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Chapter 5: Problem 6

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

Short Answer

Expert verified
The probability that Smith is not the last customer is \(P\{\text{Smith is not last}\} = \left(\frac{\lambda_1}{\lambda_1+\lambda_2}\right)^2 + \left(\frac{\lambda_2}{\lambda_1+\lambda_2}\right)^2\).

Step by step solution

01

Find the probability Smith is served by server 1

Since Smith can be served by either server 1 or server 2, the probability of Smith being served by server 1 would be given by the ratio of the server's serving rate to the combined serving rates of both servers as follows: \[\frac{\lambda_1}{\lambda_1+\lambda_2}\]
02

Find the probability Smith is served by server 1 and not the last customer

We want to find the probability that Smith is served by server 1 and not the last customer. If Smith is not last, it means server 2 finishes serving its other customer after server 1 finishes serving Smith. This probability can be obtained by multiplying the probability from step 1 by the probability that server 1 finishes first (before server 2). \[\frac{\lambda_1}{\lambda_1+\lambda_2} \times \frac{\lambda_1}{\lambda_1+\lambda_2} = \left(\frac{\lambda_1}{\lambda_1+\lambda_2}\right)^2\]
03

Find the probability Smith is served by server 2

Similarly, we can find the probability that Smith is served by server 2 as the ratio of server 2's serving rate to the combined serving rate of both servers: \[\frac{\lambda_2}{\lambda_1+\lambda_2}\]
04

Find the probability Smith is served by server 2 and not the last customer

To find the probability that Smith is served by server 2 and not the last customer, we do the same as we did in step 2 for server 1 but now for server 2. We multiply the probability that he is served by server 2 by the probability that server 1 finishes after server 2. \[\frac{\lambda_2}{\lambda_1+\lambda_2} \times \frac{\lambda_2}{\lambda_1+\lambda_2} = \left(\frac{\lambda_2}{\lambda_1+\lambda_2}\right)^2\]
05

Find the probability Smith is not the last customer

Finally, we add the probabilities from step 2 and step 4 to find the total probability that Smith is not the last customer. \[\left(\frac{\lambda_1}{\lambda_1+\lambda_2}\right)^2 + \left(\frac{\lambda_2}{\lambda_1+\lambda_2}\right)^2\] Therefore, the probability that Smith is not the last customer is: \[P\{\text{Smith is not last}\} = \left(\frac{\lambda_1}{\lambda_1+\lambda_2}\right)^2 + \left(\frac{\lambda_2}{\lambda_1+\lambda_2}\right)^2\]

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