What is the generator and stationary distribution

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Probability With Applications Assignment -

Q1. Suppose a frog jumps among lily pads 1, 2 and 3 spends an exponentially distributed length of time on lily pad i with mean 4 - i hours. The frog jumps to the higher numbered unoccupied lily pad if the frog rolls anything but a 6. If a 6 comes up, the frog jumps to the lower numbered unoccupied lily pad. What is the generator and stationary distribution for X_t where X_t is the frog's location at time t?

Q2. Suppose we have 3 machines. Each machine breaks down according to a Poisson process with rate α = 2 per hour. The time it takes to repair a machine is exponentially distributed with rate µ = 3 per hour. A broken machine is costing us $100 per hour. Each repair-person on staff costs us $50 per hour whether working or idle. Would it be better to have 1 or 2 people repairing machines?

Q3. Suppose we have a gas station with two gas pumps. Cars arrival according to a Poisson process with rate 12/hour, but they may not stop for gas if the station is too busy. The length of time for a car to get gas is exponentially distributed with mean 5 minutes. If an arriving driver sees either 0 or 1 customers in the system, the driver stops for gas. Drivers who see 2 or 3 customers in the system stop for gas with probability 1/2. Drivers who see 4 or more customers in the system, do not stop for gas. Let X_t be the number of cars in the gas station at time t.

(a) What is the state space and generator?

(b) Determine L and L_q.

(d) What fraction of customers are lost?

Q4. Suppose we have a large parking lot-pretend it can hold an unlimited number of vehicles. Suppose cars arrive according to a Poisson process with arrival rate λ > 0. Suppose each car remains in the lot for an exponentially distributed length of time with mean 1/µ. When does the stationary distribution exist, and when it exists, what is the stationary distribution of the number of cars in the lot?

Reference no: EM132287665

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