Exam of Discrete Event Systems - 04.02.2016
Exercise 1
A molecule can switch among three equilibrium states, denoted by A, B and C. Feasible state
transitions are from A to B, from C to A, and from B to both other states. The initial state is A.
1. Assume that state holding times in B are all equal to tB > 0, and transitions from B to A
and from B to C alternate (the first time from B to A, the second time from B to C, the
third time from B to A, and so on). Moreover, assume that state holding times in A take
values 8.0, 3.0, 5.0, 12.0, 8.0, 6.0, 5.0, 8.0, 4.0 and 9.0 ms, and state holding times in C take
values 10.0, 6.0, 9.0, 6.0 and 8.0 ms. Determine tB so that the fraction of time spent by the
molecule in B over a time horizon of length T = 100 ms is 40%.
Now assume that, whenever the molecule leaves B, transition to A occurs with probability q = 1/3.
Moreover, state holding times in A follow a uniform distribution over the interval [6, 12] ms, state
holding times in B are all equal to 8 ms, and state holding times in C follow a uniform distribution
over the interval [5, 10] ms.
2. Model the dynamics of the molecule through a stochastic timed automaton (E, X , Γ, p, x0 , F ).
3. Compute the probability that, after the sixth event, the molecule is in A.
4. Assume that the molecule enters B from A. Compute the probability that the molecule
returns to A within T = 15 ms.
5. Compute the probability that the molecule is in A at time t = 18 ms.
Exercise 2
A manufacturing cell is composed by two machines M1 and M2 , as shown in the figure.
M1
M2
Arrivals of raw parts are generated by a Poisson process with rate 8 arrivals/hour. Raw parts
arriving when M1 is busy, are rejected. Raw parts are processed in M1 . When M1 terminates a
job:
• if M2 is available, the finished product leaves the system with probability q = 9/10, otherwise
it is sent to M2 for inspection;
• if M2 is busy, the finished product leaves the system.
After inspection in M2 , inspected products turn out to be non-defective with probability p = 15/16,
and leave the system. Defective products are sent again to M1 for rectification. If M1 is busy,
the defective product is kept by M2 until M1 terminates the ongoing job. Processing of a part
in M1 takes a random time following an exponential distribution with expected value 5 minutes.
Inspections in M2 have random durations following an exponential distribution with expected value
3 minutes. The manufacturing cell is initially empty.
1. Model the manufacturing cell through a stochastic timed automaton (E, X , Γ, p, x0 , F ).
2. Assume that M1 is processing a part and M2 is inspecting a product. Compute the probability
that the manufacturing cell is emptied while avoiding a situation where M1 is idle and M2 is
inspecting a product.
Only first part
3. Compute the average state holding time when M1 is processing a part and M2 is inspecting
a product.
4. Assume that M1 is idle and M2 is inspecting a product. Compute the probability that the
inspection terminates successfully within T = 5 minutes, and exactly one arrival of a raw
part occurs.
Exercise 3
Consider the system of Exercise 2.
1. Verify the condition λef f = µef f for the system at steady-state.
2. Compute the utilization of M1 and M2 at steady state.
3. Compute the average time spent by a generic product in M2 upon a single inspection at
steady state.
Exercise 4
A small shop keeps a stock of a maximum of three smartphones. The daily demand D of smartphones follows a Poisson distribution
P (D = n) =
λn −λ
e ,
n!
n = 0, 1, 2, 3, . . .
where λ = 0.5. Demand exceeding the availability is not satisfied. Every morning, at opening, the
shop manager checks the availability of smartphones, and orders three of them if and only if the
stock is empty. Delivery of the new smartphones occurs the morning after with probability p = 0.7,
and the second day otherwise.
1. Model the stock size at shop opening through a discrete-time homogeneous Markov chain.
Assume the stock initially full.
2. Assume that the stock is full. Compute the probability that no order of smartphones is
carried out during the next five days.
3. Assume that the stock is full. Determine how often, on average, the shop manager carries
out an order of smartphones.