Consider a lost sales production-inventory system where demands arrive according to a Poisson process with rate 4 per hour. For each arrival, demand size is either 1 or 2 with probabilities 0.25 and 0.75, respectively. Production is one-at-a-time and it continues until 4 items is accumulated in the inventory. Production times are Exponential with rate 2 per hour. If the current inventory level is 0, what is the mean required time until the inventory level changes?

Consider a very simplified version of batch processing systems: Jobs arrive at a system according to a Poisson process with rate 4 per hour. Processing starts when there are 2 jobs accumulated in the system, and both jobs are processed at the same time. All processing times are Exponential with rate 5. No new jobs are accepted during processing. What is the steady-state probability that the processor is idle?

Consider a two-state Birth and Death process with states 0 and 1. The birth rate is 2 per hour and the death rate is 3 per hour. Given that the current state is 0, what is the probability that the remaining time until the next transition is greater than 1.5 hours?

Number of COVID-19 infected patients arriving to an hospital follows Poisson distribution with a rate of 3 patients per day. What is the probability that at least one COVID-19 infected patients arrive in a day?

Number of meteors entering the atmosphere of the earth follows a Poisson distribution with rate 3 per week. What is the variance of number of meteors entering earths atmosphere in a day?

These are seperate question each of them.