# Excessive backlog probabilities of two parallel queues

2020-10-01
Unlu, Kamil Demirberk
Sezer, Ali Devin
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.i and service rates mu i, i = 1, 2; we assume.i < mu i, i = 1, 2, i.e., X is assumed stable. Without loss of generality we assume.1 =.1/mu 1 similar to.2 =.2/mu 2. Let tn 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 t be the first time Y hits. B. The probability pn = Px (tn < t0) 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 = similar to nx similar to, x. R2+, x(1) + x(2) similar to 1, x(1) > 0, P( n- xn (1),xn (2))(t < 8) approximates Pxn (tn < t0) with exponentially vanishing relative error. Let r = (.1+.2)/(mu 1+mu 2); for r 2 <.2 and.1 similar to=.2, we construct a class of harmonic functions from single and conjugate points on a related characteristic surface for Y with which the probability Py(t < 8) can be approximated with bounded relative error. For r 2 =.1.2, we obtain the exact formula Py(t < 8) = r y(1)-y(2) + r (1-r) r-.2 similar to. y(1) 1 - r y(1)- y(2). y(2) 1 similar to.
ANNALS OF OPERATIONS RESEARCH

# Suggestions

 APPROXIMATION OF EXCESSIVE BACKLOG PROBABILITIES OF TWO TANDEM QUEUES Sezer, Ali Devin (2018-09-01) Let X be the constrained random walk on Z(+)(2) having increments (1, 0), (-1, 1), and (0, -1) with respective probabilities A lambda,mu 1, and mu 2 representing the lengths of two tandem queues. We assume that X is stable and mu 1 not equal mu 2. Let tau(n) be the first time when the sum of the components of X equals n. Let Y be the constrained random walk on Z x Z(+) having increments (-1, 0), (1, 1), and (0, -1) with probabilities lambda, mu(1), and mu(2). Let tau be the first time that the components of...
 Approximation of the exit probability of a stable Markov modulated constrained random walk Kabran, Fatma Basoglu; Sezer, Ali Devin (2020-06-01) Let X be the constrained randomwalk on Z(+)(2) having increments (1, 0), (- 1, 1), (0,- 1) with jump probabilities lambda(M-k), mu(1)(M-k), and mu(2)(M-k) where M is an irreducible aperiodic finite state Markov chain. The process X represents the lengths of two tandem queues with arrival rate lambda(M-k), and service rates mu(1)(M-k), and mu(2)(M-k); the process M represents the random environment within which the system operates. We assume that the average arrival rate with respect to the stationary measur...
 Effective Mass Quantum Systems with Displacement Operator: Inverse Square Plus Coulomb-Like Potential Arda, Altug; Sever, Ramazan (2015-10-01) The Schrodinger-like equation written in terms of the displacement operator is solved analytically for a inverse square plus Coulomb-like potential. Starting from the new Hamiltonian, the effects of the spatially dependent mass on the bound states and normalized wave functions of the "usual" inverse square plus Coulomb interaction are discussed.
 Effective-mass Dirac equation for Woods-Saxon potential: Scattering, bound states, and resonances AYDOĞDU, OKTAY; Arda, Altug; Sever, Ramazan (2012-04-01) Approximate scattering and bound state solutions of the one-dimensional effective-mass Dirac equation with the Woods-Saxon potential are obtained in terms of the hypergeometric-type functions. Transmission and reflection coefficients are calculated by using behavior of the wave functions at infinity. The same analysis is done for the constant mass case. It is also pointed out that our results are in agreement with those obtained in literature. Meanwhile, an analytic expression is obtained for the transmissi...
 Factorization of Joint Probability Mass Functions into Parity Check Interactions Bayramoglu, Muhammet Fatih; Yılmaz, Ali Özgür (2009-07-03) We show that any joint probability mass function (PMF) can be expressed as a product of parity check factors an d factors of degree one with the help of some auxiliary variables, if the alphabet size is appropriate for defining a parity chec k equation. In other words, marginalization of a joint PMF is equivalent to a soft decoding task as long as a finite field can be constructed over the alphabet of the PMF. In factor graph terminology this claim means that a factor graph representing such a joint PMF alw...
Citation Formats
K. D. Unlu and A. D. Sezer, “Excessive backlog probabilities of two parallel queues,” ANNALS OF OPERATIONS RESEARCH, pp. 141–174, 2020, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/32637.