Question 1: System with replacement
The lifetime L of a system has an exponential distribution with parameter λ. After a failure, the system is replaced by a new one. A replacement takes a random time Z which is exponentially distributed with parameter µ. All life- and replacement times are assumed to be independent. Define the stochastic process by {Xt : t ≥ 0} with Xt = 1 is the system is operating and Xt = 0 if the system is being replaced.
a) Explain why {Xt : t ≥ 0} is a continuous time Markov chain.
b) Denote with p1(t) = P (Xt = 1) and p0 = P (Xt = 0). Show that the stochastic process is goverend by the master equations
p´0(t) = -µp0(t) + λp1(t)
p´1(t) = µp0(t) - λp1(t)
c) Find the exact solution p0(t), p1(t) to the above equations for the intitial condition p1(0) = 1.
d) Find the long-time limit p1∞ = limt→∞ from the above solution. Compare p1∞ to the stationary probability measure Π1.
Question 2: A dying population
Consider the stochastic process {N (t) : t ≥ 0}, where N (t) denotes the size of the population of a certain species at time t. Initially, at time t = 0, the population has M members. For some unknown reasons, no offsprings are born for t ≥ 0. Each member of the population has the same probability γh to die in a small time interval h.
a) What is the state space S of this process?
b) Explain why pj(t + h) = pj(t) - jγhpj(t) + (j + 1)γhpj+1(t) + o(h) for pj (t) = P (N (t) = j) and derive the master equation for p?j(t).
c) For M = 3, give the matrix of transition rates wij of this process.
d) Find the stationary probability measure Π of this process for general M .
e) Define the time-dependent mean size of the population by µ1(t) = Σn=0M npn(t). Show that µ1(t) = Me-γt.
f)Check that the long-time limit µ1(t → ∞) agrees with µ1 calculated with the invariant probability Π.
Question 3: Swimming algae
The trajectories of swimming microorganisms like algae show random reorientations that in the case of phototaxis can be described by the Fokker-Planck equation
∂/∂t.p(θ, t) = -∂/∂θ [v(θ)p(θ, t)] +D/2.∂2/∂θ2p(θ, t)
for the periodic probability density p(θ, t) with -Π ≤ θ ≤ Π and p(-Π, t) = p(Π, t) with v(θ) = -uθ for -Π < θ < Π and v(θ) = 0 for θ = ±Π.
a) Under which conditions does the Fokker-Planck equation conserve the norm of p?
b)Find the stationary probability density peq(θ) under the conditions found in a).
c)Assume that the algae swim with constant absolute magnitude of velocity v0 in a two-dimensional plane, Vx = v0 sin θ, Vy = v0 cos θ with v0 > 0 a constant. Define functions Kn(t) = ∫-ΠΠ θnsinθ p(θ, t)dθ and Ln(t) = ∫-ΠΠ θncosθ p(θ, t)dθ for n = 0, 1, 2, . . .. Find the time evolution equation for the average velocities E[Vx], E[Vy] in terms of Kn and Ln.