Question: 2. A certain queuing system has two types of customers and two types of servers. Type A customers arrive according to a Poisson process

2. A certain queuing system has two types of customers and two

2. A certain queuing system has two types of customers and two types of servers. Type A customers arrive according to a Poisson process with rate 3, and, independently Type B customers arrive according to a Poisson process with rate 2. If Server A is free, then an arriving Type A customer begins service with Server A. If Server A is busy but Server B is free, then an arriving Type A customer will begin service with Server B. If an arriving Type A customer finds both servers busy, they will leave the system. If Server B is free, then an arriving Type B customer will be served by Server B, and otherwise will leave the system. Server A takes an exponential rate 2 time to finish a service, Server B takes an exponential rate 1 time to finish a service, and all service times are independent and independent of arrivals (a) Model the system as a four state Markov chain and write down its generator (b) Find the stationary distribution of the Markov chain (c) What is the stationary average number of customers in the system? (d) What is the average time an entering customer spends in the system? (e) What is the long-run proportion of time is there a Type A customer being served by Server B?

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