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Approximation of excessive backlog probabilities of two parallel queues
Date
2018-07-25
Author
Ünlü, Kamil Demirberk
Sezer, Ali Devin
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Let X be the constrained random walk on Z 2 + with increments (1, 0), (−1, 0), (0, 1) and (0, −1) representing the lengths at service completion times of two queues with exponentially distributed interarrival and service times running in parallel. Denote the arrival and service rates by λ i , µ i , i = 1, 2; we assume λ i < µ i , i = 1, 2, i.e., X is assumed stable. Without loss of generality we assume ρ 1 ≥ ρ 2. Let τ n be the first time X hits the line ∂A n = {x ∈ Z 2 : x(1)+x(2) = n}, i.e., when the sum of the components of X equals n for the first time. Let Y be the same random walk as X but only constrained on {y ∈ Z 2 : y(2) = 0} and its jump probabilities for the first component reversed. Let ∂B = {y ∈ Z 2 : y(1) = y(2)} and let τ be the first time Y hits ∂B. The probability p n = P x (τ n < τ 0) is a key performance measure of the queueing system represented by X (suppose the queues share a common buffer, then p n is the probability that this overflows overflows during the systems first busy cycle). We show that, for x n = ⌊nx⌋, x ∈ R 2 + , x(1) + x(2) ≤ 1, x(1) > 0, P (n−xn(1),xn(2)) (τ < ∞) approximates P xn (τ n < τ 0) with exponentially vanishing relative error. Let r = (λ 1 + λ 2)/(µ 1 + µ 2); for r 2 < ρ 2 and ρ 1 = ρ 2 , we construct a class of harmonic functions for Y with which the probability P y (τ < ∞) can be approximated with bounded relative error. For r 2 = ρ 1 ρ 2 , we obtain the exact formula P y (τ < ∞) = r y(1)−y(2) + r(1−r) r−ρ2 ρ y(1) 1 − r y(1)−y(2) ρ y(2) 1. The Y-harmonic functions are constructed from single points and pairs of conjugate points on a characteristic surface associated with X. The results are obtained using the approach of (Sezer, Exit Probabilities and Balayage of Constrained Random Walks, 2015).
URI
https://hdl.handle.net/11511/75269
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Graduate School of Applied Mathematics, Conference / Seminar
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K. D. Ünlü and A. D. Sezer, “Approximation of excessive backlog probabilities of two parallel queues,” presented at the International Conference on Queueing Theory and Network Applications QTNA2018, (25 - 27 Temmuz 2018), Tsukuba, Japan, 2018, Accessed: 00, 2021. [Online]. Available: https://hdl.handle.net/11511/75269.