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 1. V. V. Rykov, D. V. Kozyrev, Reliability model for hierarchical systems: Regenera- tive approach, Automation and Remote Control (2010), Vol. 71, Issue 7, pp. 1325– 1336. 2. V. M. Vishnevsky, D. V. Kozyrev, O.V̇. Semenova, Redundant queuing system with unreliable servers, International Congress on Ultra-Modern Telecommunications and Control Systems and Workshops, IEEE Xplore (2015), pp. 283–286. 3. D. V. Kozyrev, Analysis of probability-time characteristics of highly reliable telecommunication systems: the dissertation ... The candidate of physical and mathematical sciences Moscow, 2013, 128 p. The RSL OD, 61 13-1 / 1005 [in Russian]. 4. B. V. Gnedenko, On an unloaded reservation, Izv. Academy of Sciences of the USSR. Techn. cybernetics. (1964), no. 4. pp. 3–12 [in Russian]. 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 5. B. V. Gnedenko, On reservation with restoration, Izv. Academy of Sciences of the USSR. Techn. cybernetics (1964), no. 5, pp. 111–118 [in Russian]. 6. A. D. Soloviev, Reservations with fast restoration, Izv. Academy of Sciences of the USSR, Techn. cybernetics (1970), no. 1, pp. 56–71 [in Russian]. 7. V. V. Rykov, Tran Anh Ngia. On Sensitivity of Systems Reliability Characteristics to the Shape of Their Elements Life and Repair Time Distributions, Bulletin of Peo- ples’ Friendship University of Russia, Series: Mathematics. Information Sciences. Physics (2014), no. 3, pp. 65–77 [in Russian]. 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