I have this multiple choice problem that is testing my understanding of distribution models. I cannot come up with the correct one to solve the problem. Any help would be greatly appreciated!
Your website provides valuable information to the public. Due to your encryption protocols, service requirements and speed, and frequency of internet traffic, you must design your server such that it can handle an average of 2 fee-paying connections at any given time. The design and systems needed to achieve the average of 2 (as well as satisfy your budget constraints) will permit only a maximum of 4 fee-paying connections at any given time. Thus, when there are 4 fee-paying connections, the system blocks others from signing on. What percentage of time will the system block customers from signing on?
a. 6.25%
b. 27%
c. 9%
d. 4.5%
e. cannot be determined with the material we covered in class this semester
f. 22.5%
g. 18%
h. 13.5%
i. 2.5%
The right answer is "c", 9%. I have no idea how to arrive at this answer.
In a M/M/c/c, in equilibrium!, the expected value of customers in the queue is given by: $$\mathbb{E}[n]=\rho (1-P_c),$$ where $\rho$ is the service rate and $P_c$ is the probability of full system ($n=c$) - blocking! From literature, we know $$P_n = \frac{\rho^n}{n!} \left[ \sum_{i=0}^{c} \frac{\rho^n}{n!} \right]^{-1}.$$
So, numerically evaluating them, the only solution $(\rho,P_4) \in \mathbb{R}_{+} \times (0,1)$ is $$(\rho,P_4) = (2.289428485,\, 0.12641953).$$
Which is none of the available answers.
Probably your teacher was expecting and "e", and never had put the time check it. =P