Final answer:
The infinitesimal generator matrix for the described continuous-time Markov chain will have diagonal elements reflecting the negative sum of the arrival and departure rates and off-diagonal elements based on specific transition rates between states.
Step-by-step explanation:
Infinitesimal Generator Matrix of a Continuous-Time Markov Chain
The scenario described corresponds to a continuous-time Markov chain with a constant arrival rate of customers and a unique rate at which the store owner kicks all customers out regardless of their number. This Markov chain has a countable state space, where each state represents the number of customers in the store. Given that customers arrive at a rate of 3 per minute, and that the rate at which all customers are kicked out is 2 per minute, we can build the infinitesimal generator matrix (Q-matrix).
Since customers arrive at a rate of 3 per minute, the off-diagonal entries qij for i to i+1 will be 3. The diagonal entry for state i, qii, is the negative sum of the rates out of state i. Here, it's −(3+2)=−5, because customers can either arrive, or all customers leave at once, which happens at a rate of 2 per minute no matter the current state of the system. The transitions from every state to the 0 state, which represents an empty store, will have a rate of 2. This means for each state i, where i > 0, the element qi0 will be 2.
An example of the Q-matrix for this system would be:
Q = [
[-3, 3, 0, 0, ...]
[ 2, -5, 3, 0, ...]
[ 0, 2, -5, 3, ...]
[...]
]
Note that this matrix is infinitesimal as all rows sum up to 0, which is a requirement for any infinitesimal generator matrix of a continuous-time Markov chain.