M/D/1 queue: Difference between revisions

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In [[queueing theory]], a discipline within the mathematical [[probability theory|theory of probability]], an '''M/D/1 queue''' represents the queue length in a system having a single server, where arrivals are determined by a [[Poisson process]] and job service times are fixed (deterministic). The model name is written in [[Kendall's notation]].<ref>{{Cite journal | last1 = Kendall | first1 = D. G. | authorlink1 = David George Kendall| title = Stochastic Processes Occurring in the Theory of Queues and their Analysis by the Method of the Imbedded Markov Chain | doi = 10.1214/aoms/1177728975 | jstor = 2236285| journal = The Annals of Mathematical Statistics | volume = 24 | issue = 3 | pages = 338 | year = 1953| url = http://projecteuclid.org/euclid.aoms/1177728975 | pmid = | pmc = | doi-access = free }}</ref> [[Agner Krarup Erlang]] first published on this model in 1909, starting the subject of [[queueing theory]].<ref>{{Cite journal | last1 = Kingman | first1 = J. F. C. | authorlink1 = John Kingman | title = The first Erlang century—and the next | journal = [[Queueing Systems]] | volume = 63 | pages = 3–4 | year = 2009 | doi = 10.1007/s11134-009-9147-4}}</ref><ref>{{cite journal|title=The theory of probabilities and telephone conversations |journal=Nyt Tidsskrift for Matematik B |volume=20 |pages=33–39 |first=A. K. |last=Erlang |url=http://oldwww.com.dtu.dk/teletraffic/erlangbook/pps131-137.pdf |year=1909 |url-status=dead |archiveurl=https://web.archive.org/web/20111001212934/http://oldwww.com.dtu.dk/teletraffic/erlangbook/pps131-137.pdf |archivedate=October 1, 2011 }}</ref> An extension of this model with more than one server is the [[M/D/c queue]].

==Model definition==

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==Application==

Includes applications in wide area [[network planning and design|network design]] <ref>{{cite journal|last1=Kotobi|first1=Khashayar|last2=Bilén|first2=Sven G.|title=Spectrum sharing via hybrid cognitive players evaluated by an M/D/1 queuing model|journal=EURASIP Journal on Wireless Communications and Networking|volume=2017|date=2017|pages=85 |doi=10.1186/s13638-017-0871-x|url=https://jwcn-eurasipjournals.springeropen.com/articles/10.1186/s13638-017-0871-x |accessdate=2017-05-05|doi-access=free}}</ref>, where a single central processor to read the headers of the packets arriving in exponential fashion, then computes the next adapter to which each packet should go and dispatch the packets accordingly. Here the service time is the processing of the packet header and cyclic redundancy check, which are independent of the length of each arriving packets. Hence, it can be modeled as a M/D/1 queue.<ref>{{Cite book|title=Wide Area Network Design: Concepts and Tools for optimization.|last=Chan|first=Robert S.|publisher=Morgan Kaufmann Publishers Inc.|year=1998|isbn=1-55860-458-8|___location=|pages=319}}</ref>

==References==