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A university shuttle travels around campus and serves four stops every 5 minutes.
The diagram below represents the probability that the bus travels to each stop from it's current position:
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- [3]
Write the transition matrix P for this Markov chain.
The shuttle is at the main library at noon.
- [2]
Find the probability (rounded to two decimal places) the shuttle is at the Residence Halls at 12:10 PM.
Let the column vector p be the long run proportion of time spent at each stop.
- [5]
Find p.
The university sets a policy that no stop should handle more than 35% of shuttle arrivals in the long run.
- [1]
Using your answer from (c), state whether the current routing meets this goal.
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A network engineer tracks a web-server's condition hour by hour. At any given hour the server is either
O - online, or
F, offline.
The evolution of the probabilities is modeled by (On+1Fn+1)=Q(OnFn), n∈N, where On is the chance the server is online after n hours and Fn the chance it is offline.
Experimental data gives Q=(0.950.05d0.90). Initially, the server is online, so O0=1,F0=0.
- [1]
Determine the value of d.
- [1]
Explain what d represents in this context.
- [4]
Find the eigenvalues of Q.
- [2]
Find the eigenvectors of Q.
- [3]
Determine the probability (to two decimal places) that the server is online after 6 hours.
- [2]
State the long-term probability that the server is online.
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