Question

a hummingbird feeder has four ports where birds may feed. birds arrive at the feeder at a rate of two per minute. if all of the ports are occupied, an arriving hummingbird will leave. otherwise, it will begin feeding at an available port and eat for a time that has an exponential distribution with a mean of 20 seconds. consider this as a continuous time markov chain (ctmc) process where the state is the number of occupied ports. (a) draw the transition diagram.

Answer 1

To answer the student's question about constructing a CTMC transition diagram for hummingbirds feeding at a feeder, we draw a state diagram with transitions representing birds arriving and leaving, considering the given arrival rates and feeding times.

Step-by-step explanation:

The student's question relates to constructing the transition diagram for a continuous time Markov chain (CTMC) process where the state represents the number of occupied ports at a hummingbird feeder. In the given scenario, hummingbirds arrive at the feeder at a rate of two per minute, and they feed for an average of 20 seconds, which is exponentially distributed. The feeder has four ports, and if all ports are occupied, an arriving bird will leave without feeding.

To draw the transition diagram for this CTMC, depict the states (0, 1, 2, 3, 4) as nodes. Draw directed edges between these nodes to represent transitions from one state to the next. For arrivals, transitions will occur from state 'n' to 'n+1' if 'n' is less than 4. For departures, transitions will occur from state 'n' to 'n-1' if 'n' is greater than 0. The arrival rate (2 per minute) should be assigned to transitions increasing the number of birds, and the departure rate, which is the reciprocal of the mean feeding time (1/20 per second or 3 per minute), should be assigned to transitions decreasing the number of birds.