, we get:
Z∞
α · p0 =
p1 (x)δ(x)dx,
0
dp1 (x)
= −(α + δ(x)) · p1 (x),
dx
dp2 (x)
= −(α + δ(x)) · p2 (x),
dx
dp3 (x)
= −δ · p3 (x) + αp2 (x),
dx
Z∞
p (0)dx = α · p + p2 (x)δ(x)dx,
1 0
0
Z∞ Z∞
p2 (t0)dx = p1 (x)αdx + p3 (x)δ(x)dx,
0 0
Houankpo Hector G. K., Kozyrev Dmitry V. 109
and under assumption that the process has a stationary distribution when t → ∞, we
get:
Z∞
α · p0 =
p1 (x)δ(x)dx,
0
dp1 (x)
= −(α + δ(x)) · p1 (x),
dx
dp2 (x)
= −(α + δ(x)) · p2 (x),
dx
dp3 (x)
= −δ · p3 (x) + αp2 (x),
dx
Z∞
p (0)dx = α · p + p2 (x)δ(x)dx,
1 0
0
Z∞ Z∞
p2 (t0)dx = p1 (x)αdx + p3 (x)δ(x)dx.
0 0
From here, we move on to the solution obtained system of Kolmogorov differen-
tial equations using the method of variation of constants, and obtain the stationary
probabilities of the system states
b̃2 (α) b̃(α)(1 − b̃(α))
p0 = , p1 = ,
ρ− 1(1 − b̃(α)) + b̃(α) ρ− 1(1 − b̃(α)) + b̃(α)
(1 − b̃(α))2 (1 − b̃(α))(ρ− 1 − 1 + b̃(α))
p2 = , p3 = , where ρ−1 = b · α.
ρ− 1(1 − b̃(α)) + b̃(α) ρ− 1(1 − b̃(α)) + b̃(α)
Obviously, there is a dependency of stationary probabilities of states of the system
on the type of distribution of repair time.
Figure 1 shows plots of the stationary probability of failure-free operation of the
system on the relative speed of recovery.
Evidently, this dependence becomes vanishingly small with a “quick” recovery.
Explicit analytic expressions for the stationary distribution of the system under
consideration cannot always be obtained. Therefore, the problem arose of constructing a
simulation model that would adequately approximate the analytic model of the system.
3. Comparison and Analysis of the Results of Mathematical
and Simulation Modeling
Define the following states of the modeled system:
— state 0: one (main) element is running, the second — in cold reserve;
— state 1: one device failed and is under repair, the second — works;
— state 2: one device refused, one — in repair, the second is waiting for its turn for
repair, the third is running;
— state 3: all appliances refused, one — in repair, the others wait their turn for
repair.
110 ITTMM—2017
Figure 1. Plots of the stationary probability of failure-free operation 1 − p3 of ρ the various
functions of the distribution of repair time
For clarity, the simulation model is represented graphically in Figure 2 in the form
of a block diagram.
The criterion for stopping the main cycle of the model is to achieve the maximum
model execution time T . The simulation was carried out with the limitation for the
maximum model time T = 10000 runs.
Figure 3 shows plots of the probability Failure-free operation of the system from the
model parameter ρ, constructed from the results of simulation modeling.
Evidently, the differences between the curves with growth become vanishingly small.
The constructed simulation model well approximates the analytical model.
Table 1 shows the values of the stationary state probabilities calculated by a simu-
lation and the analytical formulas.
Evidently, the simulation results are in good agreement with the results obtained by
explicit analytical formulas. It also evidently that with increasing ρ differences in the
values of pi disappear.
4. Conclusions
Explicit analytical expressions for the stationary probability distribution of states
of the system and a fixed probability of failure of the system were obtained in the gen-
eral case, and for some special cases of distributions. These formulas show the clear
dependence of these characteristics on the form of the distribution function. However,
numerical research and analysis charting shown that this dependence becomes vanish-
ingly small for a “fast” recovery, that is, relative to the growth rate of recovery ρ.
It was conducted simulation system hM3 |GI|1i based on discrete-event approach.
Numerical and graphical comparison of results obtained using both approaches, shows
a high degree of similarity, it can be used as the analytical solution and the simulation
model (e.g., as part of a more complex simulation models).
Houankpo Hector G. K., Kozyrev Dmitry V. 111
BEGIN
Declaring and initializing variables and arrays
First event (failure) generation
yes
Checking stop condition t=T
(t>T)
no
Determine the next event
System state change
Simulation clock update
Collect statistics
Return simulation results
END
Figure 2. Scheme of the simulation system
Acknowledgments
The publication was financially supported by the Ministry of Education and Sci-
ence of the Russian Federation (the Agreement number 02.A03.21.0008), and RFBR
according to the research projects No. 17-07-00142 and No. 17-01-00633.
References
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112 ITTMM—2017
Figure 3. Graphs of the dependence of the probability of failure-free operation of the
system 1 − p3 on the model parameter ρ per the results of simulation
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Houankpo Hector G. K., Kozyrev Dmitry V. 113
Table 1
Stationary probabilities pi of system states < M3 /GI/1 >, imitation calculated analytically
for different values of the model parameter ρ = 1, 10, 100