1. **Problem Statement:** We want to model a computer using a time-slotted shared medium network as a Discrete-Time Markov Chain (DTMC). The computer can be in several states: idle, generating a packet, checking the channel, transmitting, and returning to idle. We need to identify the states, sketch the DTMC with transitions and probabilities, and find the long-run fraction of time the system is transmitting. 2. **States Identification:** The system has multiple states: - $I$: Idle state (computer doing nothing). - $G$: Packet generated, ready to check channel next slot. - $C$: Checking channel state. - $T_1$: Transmitting packet, first timeslot. - $T_2$: Transmitting packet, second timeslot. 3. **Transition Probabilities:** - From $I$: - With probability 0.1, generate a packet and move to $G$. - With probability 0.9, remain in $I$. - From $G$: - Move to $C$ (start checking channel). - From $C$: - If channel in use (probability 0.3), remain in $C$. - If channel free (probability 0.7), move to $T_1$. - From $T_1$: - Move to $T_2$. - From $T_2$: - Move to $I$ (transmission finished). 4. **DTMC Sketch:** $$ I \xrightarrow{0.1} G \rightarrow C \xrightarrow{0.7} T_1 \rightarrow T_2 \rightarrow I $$ $$ I \xrightarrow{0.9} I, \quad C \xrightarrow{0.3} C $$ 5. **Long-run fraction of time transmitting:** Let $\pi_I, \pi_G, \pi_C, \pi_{T_1}, \pi_{T_2}$ be the steady-state probabilities. Balance equations: $$ \pi_I = 0.9 \pi_I + \pi_{T_2} $$ $$ \pi_G = 0.1 \pi_I $$ $$ \pi_C = \pi_G + 0.3 \pi_C \Rightarrow \pi_C = \frac{\pi_G}{0.7} = \frac{0.1 \pi_I}{0.7} = \frac{1}{7} \pi_I $$ $$ \pi_{T_1} = 0.7 \pi_C = 0.7 \times \frac{1}{7} \pi_I = 0.1 \pi_I $$ $$ \pi_{T_2} = \pi_{T_1} = 0.1 \pi_I $$ Normalization: $$ \pi_I + \pi_G + \pi_C + \pi_{T_1} + \pi_{T_2} = 1 $$ $$ \pi_I + 0.1 \pi_I + \frac{1}{7} \pi_I + 0.1 \pi_I + 0.1 \pi_I = 1 $$ $$ \pi_I \left(1 + 0.1 + \frac{1}{7} + 0.1 + 0.1\right) = 1 $$ $$ \pi_I \times 1.442857 = 1 \Rightarrow \pi_I = \frac{1}{1.442857} \approx 0.693 $$ Then, $$ \pi_{T_1} = 0.1 \times 0.693 = 0.0693, \quad \pi_{T_2} = 0.0693 $$ Long-run fraction transmitting = $\pi_{T_1} + \pi_{T_2} = 0.1386$ (about 13.86%).