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Declaration: I have read the Department of Statistical Science’s guidance on academic integrity and understand what constitutes academic misconduct. I hereby affirm that the work I am submitting for this assessment is entirely my own, except where indicated by referencing.

Questions

Question 1 [10 points]

    1. [2 points] Let \(n\geq1\) be an integer. Consider the Markov chain on the integers states \(\{-n, \ldots, 0 \ldots, n\}\) with the transitions given by \(P(n,n-1)=1=P(-n,-n+1)\) and for the interior \(|i| <n\), we have \(P(i, i+1) = P(i, i-1) = 1/2\). Compute the stationary distribution for the case \(n=5\).
    1. [3 points] Find the stationary distribution for the case of general \(n\).
    1. [2 points] Consider the two-dimensional case of a Markov chain on the integer lattice points \((x,y)\) with \(|x|, |y| \leq n\), excluding the corners where both \(|x|=|y|=n\). For an interior point, we move left, right, up or down with equal probability; if we reach a boundary point, we step back to the interior with probability \(1\). Make an educated case as to what the stationary distribution must be.
    1. [3 points] Verify your guess in part (c) for the case \(n=5\), by running simulations.

Question 2 [13 points]

You are given \(\Gamma\)–a Poisson point process of intensity \(3\) on the interval \([0, \pi]\). Consider the function \(g(t) = \sin(t) +2\) and the following thinning procedure to produce another point process \(\Gamma'\). Suppose the points of \(\Gamma\) are given by \(\{t_1, \ldots, t_n\}\). We consider independent \(\mathrm{Bern}(p_i)\) random variables, \(X_1, \ldots, X_n\), with \[p_i = g(t_i)/3.\] We let \(\Gamma' \subseteq \Gamma\), where \(t_i\) is a point of \(\Gamma'\) if and only if \(X_i =1\). Let \(M(t) = \Gamma'[0,t)\) be the number of points of \(\Gamma'\) in the interval \([0, t)\).

    1. [2 points] Let \(0<t< \pi\). Prove that \[\lim_{h \to 0 } \frac{1}{h}\mathbb{P}[M(t+h) - M(t) \geq 2] = 0.\]
    1. [4 points] Let \(0<t< \pi\). Prove that \[\lim_{h \to 0} \frac{1}{h}\mathbb{P}[M(t+h) - M(t) = 1] = g(t).\]
    1. [4 points] Demonstrate by simulations, that \[\mathbb{E}M(\pi) = \int_0 ^{\pi} g(t) dt=2\pi +2.\]
    1. [3 points] Identity the distribution of the random variable \(M(\pi)\) via simulations.

Question 3 [15 points]

Suppose items arrive at an exponential rate \(\lambda\) to a queue with two independent servers, each of which serve at the same rate \(\mu > \lambda\). Items will go to the first available server and if both servers are available, we can designate one to be the default choice; this is known as a \(M/M/2\) queue.

    1. [7 points] Following a similar analysis as in notes, calculate the stationary distribution of this queuing system.
    1. [3 points] What is the average time an item spends in this (stationary) system?
    1. [5 points] Verify your answers via simulations with \(\lambda =3\) and \(\mu=5\).

Question 4 [13 points]

Consider a Poisson arrival process on \([0, \infty)\) of intensity \(\lambda\). Suppose we colour the first arrival blue, and then next arrival red, and continue colouring the points in this alternating fashion. Consider the arrival process formed by considering only the red points. Let \(N(t)\) be the number of arrivals by time \(t >0\).

    1. [2 points] Is \(N\) a renewal process? Explain
    1. [3 points] Manually compute \[\lim_{t \to \infty} \mathbb{E}[N(t+1) - N(t)].\]
    1. [4 points] For \(k=0,1,2\), manually compute,
      \[\lim_{t \to \infty} \mathbb{P}[N(t+1) - N(t)=k].\] Here, you maybe able to do the calculation without writing down an integral.
    1. [1 points] Referring back to part (c), write down a formula for the case of general \(k\).
    1. [3 points] Verify both limits above in parts (b) and (c) via simulations, taking \(\lambda =2\).

Question 5 [5 points]

Consider a queue with exponential arrival rate \(\lambda =1\) and exponential service rate \(\mu=2\); this would be a \(M/M/1\) first come first serve queue, except for the following proviso. Suppose that each item is independently assigned a colour red or blue, with probability \(3/4\) and \(1/4\), respectively. The red points get priority over the blue item in the line up; in particular, if there is one blue item being served and there is one blue item waiting to be served and a red item arrives, the red item moves ahead of the blue item and will be served next. Argue that the blue items do eventually get served in finite time.

Endnotes