Boğaziçi University Mathematics Colloquium

Approximation of the Exit Probability of a Stable Markov Modulated Constrained Random Walk
Ali Devin Sezer
ODTÜ, Turkey
Özet : Let X be the constrained random walk on Z 2 + having increments (1,0), (−1,1), (0,−1) with jump probabilities λ(Mk), μ1(Mk), and μ2(Mk) where {Mk} is an irreducible aperiodic finite state Markov chain. X represents the lengths of two tandem queues with arrival rate λ(Mk), and service rates μ1(Mk), and μ2(Mk). We assume that the average arrival rate with respect to the stationary measure of M is less than the average service rates, i.e., X is assumed stable. Let τn be the first time X hits the line ∂An = {x : x(1) + x(2) = n}, i.e., the first time the sum of the components of X equals n. Let Y be the random walk on Z × Z+ (i.e., constrained only on ∂2 = {y ∈ Z × Z+ : y(2) = 0}) again modulated by M and having increments (−1,0), (1,1), (0,−1) with probabilities λ(Mk), μ1(Mk), and μ2(Mk). Let B = {y ∈ Z 2 : y(1) = y(2)} and let τ be the first time Y hits B. Let Tn : Z 2 7→ Z 2 be the affine map y 7→ (n − y(1), y(2) and let m denote the initial point of M. For x ∈ R 2 +, x(1) + x(2) < 1, x(1) > 0, and xn = bnxc, we show that P(Tn(xn),m) (τ < ∞) approximates P(xn,m) (τn < τ0) with exponentially vanishing relative error as n → ∞. For the analysis we define a characteristic matrix in terms of the jump probabilities of (X,M). The 0-level set of the characteristic polynomial of this matrix defines the characteristic surface H ⊂ C 2 for the problem. Conjugate points on H and the associated eigenvectors of the characteristic matrix are used to define (sub/super) harmonic functions which play a fundamental role both in our analysis and the computation / approximation of P(y,m) (τ < ∞).
  Tarih : 20.03.2019
  Saat : 13:30
  Yer : TB 130, Bogaziçi University
  Dil : English