68 suppose customers arrive in one server qucue according to poisson distribution with rate a 1

First, we need to find the expected service time for a customer. Since there are three possible

68 suppose customers arrive in one server qucue according to...

6.8 Suppose customers arrive in a one-server queue according to a Poisson distribution with rate λ = 1 (in hours). Suppose that the service times equal 1/4 hour, 1/2 hour, or one hour each with probability 1/3. (a) Assume that the queue is empty and a customer arrives. What is the expected amount of time until that customer leaves? (b) Assume that the queue is empty and a customer arrives. What is the expected amount of time until the queue is empty again? (c) For a large time, what is the probability that there are no customers in the queue?

Transcript

00:01 In this question, the motive is to solve three sub -problems and in the first part, we can say that customers arrive according to poison distribution with the rate of 1 per hour.

00:13 So, let's write over here hour.

00:15 Now, we are also given that service time s is equals to 1 by 4 hour or it can be here 1 by 2 or it can be here 1 hour.

00:27 And the probabilities are also given.

00:30 So what are these probability values? we can write them as s1.

00:34 So s1 value is equal to 1 divided by 4 and the corresponding probability value is probability of s1.

00:42 So this is equal to 1 divided by 3 and it is suppose s2 then it will be equal to 1 divided by 2 and the probability of service time which is equals to 1 divided by 2 so it will also be 1 divided by 3 now s3 value will be equals to 1 hour and the corresponding probability will be probability of s3 and it is also 1 divided by 3 now we need to find the expected value of that is it will be here expected value of service time which means the expected amount of time until the customer leaves so it will be equals to summation of it is si multiplied with probability of si.

01:29 So this value will be here 1 divided by 4 and then it is multiplied with 1 divided by 3.

01:37 Now it will be plus 1 divided by 3 and then it is multiplied with 1 divided by 2.

01:44 Now it will be plus 1 multiplied with 1 divided by 3.

01:49 And hence the expected value of service time will be equals to it is 7 divided by 12 hours or if we convert it into minutes then it will be here 35 minutes.

02:03 So we have obtained the answer and we can put the answer inside the box and then we will proceed towards the next part of this very problem.

02:12 So, in the next part, we need to find the expected amount of time until the queue is empty.

02:20 So, here let us write part b.

02:23 Now, we can say it will be expected...

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