Arrivals to a three-server system are according to a Poisson process with rate X

Question

Arrivals to a three-server system are according to a Poisson process with rate X Arrivals finding server 1 free enter service with 1. Arrivals finding 1 busy but 2 free enter service with 2. Arrivals finding both 1 and 2 busy do not join the system. After completion of service at either 1 or 2 the customer will then either go to server 3 if 3 is free or depart the system if 3 is busy. After service at 3 customers depart the system. The service times at i are exponential with rate ??, 1, 2, 3 (a) Define states to analyze the above system. (b) Give the balance equations (c) In terms of the solution of the balance equations, what is the average time that an entering customer spends in the system? Find the probability that a customer who arrives when the system is empty is served by server 3 (d)

Explanation / Answer

you want, you can watch the section of the lecture, think about it, and wrap your head around this to understand how those two problems are equivalent. Note that at time 19:26 in the video above, Joe explains an example of your identical problem, except with 4 boxes and 6 balls.

The formula:

For problems of this structure, the formula to use is:

(n+k?1k)(n+k?1k)

So for this problem, n = 3 boxes, k = 5 balls, so our answer is:

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