probability matrix for a Markov chain

i have a homework on data privacy course and its related to python i need a solution for iti have a homework on data privacy course and its related to python i need a solution for it

Additional Problems for Test 2Note: there was a request for some additional problems, so here they are. You should ‘crowdsource’ the solutions among yourselves by sharing information in the Chat room for the course(follow the link labeled ‘Chat’). I can assist if solutions remain elusive.Problems:1)Customers arrive at a service center as a Poisson process with rate λ. There is one server and onequeue. When there are 0 or 1 customers in the queue the mean service time is µ−1. When there are2 or more customers in the queue the mean service time is reduced to µ−1/2 seconds. Assume theservice time is exponential in all cases, and that the queueing capacity is infinite.a) Write down the rate-balance equations for the stationary distribution of the system.b) Solve the rate balance equations and compute the stationary probability pn for all n ≥ 0.c) Assuming λ = 3 per minute and µ−1 = 30 seconds, find the mean number of customers waitingin queue.2)Consider an irreducible chain on 3 states with transition matrix P. Either prove that (P3)jj mustbe positive for some state j, or give an example where (P3)jj = 0 for every state j.3)Consider the following transition probability matrix for a Markov chain on 4 states:P =0.5 0 0 0.50 0 1 00.5 0.25 0 0.250.75 0.25 0 0Number the states {1, 2, 3, 4} in the order presented.Given that the chain starts in state 1, find the expected number of steps until the first visit to state2.14)A fair die is rolled repeatedly. Let Sn be the sum of the first n rolls, and let Xn be the remainderof Sn divided by 5. Write down the transition matrix for the Markov chain {Xn}, and compute itsstationary distribution. Using this, answer the question “what is the long-run fraction of rolls whereSn is divisible by 5?”.5)Suppose that women enter a supermarket as a Poisson process with rate λ = 5 per minute, and menenter as an independent Poisson process with rate µ = 4 per minute.a) Find the probability that there are at least 2 men among the first 5 customers who enter.b) Find the expected time when the 50th customer enters.6)The law offices of Dodge & Weave have three customer service representatives. Arriving clients mustwait in a waiting room if all representatives are busy with other clients. The waiting room can holdat most 2 people. If the waiting room is full any new arriving client is told to come back anotherday. Assume that this model is described by a M/M/3/5 queue, with arrival rate λ = 8 clients perhour, and mean service time µ−1 = 20 minutes for each customer service representative. Find themean number of clients in the waiting room, and find the mean time spent in the waiting room bya client.2

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