dtmc - Class Problems DTMC 1 A company uses two forecasting...

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Class Problems: DTMC 1. A company uses two forecasting tools to predict the demand of its product. Tool i is effective w.p. p i (for i = 1 , 2). If the n th prediction uses tool i and it is observed to be effective, then the ( n + 1) st prediction is also done using the same tool; if it is observed to be ineffective, then the ( n + 1) st prediction is made using the other tool. Let X n be the tool used for the n th prediction. Show that { X n , n 0 } is a DTMC, find its transition probability matrix, and draw the transition diagram. 2. There are two identical routes for messages to traverse from node A to B . If a route was congested when a message needed to be sent, then it will be congested w.p. q when the next message needs to be sent. Likewise, if a route was congestion-free when a message needed to be sent, then it will be congestion-free w.p. p when the next message needs to be sent. The routes behave independently of each other. Let X n be the number of routes available when the n th message is attempted. Show that { X n , n 0 } is a DTMC. 3. Consider a machine tool that is used on an NC-machine. The time to produce a part and inspect it is 1 hour. If the tool has produced
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