课程名称︰ 高等计算机网路 课程性质︰ 选修课程 课程教师︰ 逄爱君 开课学院： 电机资讯学院 开课系所︰ 资讯工程研究所 考试日期（年月日）︰ 2010/01/16 考试时限（分钟）： 180 是否需发放奖励金： 是 （如未明确表示，则不予发放） 试题 : # = 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). -.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-. .-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- -.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-. .-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- -.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-. -- 没有不可能的事, 只有不愿做的事 --

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