1. **Problem statement:** We have a call center with two telephone lines. Calls arrive following a Poisson process with rate $\lambda > 0$ calls per minute. Each call duration is exponentially distributed with parameter $\mu > 0$. The system states are: - State 0: no line busy - State 1: one line busy - State 2: both lines busy We want to model this as a continuous-time Markov chain (CTMC), find the transition graph and matrix, and simulate the system. 2. **Transition graph and matrix:** - From State 0: - Arrival of a call (rate $\lambda$) moves to State 1 - From State 1: - Arrival of a call (rate $\lambda$) moves to State 2 - Call ends (rate $\mu$) moves to State 0 - From State 2: - Call ends (rate $2\mu$) moves to State 1 - Arrival of a call is lost (no state change) The transition rate matrix $Q$ is: $$ Q = \begin{pmatrix} -\lambda & \lambda & 0 \\ \mu & -(\lambda + \mu) & \lambda \\ 0 & 2\mu & -2\mu \end{pmatrix} $$ 3. **Simulation algorithm for the CTMC:** - Initialize state $s = 0$ and event count $n = 0$ - While $n < 50$: 1. Determine rates of possible transitions from current state $s$ 2. Compute total rate $r = -Q_{ss}$ 3. Sample time to next event $\Delta t$ from exponential distribution with parameter $r$ 4. Choose next state based on transition probabilities $p_{s \to j} = Q_{sj}/r$ 5. Update state $s$ and increment $n$ 6. Record if a call is lost (when in state 2 and arrival occurs) 4. **Python simulation code:** ```python import numpy as np lambda_rate = 1.0 # example arrival rate mu = 0.5 # example service rate Q = np.array([[-lambda_rate, lambda_rate, 0], [mu, -(lambda_rate + mu), lambda_rate], [0, 2*mu, -2*mu]]) state = 0 n_events = 0 lost_calls = 0 while n_events < 50: rates = Q[state].copy() rates[state] = 0 total_rate = -Q[state,state] if total_rate == 0: break dt = np.random.exponential(1/total_rate) probs = rates / total_rate next_state = np.random.choice([0,1,2], p=probs) if state == 2 and next_state == 2: lost_calls += 1 else: state = next_state n_events += 1 proportion_lost = lost_calls / 50 print(f"Proportion of lost calls: {proportion_lost}") ``` 5. **Explanation:** - The CTMC models the number of busy lines. - Transitions correspond to call arrivals and call completions. - Lost calls occur only when both lines are busy and a new call arrives. - The simulation runs 50 events and estimates the proportion of lost calls. **Final answer:** The transition matrix is given above, and the simulation algorithm and code estimate the lost call proportion after 50 events.