课程名称︰应用随机过程
课程性质︰选修
课程教师︰吕学一
开课学院：电资学院
开课系所︰资工所、生医电资所
考试日期（年月日）︰2012.06.21
考试时限（分钟）：180
试题 :
台大资工 随机过程 期末考
2012年6月21日 下午两点廿分起三个小时
说明：共十二题，每题十分，可按任何顺序答题。
每题难度不同，审慎判断恰当的解题顺序。
第一题
请定义何谓 Poisson process，并请说明为什么它是一种 continuous-time Markov
chain。
第二题
A server works for an exponentially distributed time with rate μ and then
fails. A TA checks the server at times distributed according to a Poisson
process with rate λ. If the server is found to have failed, then it is
immediately replaced. Find the expected time between two replacements of
servers.
第三题
Customers arrive at a two-server service station according to a Poisson process
with rate λ. Whenever a new customer arrives, any customer in the system
immediately leaves. A new arrival enters service first with server 1 and then
with server 2. Suppose that the service times at the servers are independent
exponentials with respective rates μ_1 and μ_2. Compute the probability that
an entering customer completes the whole service process.
第四题
猫味抓到老鼠的过程是个 Poisson process with rate λ。老鼠有黑、灰、白三种，出
现的机率分别为0.6、0.3、0.1。请证明猫味完整收集三种老鼠所需时间的期望值，洽好
等于完整收集到三种老鼠时总共抓到老鼠数量之期望值的1/λ倍。
第五题
Power surges occur according to a Poisson process with rate λ, and each surge
independently causes our server to fail with probability p. Let T be the time
at which the server fails and let N be the number of surges that a server
failure takes.
(a) Explain why the distribution of T conditioned on N = n is a gamma with
parameter (n, λ).
(b) Explain why the distribution of N conditioned on T = t is 1 plus a
Poisson with parameter λ(1 - p)t.
第六题
Consider a non-homogeneous Poisson process whose intensity function λ(t) is
upper-bounded by some finite number λ. Argue that such a process is equivalent
to a process of "counted events" from a (homogeneous) Poisson process with
rate λ, where an event at time t is "counted" (independent of the past) with
probability λ(t)/λ.
第七题
Consider two servers maintained by one TA. For each i = 1, 2, server i
functions for an exponential time with rate μ_i. The repair time for either
server is exponential with parameter μ. Can we analyze this as a
birth-and-death process? If so, what are the parameters (i.e., birth rate β_i
and death rate δ_i for each integral state i)? If not, how do we characterize
this continuous-time Markov chain? (That is, what are the state space, the
transition rate λ_i of each state i, and the transition probability Ｐ[i, j]
for any two distinct states i and j?)
第八题
There are n individuals in a population, some of whom have a certain infection
that spreads as follows. Contacts between two members of this population occur
in accordance with a Poisson process having rate λ. When a contact occurs, it
╭n╮
is equally likely to involve any of the │ │ pairs of individuals in the
╰2╯
population. If a contact involves an infected and a noninfected individual,
then with probability p the noninfected individual becomes infected. Once
infected, an individual remains infected throughout. Let X(t) denote the
number of infected members of the population at time t. Let X(0) = 1. What is
the expected length of time when all n individuals of the population are
infected.
第九题
Prove Kolmogorov's backward differential equation
P' (t) = Σ q P (t) - λ P (t)
i,j k∈S ik kj i ij
for finite continuous-time Markov chain.
第十题
Consider a taxi station where taxis and customers arrive in accordance with
Poisson processes with respective rates of 1 and 2 per minute. A taxi will
wait no matter how many other taxis are present. However, an arriving customer
who does not find a taxi waiting leaves. Find
(a) the expected number of taxis waiting, and
(b) the long-run proportion of arriving customers who get taxis.
第十一题
Explain why an irreducible positive recurrent continuous-time Markov chain Ｘ
is time-reversible if π_i．q_ij = π_j．q_ji holds for any two states i and j
of Ｘ, where π_i (respectively, π_j) is the limiting probability of state i
(respectively, j) and q_ij (respectively, q_ji) is the instantaneous rate of
i-to-j (respectively, j-to-i) transitions.
第十二题
Characterize the uniformized version of the continuous-time Markov chain of
第七题.