3. Suppose we have an M/M/1/2 queueing system. That is, suppose customers arrive according to a...

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3. Suppose we have an M/M/1/2 queueing system. That is, suppose customers arrive according to a Poisson process with rate > 0. There is a single server, and the service times are exponentially distributed with mean 1/ > 0. In addition to the server, there is a waiting area that can hold one customer. If an arriving customers sees 2 customers in the system, then the arriving customers is blocked, i.e., lost. Let X be the number of customers in the system just before the nth arrival for n N. Let Y, be the nth state visited, i.e., the jump chain. Let Zn be the uniformized chain. (a) Please give the 3 transition matrices. The elements on the diag- onal of the jump chain transition matrix must be what? (b) Each time the jump chain enters state i, the continuous time queue length process remains in state i for a random length of time that has what distribution what mean? Let (i be that mean. (c) Find the stationary distribution for the jump chain, and call it . (d) The long run proportion of time that the continuous time queue length process spends in state i should be proportional to (i)(i). That is, it should be proportional to how often the jump chain enters state i times the mean length of time that the queue length process remains in state i before the next jump. Determine the distribution that gives the long run proportion of time that the continuous time queue length process is in state i? (e) Find the stationary distribution for the uniformized version, and call it . (f) A similar argument should allow us to obtain from the station- ary distribution of the uniformized process weighted by the mean length of time that the queue length remains in that state until the next state of the uniformized process. For the uniformized process, that mean length of time is the same for all states. Thus, and should be equal. Are they? (If they are, let's use for both.) (g) Pretend that we do not know that PASTA tells us that should also be the stationary distribution embedded just before arrivals? Is C (or if they are equal) the stationary distribution for Xo, X1,...? (BTW, it's much easier to check if a distribution is a stationary distribution than it is to find a stationary distribution. So, if you're approaching this problem by finding the stationary distribution imbedded just before arrivals, then you're making things harder for yourself.) 3. Suppose we have an M/M/1/2 queueing system. That is, suppose customers arrive according to a Poisson process with rate > 0. There is a single server, and the service times are exponentially distributed with mean 1/ > 0. In addition to the server, there is a waiting area that can hold one customer. If an arriving customers sees 2 customers in the system, then the arriving customers is blocked, i.e., lost. Let X be the number of customers in the system just before the nth arrival for n N. Let Y, be the nth state visited, i.e., the jump chain. Let Zn be the uniformized chain. (a) Please give the 3 transition matrices. The elements on the diag- onal of the jump chain transition matrix must be what? (b) Each time the jump chain enters state i, the continuous time queue length process remains in state i for a random length of time that has what distribution what mean? Let (i be that mean. (c) Find the stationary distribution for the jump chain, and call it . (d) The long run proportion of time that the continuous time queue length process spends in state i should be proportional to (i)(i). That is, it should be proportional to how often the jump chain enters state i times the mean length of time that the queue length process remains in state i before the next jump. Determine the distribution that gives the long run proportion of time that the continuous time queue length process is in state i? (e) Find the stationary distribution for the uniformized version, and call it . (f) A similar argument should allow us to obtain from the station- ary distribution of the uniformized process weighted by the mean length of time that the queue length remains in that state until the next state of the uniformized process. For the uniformized process, that mean length of time is the same for all states. Thus, and should be equal. Are they? (If they are, let's use for both.) (g) Pretend that we do not know that PASTA tells us that should also be the stationary distribution embedded just before arrivals? Is C (or if they are equal) the stationary distribution for Xo, X1,...? (BTW, it's much easier to check if a distribution is a stationary distribution than it is to find a stationary distribution. So, if you're approaching this problem by finding the stationary distribution imbedded just before arrivals, then you're making things harder for yourself.)