91Ó°ÊÓ

In Example \(5.3\) if server \(i\) serves at an exponential rate \(\lambda_{i}, i=1,2\), show that \(P\\{\) Smith is not last \(\\}=\left(\frac{\lambda_{1}}{\lambda_{1}+\lambda_{2}}\right)^{2}+\left(\frac{\lambda_{2}}{\lambda_{1}+\lambda_{2}}\right)^{2}\)

Let \(X\) be an exponential random variable with rate \(\lambda .\) (a) Use the definition of conditional expectation to determine \(E[X \mid Xc] P[X>c \mid $$

One hundred items are simultaneously put on a life test. Suppose the lifetimes of the individual items are independent exponential random variables with mean 200 hours. The test will end when there have been a total of 5 failures. If \(T\) is the time at which the test ends, find \(E[T]\) and \(\operatorname{Var}(T)\).

Let \(X_{1}\) and \(X_{2}\) be independent exponential random variables, each having rate \(\mu .\) Let $$ X_{(1)}=\operatorname{minimum}\left(X_{1}, X_{2}\right) \text { and } X_{(2)}=\operatorname{maximum}\left(X_{1}, X_{2}\right) $$ Find (a) \(E\left[X_{(1)}\right]\) (b) \(\operatorname{Var}\left[X_{(1)}\right]\) (c) \(E\left[X_{(2)}\right]\) (d) \(\operatorname{Var}\left[X_{(2)}\right]\)

In a certain system, a customer must first be served by server 1 and then by server \(2 .\) The service times at server \(i\) are exponential with rate \(\mu_{i}, i=1,2 .\) An arrival finding server 1 busy waits in line for that server. Upon completion of service at server 1 , a customer either enters service with server 2 if that server is free or else remains with server 1 (blocking any other customer from entering service) until server 2 is free. Customers depart the system after being served by server \(2 .\) Suppose that when you arrive there is one customer in the system and that customer is being served by server \(1 .\) What is the expected total time you spend in the system?

There are two servers available to process \(n\) jobs. Initially, each server begins work on a job. Whenever a server completes work on a job, that job leaves the system and the server begins processing a new job (provided there are still jobs waiting to be processed). Let \(T\) denote the time until all jobs have been processed. If the time that it takes server \(i\) to process a job is exponentially distributed with rate \(\mu_{i}, i=1,2\), find \(E[T]\) and \(\operatorname{Var}(T)\)

Each entering customer must be served first by server 1 , then by server 2 , and finally by server \(3 .\) The amount of time it takes to be served by server \(i\) is an exponential random variable with rate \(\mu_{i}, i=1,2,3 .\) 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.

Consider \(n\) components with independent lifetimes, which are such that component \(i\) functions for an exponential time with rate \(\lambda_{i} .\) Suppose that all components are initially in use and remain so until they fail. (a) Find the probability that component 1 is the second component to fail. (b) Find the expected time of the second failure. Hint: Do not make use of part (a).

Let \(X\) and \(Y\) be independent exponential random variables with respective rates \(\lambda\) and \(\mu\), where \(\lambda>\mu .\) Let \(c>0\) (a) Show that the conditional density function of \(X\), given that \(X+Y=c\), is $$ f_{X \mid X+Y}(x \mid c)=\frac{(\lambda-\mu) e^{-(\lambda-\mu) x}}{1-e^{-(\lambda-\mu) c}}, \quad 0

The lifetimes of A's dog and cat are independent exponential random variables with respective rates \(\lambda_{d}\) and \(\lambda_{c} .\) One of them has just died. Find the expected additional lifetime of the other pet.