課程名稱︰系統效能評估
課程性質︰選修
課程教師︰周承復
開課學院：電資學院
開課系所︰資工所、網媒所
考試日期（年月日）︰
考試時限（分鐘）：
試題 :
Mid-term exam. Perf. 2014
1. (10%) Please show the inter-arrival time dist. of a Poisson process with
rate λ is an exponential dist. with parameter λ.
2. (15%) 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 when the service time is exponential with mean 1/μ for
job a and the service time is exponential with mean 1/λ for job b?
3. (25%) Consider the failure of a link in a communication network. Failures
occur according to a Poisson process with rate 9.6 per day. Find
a) P[time between failures ≦ 20 days]
b) P[5 failures in 10 days]
c) Expected time between 2 consecutive failures
d) P[0 failures in next day]
e) Suppose 12 hours have elapsed since last failure, find the expected time
to next failure
4. (15%) Please show that, for a Poisson process with rate λ, its renewal
equation m(t) = λt by the following definition
t
m(t) = F(t) + ∫ m(t-x) f(x) dx
0
5. (15%) The CPU time requirement X of a typical job can be modeled by the
following hyerexponential distribution:
-tλ_1 -tλ_2
P(X ≦ t) = α(1 - e ) + (1 - α)(1 - e )
where α = 0.6, λ_1 = 10, and λ_2 = 1. Compute
(a) the probability density function of X,
(b) the mean service time E[X], and
(c) the variance of service time Var[x].
1 - 3s
6. (20%) Find the function f(t) that satisfies F*(s) = ────────
(1-2s)(1-4s)^2
7. (10%) Customers arrive at a fast-food restaurant at a rate of 5 per minutes
and wait to receive their order for an average 3 minutes. Customers eat in
the restaurant with prob. 0.5 and carry out their order without eating with
prob. 0.5. A meal requires an average of 20 minutes. What is the average
number of customers in the restaurant?
8. (10%) For Geometric dist., please show that it has markov property or it is
memoryless: P[x = i+n│x > n] = P[x = i], where P(i) = P(x=i) = p(1-p)^(i-1)