Show/Hide Menu
Hide/Show Apps
anonymousUser
Logout
Türkçe
Türkçe
Search
Search
Login
Login
OpenMETU
OpenMETU
About
About
Open Science Policy
Open Science Policy
Communities & Collections
Communities & Collections
Help
Help
Frequently Asked Questions
Frequently Asked Questions
Guides
Guides
Thesis submission
Thesis submission
MS without thesis term project submission
MS without thesis term project submission
Publication submission with DOI
Publication submission with DOI
Publication submission
Publication submission
Supporting Information
Supporting Information
General Information
General Information
Copyright, Embargo and License
Copyright, Embargo and License
Contact us
Contact us
Excessive backlog probabilities of two parallel queues
Date
2020-10-01
Author
Unlu, Kamil Demirberk
Sezer, Ali Devin
Metadata
Show full item record
This work is licensed under a
Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License
.
Item Usage Stats
147
views
0
downloads
Cite This
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.
Subject Keywords
Approximation of Probabilities of Rare Events
,
Exit Probabilities
,
Constrained Random Walks
,
Queueing Systems
,
Large Deviations
URI
https://hdl.handle.net/11511/32637
DOI
https://doi.org/10.1007/s10479-019-03324-w
Collections
Graduate School of Applied Mathematics, Article
Citation Formats
IEEE
ACM
APA
CHICAGO
MLA
BibTeX
K. D. Unlu and A. D. Sezer, “Excessive backlog probabilities of two parallel queues,” pp. 141–174, 2020, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/32637.