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huktonfonix
Oct 03, 2018
In Random Processes
Hi again. one more question, and then I'll be out of your hair. Let N = (0,1,2,...) be the random variable equal to the number of messages in two M/M/1 queueing systems, each of which operates independently with Poisson input rrates L1, L2 and exponential service rates U1, U2. determine the steady-state probability of n messages in the system as a whole for n = 0,1,2,... and determine the expected value E{N} of the number of messages in the system as a whole. so I believe that I need to tackle this as such: probability of n messages, p(n) = X + Y, where X and Y are independent random variables? I know the probability of X messages in a single queue is usually (1-p)p^X, where p = L/U. but this formula doesn't take into account different distributions as the input and output rates. How do I work distributions into the equation? and I assume once you get the probability of n messages, then you find the expected value the way you would normally calculate such a thing (average of all probabilities)? Thanks.
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huktonfonix
Oct 03, 2018
In Random Processes
Hi, I have a couple of queue related questions I hoped you'd be able to provide guidance on. Let N be the total number of messages in two identical M/M/1 queues, each operating independently with input rate L and service rate U. Determine the steady-state probability of n messages in the system as a whole, and determine the average number of message in the system as a whole. I know the probability of m messages, p(m), in a single, M/M/1 queue is = (1-p)p^m, where p = L/U. so, since I have two queues, do I treat the n messages in the system as a whole as a two-random variable sum (Z = X+Y), or do I treat it as a single, since the I/O rates match? I believe I know the average number of messages in the system: since the average number of messages in a single queue is r/(1-r), the average in a two-queue system is 2r/(1-r). is this correct?
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huktonfonix
Oct 03, 2018
In DT Signals and Systems
I can only do this problem partially. My instructor has explained it to me twice, and I'm still having issues. assume a filter ho[n] = h[n] * h[-n] and H(z) = 1/(1-0.9z^-1) is the z transform of h[n], and that h[n] is causal. find Ho(z), the ROC of Ho(z) and ho[n]. I believe that since ho[n] = h[n] * h[-n], Ho(z) = H(z) H(z^-1) = (1/1-0.9(z^-1))(-1/1-0.9(z^-1)) = (1/1-0.9(z^-1))(1/1-0.9(z^-1)), where the left term has a ROC outside of the unit circle radius 0.9, and the right term has a ROC inside of the unit circle radius of 0.9. (These ROCs match the table from my textbook.) these ROCs end up cancelling out. my professor says that the ROC is actually a ring from 0.9 to 1+0.9. my textbook examples all show the addition of two terms giving a ring-shaped ROC, but nothing of multiplication of two terms. my professor did some really quick math to prove it, but I wasn't able to follow it. Also, now I have to repeat the problem for H(z) = 1 - 0.9z^-1. I don't have any charts with a transfer function which looks like this. any ideas what I should do? Thanks.
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