## 2 Materials and Methods
The overall availability of a communication system without standby in a one-shaft underground mine is based on Markov analysis and system dynamics modeling, using data from the availability ratings provided by the manufacturer if the equipment was not damaged during this period. Availability model of the communication system without standby in a one-shaft underground mine which was shown in Figure 6 (see Supplementary Materials) and comparing results of the model were shown in Figure 11 (see Supplementary Materials) in the Section 3. Figure 5: Site link model. Figure 6: Availability of a communication system without standby in a one-shaft underground mine. Increasing the Probability of Communication System Availability or the Reliability of a Single Standby System  
The standby system consists of two independently operating communication devices with parallel connections. It is believed that one device failure does not affect the other device. The system is defined by four states: (0, 0), (1, 0), and (0, 1) are operating states, and (1, 1) is a failure state. Figure 7 shows all possible states and transitions between them. Description of probability is shown in Figure 7:  
\(\mathrm{P_{t}}[0,0]\) is the probability that the system will run at time (t). \(\mathrm{P_{t}}[1,0]\) is the probability of the second device functioning and failure of the first device at time (t). \(\mathrm{P_{t}}[0,1]\) is the probability of the first device functioning and failure of the second device at time (t). \(\mathrm{P_{t}}[1,1]\) is the probability that the system will not function at time (t). This system is modeled as the following system of four ordinary differential equations based on Markov analysis:  
\[\frac{\mathrm{dP_{t}}[0,0]}{\mathrm{dt}}=-[\lambda_{1}(t)+\lambda_{2}(t)+ \lambda_{5}(t)]\mathrm{P_{t}}[0,0]+\mu_{1}(t)\mathrm{P_{t}}[1,0]+\mu_{2}(t) \mathrm{P_{t}}[0,1]+\mu_{5}(t)\mathrm{P_{t}} \tag{3}\]  
\[\frac{\mathrm{dP_{t}}[1,0]}{\mathrm{dt}}=-[\lambda_{3}(t)+\mu_{1}(t)]\mathrm{ P_{t}}[1,0]+\lambda_{1}(t)\mathrm{P_{t}}[0,0]+\mu_{3}(t)\mathrm{P_{t}}[1,1] \tag{4}\]  
\[\frac{\mathrm{dP_{t}}[0,1]}{\mathrm{dt}}=-[\lambda_{4}(t)+\mu_{2}(t)]\mathrm{ P_{t}}[0,1]+\lambda_{2}(t)\mathrm{P_{t}}[0,0]+\mu_{1}(t)\mathrm{P_{t}}[1,1] \tag{5}\]  
\[\frac{\mathrm{dP_{t}}[1,1]}{\mathrm{dt}}=-[\mu_{4}(t)+\mu_{3}+\mu_{5}(t)] \mathrm{P_{t}}[1,1]+\lambda_{5}(t)\mathrm{P_{t}}[0,0]+\lambda_{3}(t)\mathrm{P_ {t}}[1,0]+\lambda_{4}(t)\mathrm{P_{t}}[0,1] \tag{6}\]  
Initial condition of the system is: \(\mathrm{P_{0}}[0,0]=1\), \(\mathrm{P_{0}}[1,0]=0\), \(\mathrm{P_{0}}[0,1]=0\), \(\mathrm{P_{0}}[1,1]=0\)  
To calculate the equation above, a distribution law to model the failure and repair parameters must be chosen. As the law of exponential distribution has been used often in research papers on device reliability [6], this study has been developed on exponential, Weibull, and Poisson distribution. The development results were almost identical and were considered only in the case of the Weibull distribution law. The density function of Weibull distribution is determined using the following formula:  
Figure 3:0  
Figure 3:1 Reliability model of a communication system with a single standby device.