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. | authorlink1author-link1 = 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 | pmiddoi-access = | pmc =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. | authorlink1author-link1 = 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 |deadurlurl-status=yesdead |archiveurlarchive-url=https://web.archive.org/web/20111001212934/http://oldwww.com.dtu.dk/teletraffic/erlangbook/pps131-137.pdf |archivedatearchive-date=October 1, 2011 }}</ref> An extension of this model with more than one server is the [[M/D/c queue]].

==Model definition==

An M/D/1 queue is a stochastic process whose [[state space]] is the set {0,1,2,3,...} where the value corresponds to the number of customersentities in the system, including any currently in service.

* Arrivals occur at rate λ according to a [[Poisson process]] and move the process from state ''i'' to ''i''&nbsp;+&nbsp;1.

* Service times are deterministic time ''D'' (serving at rate ''μ''&nbsp;=&nbsp;1/''D'').

* A single server serves customersentities one at a time from the front of the queue, according to a [[first-come, first-served]] discipline. When the service is complete the customerentity leaves the queue and the number of customersentities in the system reduces by one.

* The buffer is of infinite size, so there is no limit on the number of customersentities it can contain.

The [[state space]] diagram for M/D/1 queue is as below:

[[File:Flow_chart111.png|link=https://en.wikipedia.org/wiki/File:Flow_chart111.png|thumb|602x602px|Stage Space Diagram of M/D/1 Queue|none]]

=== Transition Matrix ===

The transition probability matrix for an M/D/1 queue with arrival rate λ and service time 1, such that λ <1 (for stability of the queue) is given by P as below:<ref name=":1" />

=== Transition Matrix ===

<math>P=\begin{pmatrix} a_0 & a_1 & a_2 & a_3 & ... \\ a_0 & a_1 & a_2 & a_3 & ...\\ 0 & a_0 & a_1 & a_2 & ...\\ 0&0 & a_0 & a_1 & ...\\... & ... &...&... &...\\\end{pmatrix}</math> ， <math>a_n=\frac{\lambda^n}{n!}e^{-\lambda}</math>, n = 0,1,....

=== Classic performance metrics ===

The following expressions present the classic performance metrics of a single server queuing system such as M/D/1, with:

* arrival rate <math>=\lambda</math>,

* service rate <math>=\mu</math>,&nbsp;and

* utilization <math>=\rho=\frac{\lambda}{\mu}</math>

The average number of entities in the system, L is given by:

<math>L=\rho+\frac{1}{2}\left ( \frac{\rho^2}{1-\rho} \right );

</math>

The average number of entities in the queue (line), L_Q is given by:

<math>L_Q=\frac{1}{2}\left ( \frac{\rho^2}{1-\rho} \right );

</math>

The average waiting time in the system, ω is given by:

<math>\omega= \frac{1}{\mu}+\frac{\rho}{2\mu(1-\rho)};

</math>

The average waiting time in the queue (line), ω _Q is given by:

<math>\omega_Q=\frac{\rho}{2\mu(1-\rho)}</math>

=== Example ===

Customers arriveConsidering a Starbuckssystem linethat athas aonly rateone of 20 per hourserver, and followswith an exponentialarrival distribution.rate Thereof is20 onlyentities oneper server,hour and the service rate is at a constant of 30 per hour.

So the utilization of the server is: ρ=20/30=2/3. Using the metrics shown above, the results are as following: 1) Average number in line L_Q= 0.6667; 2) Average number in system L =1.333; 3) Average time in line ω_Q = 0.033 hour; 4) Average time in system ω = 0.067 hour.

Arrival Rate: 20 per hour

=== Relations for Mean Waiting Time in M/M/1 and M/D/1 queues ===

Service Rate: 60 per hour

===For Relationan forequilibrium MeanM/G/1 Waitingqueue, Timethe expected value of the time W spent by a customer in M/M/1the andqueue M/D/1are given by Pollaczek-Khintchine formula as queuesbelow:<ref name=":0">{{Cite book|title=Introduction to Queuing Theory|last=Cooper|first=Robert B.|publisher=Elsevier Science Publishing Co.|year=1981|isbn=0-444-00379-7|___location=|pages=189}}</ref> ===

<math>E(W)=\frac{\rho\tau}{2(1-\rho)}\left(1+\frac{\sigma^2}{\tau^2}\right)</math>

ρ=20/30=2/3

where τ is the mean service time; σ<math display="inline"sup>\sigma^2</mathsup> is the variance of service time; and ρ=λτ < 1, λ being the arrival rate of the customers.

Using the queueing theory equations, the results are as following:

For M/M/1 queue, the service times are exponentially distributed, then σ<math display="inline"sup>\sigma^2</mathsup>=<math display="inline" τ<sup>\tau^2</mathsup> and the mean waiting time in the queue denoted by WMW_M is given by the following equation:<ref name=":0" />

Average number in line= 0.6667

<math>\overline{W_M}=\frac{\rho\tau}{1-\rho}</math>

Average number in system: 1.333

Using this, the corresponding equation for M/D/1 queue can be derived, assuming constant service times. Then the variance of service time becomes zero, i.e.   σ<math display="inline"sup>\sigma^2</mathsup> = 0.   The mean waiting time in the M/D/1 queue denoted as WDW_D is given by the following equation:<ref name=":0" />

Average time in line: 0.033

<math>\overline{W_D}=\frac{\rho\tau}{2(1-\rho)}</math>

Average time in system: 0.067

=== Relation for Mean Waiting Time in M/M/1 and M/D/1 queues<ref>{{Cite book|title=Introduction to Queuing Theory|last=Cooper|first=Robert B.|publisher=Elsevier Science Publishing Co.|year=1981|isbn=0-444-00379-7|___location=|pages=189}}</ref> ===

For an equilibrium M/G/1 queue, the expected value of the time W spent by a customer in the queue are given by Pollaczek-Khintchine formula as below

<math>E(W)=\frac{\rho\tau}{2(1-\rho)}(1+\frac{\sigma^2}{\tau^2})</math>

where τ is the mean service time; <math display="inline">\sigma^2</math> is the variance of service time; and ρ=λτ < 1, λ being the arrival rate of the customers.

