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Midterm Exam - Discrete Event Systems - 19.11.2014

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Midterm Exam - Discrete Event Systems - 19.11.2014

Exercise 1

A machine makes finished products from raw parts that are always available. Occasionally, machine operation is interrupted for inspection. After inspection, the machine either resu- mes production or goes under maintenance. After maintenance, the machine starts a new production, because during maintenance semifinished parts inside the machine are removed.

1. Model the machine through a stochastic automaton (E, X , Γ, p, x

0

), assuming that the machine is initially working and the probability that the machine goes under mainte- nance after inspection is q = 1/10.

Assume that productions take 12, 25, 18, 20, 23, 16, 15, 21 and 26 minutes; inspections are scheduled at time instants 30, 75 and 120 minutes, and take 10 minutes; maintenance is required only after the third inspection and takes 30 minutes.

2. Determine the average machine throughput (number of finished products per hour) over the time interval [0, 210] minutes.

Assume that productions have random durations following a uniform distribution over the in- terval [10, 30] minutes; inspections are scheduled 45 minutes after the machine starts/restarts operation and take 10 minutes; maintenance has a random duration following an exponential distribution with expected value 30 minutes.

3. Compute the probability that at least two products are finished before the first inspec- tion starts.

4. Compute the mean time to machine restart from when an inspection starts.

Exercise 2

A dual-core processor is composed of two central processing units (CPUs) and a dedicated cache. Programs executed by the processor can be of two types. Type 1 programs are those that require sequential computing, and therefore are executed in one CPU only. Type 2 programs allow for parallel computing and therefore can be executed simultaneously in both CPUs.

Consider a dual-core processor preceded by a cache which may host only one program waiting to be executed. Programs arrive as generated by a Poisson process with rate λ = 0.05 arrivals· s

−1

. Programs are of type 1 with probability p = 2/3. The number of operations required by a generic program is a random variable L whose cumulative distribution function can be approximated by an exponential distribution with expected value 10

6

operations. The speed of each CPU is v = 10

5

operations· s

−1

. If a program arrives when the cache is full, it is rejected. A type 2 program is executed only when both CPUs are available. For a type 2 program requiring L operations, L/2 operations are executed in one CPU and L/2 operations in the other (parallel computing).

1. Model the system through a stochastic timed automaton (E, X , Γ, p, x

0

, F ), assuming

that both CPUs are initially idle.

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2. Assume that both CPUs are executing type 1 programs and the cache hosts a type 1 program. Compute the probability that both CPUs are idle when the next program arrives.

3. Assume that both CPUs are executing type 1 programs and the cache hosts a type 2 program. Compute the mean time to the start of the execution of the type 2 program.

4. Assume that both CPUs are executing type 1 programs and the cache hosts a type 2

program. Show the procedure to compute the probability that the two type 1 programs

and the type 2 program are all executed within T = 15 s.

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