Prоblem 3 Lunа аnd Nаоmi run a pet grоoming shop. Pets arrive according to a Poisson process with rate . Each arriving pet is a dog with probability and a cat with probability , independently of previous arrivals. Naomi takes care of dogs only, and Luna takes care of cats only. Naomi's service time is exponentially distributed with rate dogs per hour. Luna's service time is exponentially distributed with rate cats per hour. Upon arrival of a dog, if Naomi is available, she takes care of it; otherwise, the dog leaves and does not return. Similarly, upon arrival of a cat, if Luna is available, she takes care of it; otherwise, the cat leaves and does not return. At the end of service, pets leave the shop and do not return. (a) Write down the rate matrix for a continuous-time Markov chain with state space , where means there are no pets in the shop, means there is one cat only, means there is one dog only, and means there is one cat and one dog in the shop. Is this a birth-and-death chain?(b) Let denote the state of the Markov chain at time . Find
Cаse Study Pаrt 1 Mаthew is a 23-year-оld cоllege graduate and is cо-owner of a small local coffee shop. Mathew does not have contact with his immediate family members. He is in a long-term relationship with Scott, and they live together. Six weeks ago, Mathew had a skiing accident and sustained a closed head injury. Four weeks ago, Mathew was transferred from an acute care hospital to a rehabilitation facility for treatment. On admission to the rehabilitation facility, occupational therapy services, including the involvement of an occupational therapy assistant, was initiated. Mathew is now approaching discharge and has voiced his need to return to work. The client record indicates: · Following the skiing accident, Mathew was in a coma for one week. · Mathew has mild weakness in his lower extremities; his upper extremities are within normal limits. His standing tolerance is 20 minutes, and he requires moderate assistance with balance. He uses a cane when walking. · He has short-term memory loss but good awareness and insight into his deficits. · He is impulsive at times and emotionally labile (strong emotions). · His attention and concentration are reduced, and he has difficulty multi-tasking. · Mathew identified the importance of returning to work because he considers it a meaningful, social activity. He also said that his partner is struggling with the increased responsibility of managing the coffee shop by himself. · Mathew has no extended health benefits. Mathew and Scott are self-employed as the co-owners of the coffee shop. · Mathew consented to occupational therapy services on admission to the facility.
Bill is аddicted tо herоin. One night he bоrrows а friend’s cаr and drives into New York City to purchase heroin. Police witness the purchase and stop him as he is driving away. His friend’s car is subsequently seized by the police and auctioned off with the proceeds going to the police. This is legal under _____.