Suppose that one of five servers i = 1, , 5 can be used to process a certain req

Question

Suppose that one of five servers i = 1, , 5 can be used to process a certain request. Thereby, each server i = 1, , 5 rejects the request with probability of 1/i and gives a successful response with the probability of 1 - i/5. Suppose that the user does not know about the server characteristics and sends request to a randomly chosen server. a) Suppose that a request was sent to a server and was rejected. Then it was sent again to the same server and was rejected again. If the request will now be sent to another server (randomly chosen from the remaining four), what is the probability of a successful response? b) Suppose that a request was sent to a server and was rejected. Then it was sent to another server (randomly chosen from the remaining four) and was also rejected. Then the request is sent to another server (randomly chosen from the remaining three). Calculate the probability of a successful response.

Explanation / Answer

Given : P(failure) = 1/i ,that is 1,1/2,1/3,1/4,1/5 respectively for the 1st,2nd,3rd,4th & 5th servers.

& P(success) = 1-(i/5) , that is 4/5,3/5,2/5,1/5,0 respectively for the 1st,2nd,3rd,4th & 5th servers.  

a) Here, the request was sent to the same server & has been rejected twice. We want the probability of successful response for the same server after sending the request to the same server again & again.

In short, we are repeating the trials until we get the successful response. It follows Geometric distribution.

P(X = x) = [1-(1/i)]x-1*[1-(i/5)] , x = 3,4,5,... & i = 1,2,3,4,5

For example , if the first server is selected to send the request again & again till we get the first success, then

required probability = 1*1*4/5=4/5 ... success in the 3rd attempt

= 1*1*1*4/5 = 4/5 = 4/5*1 ... success in the 4th attempt

Therfore , for the first server probability is (1-i/5) =4/5

Now for the second server, Probability = 1/2*1/2*3/5 = 1/4*3/5 = 3/20 for the 3rd attempt

= 1/2*1/2*1/2*3/5 = 1/8*3/5 = 3/40 = 3/20*1/2 and so on

b) request sent to the server , it is rejected. Therefore sent to another server, again rejected . Now to get the probability of suceess if the request is to any of the remaing 3 servers , then the probability of success is

= 1*1/2*(1-i/5) = 1/2*(1-i/5) , i = 3,4,5

= 1/3*(1-i/5) , i = 2,4,5 and so on

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