1. 设 X1 和 X2 是独立的指数随机变量，速率为 1； 2、分别。让
X(1) = min(X1;X2); X(2) = 最大值(X1;X2)
(a) 如果 1 = 2 = ，则 nd E[X(1)];Var[X(1)];E[X(2)];Var[X(2)]。
(b) 如果 1 = 2，则 nd E[X(1)];Var[X(1)];E[X(2)];Var[X(2)]。

2.每个进入的客户必须首先由服务器1服务，然后由服务器2，最后由服务器
3. 服务器 i 提供服务所需的时间是一个指数随机变量，其中

(a) 求当您转移到服务器 2 时服务器 3 仍然忙碌的概率。
(b) 求当您转移到服务器 3 时服务器 3 仍然忙碌的概率。
(c) 找出您在系统中花费的预期时间。 （每当你遇到

(d) 假设您在系统中包含一个正在接受服务的客户时进入系统

3.若Xi，i=1； 2; 3，是独立的指数随机变量，速率为i，i = 1； 2; 3、第
(a) P fX1 < X2 < X3g，
(b) P fX1 < X2jmax(X1;X2;X3) = X3g，
(c) E[maxXijX1 < X2 < X3]，
(d) E[maxXi]。

1. Let X1 and X2 be independent exponential random variables with rates 1; 2, respectively. Let
X(1) = min(X1;X2); X(2) = max(X1;X2)
(a) If 1 = 2 = , nd E[X(1)];Var[X(1)];E[X(2)];Var[X(2)].
(b) If 1 = 2, nd E[X(1)];Var[X(1)];E[X(2)];Var[X(2)].

2. Each entering customer must be served rst by server 1, then by server 2, and nally by server
3. The amount of time it takes to be served by server i is an exponential random variable with
rate i; i = 1; 2; 3. Suppose you enter the system when it contains a single customer who is being
served by server 3.
(a) Find the probability that server 3 will still be busy when you move over to server 2.
(b) Find the probability that server 3 will still be busy when you move over to server 3.
(c) Find the expected amount of time that you spend in the system. (Whenever you encounter
a busy server, you must wait for the service in progress to end before you can enter service.)
(d) Suppose that you enter the system when it contains a single customer who is being served
by server 2. Find the expected amount of time that you spend in the system.

3. If Xi, i = 1; 2; 3, are independent exponential random variables with rates i, i = 1; 2; 3, nd
(a) P fX1 < X2 < X3g,
(b) P fX1 < X2jmax(X1;X2;X3) = X3g,
(c) E[maxXijX1 < X2 < X3],
(d) E[maxXi].