Queuing systems

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Problem: A system of differential equations must be composed for the systems, given by the graphs in the figure. \[ \begin{array}{l} P_{0}^{\prime}(t)=-0,01 P_{0}(t) \\ P_{1}^{\prime}(t)=0,01 P_{0}(t)-0,01 P_{2}(t) \\ P_{2}^{\prime}(t)=0,01 P_{1}(t)-0,01 P_{2}(t) \\ P_{3}^{\prime}(t)=0,01 P_{2}(t) \end{array} \text { for the possibilities } P_{i}(t), i=0,1, \ldots \] of the form \( P_{2}^{\prime}(t)=0,01 P_{1}(t)-0,01 P_{2}(t) \) for the possibilities \( P_{i}(t), i=0,1, \ldots \) Wherein, assume that at the initial moment \( (t=0) \) the system is in zero state. a) b) c) d)

15.4.1 Queuing systems

3.75 $

Problem: There are three machines, working in the manufactory, which break down with intensities \( \lambda_{1}, \lambda_{2}, \lambda_{3} \) (per day) respectively. There are two regulators in the staff, eliminating the breakdowns of machines with intensities \( \mu_{1}, \mu_{2} \) (per day) respectively. It is required to plot the graph of this queuing system and find the fraction of time, both regulators are busy with work. \begin{tabular}{|c|c|c|c|c|} \hline\( \lambda_{1} \) & \( \lambda_{2} \) & \( \lambda_{3} \) & \( \mu_{1} \) & \( \mu_{2} \) \\ \hline 0,3 & 0,3 & 0,1 & 0,3 & 0,1 \\ \hline \end{tabular}

15.4.2 Queuing systems

7.49 $

Problem: There are three machines, working in the manufactory, which break down with intensities \( \lambda_{1}, \lambda_{2}, \lambda_{3} \) (per day) respectively. There are two regulators in the staff, eliminating the breakdowns of machines with intensities \( \mu_{1}, \mu_{2} \) (per day) respectively. It is required to plot the graph of this queuing system and find the fraction of time, both regulators are busy with work. \begin{tabular}{|c|c|c|c|c|} \hline\( \lambda_{1} \) & \( \lambda_{2} \) & \( \lambda_{3} \) & \( \mu_{1} \) & \( \mu_{2} \) \\ \hline 0,4 & 0,15 & 0,15 & 0,4 & 0,1 \\ \hline \end{tabular}

15.4.3 Queuing systems

7.49 $