Connecting to an Overloaded Web Server: How Many Attempts Before Success?

[steve23063]

Homework Statement

A client is trying to connect to a web server on an overloaded machine that only accepts an average of one in 20 connections.

How many times would the client expect to try before getting a connection? Explain your answer.

The Attempt at a Solution

This is part 5 of a problem and I don't know where to start. I know that to find expected value you take a sum of all events times the probability of each event. I don't see how you can use that here though. Please help.

 

[Dick]

Let's get warmed up. What is the probability of connecting on the first call? What is the probability of failing the first call and connecting on the second? Probability of failing on k calls and then connecting?

 

[steve23063]

1/20 for first call
(1/20)*(19/20)^(k-1) in general to get a connection on the kth try

 

[Dick]

Great! So now with these probabilities how do you get an expectation value for k?

 

[steve23063]

I honestly don't know. I'm back to where I started. I don't know how to figure out how many times the user should expect to try before getting connected.

 

[D H]

Sometimes it helps to look at the complementary problem. For example, the probability of not getting a connection on the first attempt is 19/20. What is the probability of not getting a connection on the first and second attempts? Now generalize: What is the probability of not getting a connection on the first through n^th attempts? (Hint: These are easy probabilities to calculate.) Once you generate a general expression for this latter probability the answer to the original question is fairly easy to answer.

 

[Dick]

For expectation values, you gave the answer in your problem post. Calling your computed probability p_k for any k, then the expectation value of k is the sum k*p_k over all k. This is a 'fairly' easy infinite series.

 

[steve23063]

OK I talked to a friend of mine who says the answer is simply 20. The question was how many times would the client expect to try before connecting and since the probability of connection is 1/20 then he would expect to get connected 1 time if he makes 20 attempts.

Do others here think that infinite series is the way to go?

 

[Dick]

And 20 IS the sum of the series. Whether you HAVE to do the series depends on what you already know. If you already have a formula for this case (which was derived using the series) then no, you don't need to resum the series again.