Excessive backlog probabilities of two parallel queues
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Excessive backlog probabilities of two parallel queues Kamil Demirberk Ünlü1,2 · Ali Devin Sezer2 © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract Let X be the constrained random walk on Z2+ with increments (1, 0), (−1, 0), (0, 1) and (0, −1); X represents, at arrivals and service completions, the lengths of two queues (or two stacks in computer science applications) working in parallel whose service and interarrival times are exponentially distributed with arrival rates λi and service rates μi , i = 1, 2; we assume λi < μi , i = 1, 2, i.e., X is assumed stable. Without loss of generality we assume ρ1 = λ1 /μ1 ρ2 = λ2 /μ2 . Let τn be the first time X hits the line ∂ An = {x ∈ Z2 : 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 ∈ Z2 : y(2) = 0} and its jump probabilities for the first component reversed. Let ∂ B = {y ∈ Z2 : y(1) = y(2)} and let τ be the first time Y hits ∂ B. The probability pn = Px (τn < τ0 ) is a key performance measure of the queueing system (or the two stacks) represented by X (if the queues/stacks share a common buffer, then pn is the probability that this buffer overflows during the system’s first busy cycle). Stability of the process implies that pn decays exponentially in n when the process starts off the exit boundary ∂ An . We show that, for xn = nx, x ∈ R2+ , x(1) + x(2) 1, x(1) > 0, P(n−xn (1),xn (2)) (τ < ∞) approximates Pxn (τn < τ0 ) with exponentially vanishing relative error. Let r = (λ1 +λ2 )/(μ1 +μ2 ); for r 2 < ρ2 and ρ1 = ρ2 , we construct a class of harmonic functions from single and conjugate points on a related characteristic surface for Y with which the probability Py (τ < ∞) can be approximated with bounded relative error. For r 2 = ρ1ρ2 , we obtain the exact formula Py (τ < ∞) = r y(1)−y(2) +
r (1−r ) r −ρ2
y(1)
ρ1
y(2)
− r y(1)−y(2) ρ1
.
Keywords Approximation of probabilities of rare events · Exit probabilities · Constrained random walks · Queueing systems · Large deviations
1 Introduction This work concerns the random walk X with independent and identically distributed increments {I1 , I2 , I3 , ...}, constrained to remain in Z2+ :
B
Ali Devin Sezer [email protected]
1
Department of Statistics, Ankara University, Ankara, Turkey
2
Institute of Applied Mathematics, Middle East Technical University, Ankara, Turkey
123
Annals of Operations Research
. X 0 = x ∈ Z2+ , X k+1 = X k + π(X k , Ik ), k = 1, 2, 3, ... . v, if x + v ∈ Z2+ , π(x, v) = 0, otherwise, . Ik ∈ V = {(1, 0), (−1, 0), (0, 1), (0, −1)}, P(Ik = (1, 0)) = λ1 , P(Ik = (0, 1)) = λ2 , P(Ik = (−1, 0)) = μ1 , P(Ik = (0, −1)) = μ2 .
(1) . The dynamics of X are depicted in Fig. 1. We denote the constraining boundaries by ∂i = {x ∈ Z2 : x(i) = 0}, i = 1, 2. A well known interpretation for X is as the embedded random walk of two parallel queues with Poisson arrivals and independent and exponentially distributed service times. This random walk appears in com
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