課程名稱︰系統效能評估
課程性質︰資工系研究所選修
課程教師︰周承復
開課學院：電機資訊學院
開課系所︰資工系
考試日期（年月日）︰2016.4.20
考試時限（分鐘）：120 (mins)
試題 :
1. (25%) Consider a system with two computers. Two jobs: a and b are submitted
to that system simultaneously. Job a and b are assigned directly to each computer.
What is the probability that job a is still in a system after job b finished?
i. (5%) the service time for each job is exactly 15 mins
ii. (5%) the service time for each job are i mins with prob. 1/3, i = 1, 2, 3
iii. (5%) the service time for job a is exponential with mean 20 mins,
the service time for job b is exponential with mean 10 mins
iv.(10%) the service time for job a is uniformly distributed [0, 20 mins], and
the service time for job b is uniformly distributed [0, 10 mins]
2. (10%) Consider the failure of a link in a communication network. Failures occur
according to a Poisson process with rate 2.4 per day. Find
a) P[time between failures <= 15 days]
b) P[7 failures in 10 days]
c) Expected time between 2 consecutive failures
d) P[0 failures in next day]
3. (15%) Packets arrive at a transmission facility according to a Poisson process
with rate lambda. Each packet is independently routed with probability p to one of
two transmission lines and with probability (1 - p) to the other. Show that the
arrival processes at the two transmission lines are Poisson with rate lambda * p,
lambda * (1 - p), respectively.
4. (25%) Consider a sustem with 2 components. We observe the state of the system
every hour. A given component operating at time n has prob. 0.4 of failing before
the next observation at time n + 1. A component that was in a failed condition at
time n has a probability 0.8 of being repaired by time n + 1, independent of how long
the component has been in a failed state. The component failures and repairs are
mutually independent events. Let x_n be the number of components in operation at
time n {x_n, n = 0, 1, 2, ...} is a discrete-parameter homogeneous Markov chain with
the state space I = {0, 1, 2}. Determine its transition probability matrix P. and
draw the state diagram. Obtain the steady-state probability vector if it exists.
5. (10%) Let A(t) and Y(t) denote respectively the age and excess at t.
Find P{Y(t) > x | A(t + x) > s} for a Poisson process.
6. (15%) f''(t) - 4f'(t) + 3f(t) = t, f(0) = 0, f'(0) = 0; find out f(t)
7. (15%) Y = X_1 + X_2 + ... + X_N_tilde
X_i are i.i.d. exponentially distributed, N_tilde is geometrically distributed.
(1) (5%) Find E[Y]
(2) (18%) What is Var[Y]