i posted the qeustions please answer them.

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i posted the qeustions please answer them.

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4. (40 points] Queueing System. Consider a queueing system with a single server working at unit rate. Jobs arrive in pairs (two jobs per arrival) according to a Poisson process with rate 1. Jobs are processed by the server individually (one job at a time) and each one has an inde- pendent Exponential() distributed service requirement. (a) (5 points) Let {X(t),t 2 0} be the number of jobs in the system. Formulate this as a continuous-time Markov chain. (b) (10 points) Show that the stationary distribution (when it exists) satisfies TT-0 X T = -(13-1 + 1-2) i 22. (Hint: Do not try to solve for each beyond this recursion) (c) (10 points) Solve for to and provide the stability condition for the system (in terms of and ) (Hint: Use part (b) and 2 = 1 repeatedly. Re-index the sum as necessary, e.g.: E § = = 1 – 7o. 12 You do not need to solve for each ) (d) (10 points) Let X be the steady-state number of jobs in the system. Show that 38 E[X] = M-22 (Hint: Use part (b) and E(X) = 20e imi repeatedly. Re-index the sum as necessary.) (e) (5 points) How does the steady-state expected sojourn time of this system compare to that of an M/M/1 system with arrival rate 2X? (All other aspects of the systems being the same.) (Hint: Use Little's Law, but be careful with the arrival rate.) (10 point where White Charging Stations. Suppe that te while seiner ne ditented long a highway aconding to a press with rate pe wide (a) je postal What is the stat distance from the start of the highway to the three charging station ( open) What is the probability thwe is a station within 30 ide of the stof highway Suppose your chote wahishe has a mange of 300 miles and as soon as your wale wanke drops to 30 milyon whare to fill at the next station ( 12 points What is the probability your battery runs out before reaching the we? d) N points) What is the expected mumber of stations you pass between charge 0 U = 6 123 03 = 5 1. (20 points] CTMC. Consider the following continuous-time Markov chain on state space S = {1,2,3}: 0 1/2 1/2 P= 1/3 2/3 1 0 P is the one-step transition probability matrix for the embedded discrete-time Markov chain and v is the holding time parameter for state i. (Hint: Solve part (c) first.) (a) (5 points) Find the stationary distribution of the embedded DTMC. (b) (5 points) Find the stationary distribution of the CTMC. (c) (5 points) Formulate an equivalent CTMC with Ūj = Üy = y3 =v. (Make sure you pick and state a valid numerical value for v.) (d) (5 points) Find the stationary distribution of the embedded DTMC in (c). 2. (30 points) Bus Stop. Suppose passengers arrive to a bus stop according to a Poisson process with rate 3 per minute. When a bus comes, all waiting passengers board the bus and it departs (assume boarding is instantaneous and each bus can take an unlimited number of passengers) Suppose buses come regularly, one bus every 10 minutes. (a) (5 points) What is the probability distribution of the number of passengers on a bus? (Be sure to specify the relevant parameter(s) of the distribution.) (b) (5 points) If there are 30 passengers on a bus, what is the probability that 15 of them waited at least 5 minutes at the bus stop? (You may leave your answer as a formula.) Now suppose a bus comes as soon as there are 20 passengers and immediately departs. (c) (5 points) What is the probability distribution of the time between bus departures? (d) (10 points) Let {X(t), t > 0} be the stochastic process representing the number of passengers waiting for a bus at time t. Formulate this stochastic process as a continuous- time Markov chain on S = {0,1,...,19}. (e) (5 points) Find the long-run proportion of time that there are k passengers waiting (for each k ES). (Hint: It is possible to find the answer to part (e) very quickly and without solving balance equations. However, you must explain how you arrived at your answer.)
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