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Second Partial Exam - Discrete Event Systems - 30.01.2013

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Second Partial Exam - Discrete Event Systems - 30.01.2013

Exercise 1

One method of transport used in living cells is axonal transport, in which certain (motor) proteins carry cargo such as mitochondria, other proteins, and other cell parts, on long mi- crotubules. These microtubules can be thought of as the “tracks” of the transportation mechanism. Assume to break the microtubule into N + 1 equally sized intervals, labelled from 0 (start) to N (end). Every second, the motor protein may have moved to the next interval with probability p = 1/2, or back to the previous with probability q = 1/3.

1. Model the described transportation system of living cells by a discrete-time homoge- neous Markov chain.

2. Determine the maximum N such that the average time from start to end does not exceed 4 seconds.

3. Using N = 4, compute the probability that the motor protein never comes back to start.

Exercise 2

Consider a production system composed by a machine preceded by a unitary buffer. The machine may process parts of two types, namely type 1 and type 2. Parts arrive according to Poisson processes with rates λ

1

= 1 and λ

2

= 0.8 arrivals/hour, respectively. Arriving parts are rejected if the system is full. When the machine receives a new part, and the new part is of different type than the previous, it must be reconfigured. Duration of a reconfiguration is exponentially distributed with expected value 15 minutes. Processing times of parts of both types are also exponentially distributed with rates µ

1

= 1.4 and µ

2

= 1.2 services/hour, respectively.

1. Model the described production system by a continuous-time homogeneous Markov chain.

2. Assuming that the production system is initially empty, and the machine is configured for parts of type 1, compute the probability that a part arriving after exactly 3 hours is not accepted. Justify the answer.

3. Compute the average reconfiguration time that a generic part has to wait in the machine at steady state.

4. Compute the utilization of the machine at steady state.

5. Verify the condition λ

ef f

= µ

ef f

at steady state.

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