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課程名稱︰ 高等計算機網路
課程性質︰ 選修課程
課程教師︰ 逄愛君
開課學院： 電機資訊學院
開課系所︰ 資訊工程研究所
考試日期（年月日）︰ 2010/01/16
考試時限（分鐘）： 180
是否需發放獎勵金： 是
（如未明確表示，則不予發放）
試題 :
# = Lambda
1. Is it true that: (9%)
a. {n(t)<n} if and only if {Sn>t}?
b. {n(t)<=n} if and only if {Sn >= t } ?
c. {n(t)>n} if and only if {Sn<t}?
2. (9%)Let N(t),t>=0 be a Poisson process with rate #(Lambda). Let Sn demote the time of
the n th event.Find
a. E[S4],
b. E[S4|N(1)=2],
c. E[N(4)-N(2)|N(1)=3].
3. (10%) Let T1, T2 .....denote the interarrival times of events of a nonhomogeneous Poisson
Process having intensity function #(t).
a. Find the distribution of T1.
b. Find the distribution of T2.
4. (10%) Customers arrive at a two-server service station according to a Poisson process with
rate #. Whenever a new customer arrives, any customer that is in the system immediately
departs. A new arrival enter service first with server 1 and then with server 2. If the
service times at the servers are independent exponentials with respective rates u1 and u2,
what proportion of entering customers completes their service with server 2?
5. (10%)An insurance company pays out claims on its life insurance policies in accordance with
a Poisson process having rate #=5 per weel. If the amount of money paid on each policy is
exponentially distributed with mean $2000, what is the mean and variance of the amount of
money paid by the insurance company in a four-week span? (Please describe how to get your
answer.)
6. (10%) PLease describe why doesn't the counter-example satisfy the lack of anticipation
assumption (LAA) and (5%) what is the LAA? The counter-example is hown as follows
7. (10%) A store opens at 8 A.M. from 8 until 10 customers arrive at a Poisson rate of four
an hour. Between 10 and 12 they arriev at a Poisson rate of eight an hour. From 12 to 2
the arrival rate increases steadily from eight per hour at 12 to ten per hour at 2; and
from 2 to 5 the arrival rate drops steadily from ten per hour at 2 to four per hour at 5.
Determine the probability distribution of the number of customerts that enter the store
on a given day.
8. (13%) Busloads of customers arrive at an infinite server queu at a Poisson rate #. Let G
denote the service distribution. A bus contains j customers with probability aj, j =1...
Let X(t) denote the number of customers that have been served by time t.
a. E[X(t)] (7%)
b. Is X(t) Poisson distributed ? Please describe your reason. (6%)
9. (14%) The number of trials to be performed is a Poisson random variable with mean #. Each
trial has n possible outcomes and, independent of everything else, results in outcome number i
with probability Pi, Sum(Pi) = 1. Let Xj denote the number of outcomes that occur exactly j
times , j =0,1,2.... Compute E[Xj], Var(Xj).
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