For M/M/1 queue, the service times are exponentially distributed, then <math display="inline">\sigma^2</math>=<math display="inline">\tau^2</math> and the mean waiting time in the queue denoted by WM is given by the following equation

<math>\overline{W_M}=\frac{\rho\tau}{1-\rho}</math>

Using this, the corresponding equation for M/D/1 queue can be derived, assuming constant service times. Then the variance of service time becomes zero, i.e.  <math display="inline">\sigma^2</math>=0.  The mean waiting time in the M/D/1 queue denoted as WD is given by the following equation

<math>\overline{W_D}=\frac{\rho\tau}{2(1-\rho)}</math>

From the two equations above, we can infer that Mean queue length in M/M/1 queue is twice that in M/D/1 queue.

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==Stationary distribution==

The number of jobs in the queue can be written as an [[M/G/1 queue|M/G/1 type Markov chain]] and the stationary distribution found for state ''i'' (written π_''i'') in the case ''D''&nbsp;=&nbsp;1 to be<ref name=":1">{{cite journal| url = http://wwwkashiwa.orsjnagaokaut.orac.jp/~archivemembers/pdfnakagawa/e_magronbun/Vol.48_2_111029.pdf | journal = Journal of the Operations Research Society of Japan | volume = 48 | year = 2005 | issue = 2 | pages = 111-122111–122 | title = On the Series Expansion for the Stationary Probabilities of an M/D/1 queue | first = Kenji | last = Nakagawa | doi=10.15807/jorsj.48.111| doi-access = free }}</ref>

:<math>\begin{align}\pi_0 &= 1-\lambda \\

\pi_1 &= (1-\lambda)(e^\lambda - 1)\\

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==Delay==

Define ''ρ''&nbsp;=&nbsp;''λ''/''μ'' as the utilization; then the mean delay in the system in an M/D/1 queue is<ref>{{cite book|title=Wide Area Network Design:Concepts and Tools for Optimization|page=319|first=Robert S.|last=Cahn|year=1998|publisher=Morgan Kaufmann|isbn=1558604588}}</ref>

::<math>\frac{1}{2\mu}\cdot\frac{2-\rho}{1-\rho}.</math>

and in the queue:

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==Busy period==

The busy period is the time period measured from the instant a first customer arrives at an empty queue to the time when the queue is again empty. This time period is equal to ''D'' times the number of customers served. If ''ρ''&nbsp;<&nbsp;1, then the number of customers served during a busy period of the queue has a [[Borel distribution]] with parameter ''ρ''.<ref>{{Cite journal | last1 = Tanner | first1 = J. C. | doi = 10.1093/biomet/48.1-2.222 | title = A derivation of the Borel distribution | journal = [[Biometrika]]| volume = 48 | pages = 222–224 | year = 1961 | pmid = | jstor = 2333154| pmc = }}</ref><ref>{{Cite journal | last1 = Haight | first1 = F. A. | last2 = Breuer | first2 = M. A. | doi = 10.1093/biomet/47.1-2.143 | title = The Borel-Tanner distribution | journal = [[Biometrika]]| volume = 47 | pages = 143 | year = 1960 | pmid = | pmc = | jstor = 2332966}}</ref>

==Finite capacity==

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===Stationary distribution===

A stationary distribution for the number of customers in the queue and mean queue length can be computed using [[probability generating function]]s.<ref>{{cite journal | last1 = Brun | first1 = Olivier | last2 =Garcia | first2 = Jean-Marie | year = 2000 | title = Analytical Solution of Finite Capacity M/D/1 Queues | journal = Journal of Applied Probability | volume = 37 | issue = 4 | pages = 1092-10981092–1098 | publisher = [[Applied Probability Trust]] | jstor = 3215497 | doi = 10.1239/jap/1014843086 | url = | format = | accessdate = }}</ref>

<math>P_0(N)=\frac{1}{1+\rho b_{N-1}};</math>

<math>P_N(N)=1-\frac{b_{N-1}}{1+\rho b_{N-1}};</math>

<math>P_j(N)=\frac{b_{j}-b_{j-1}}{1+\rho b_{N-1}}</math>， j = 1,..., N-1.

===Transient solution===

The transient solution of an M/D/1 queue of finite capacity N, often written M/D/1/N, was published by Garcia et al. in 2002.<ref>{{cite journal | last1 = Garcia | first1 = Jean-Marie | last2 = Brun | first2 = Olivier | last3 = Gauchard | first3 = David | year = 2002 | title = Transient Analytical Solution of M/D/1/N Queues | journal = Journal of Applied Probability | volume = 39 | issue = 4 | pages = 853-864853–864 | publisher = Applied Probability Trust | jstor = 3216008 | doi=10.1239/jap/1037816024}}</ref>

The mean number of customers in M/D/1/N queue presented in Garcia et al. 2002 is as follows:

<math>X_N=N-\frac{\sum_{k=0}^{N-1}b_k}{1+\rho b_{N-1}};

</math>

The mean waiting time W N in the M/D/1/N queuing system presented in Garcia et al. 2002 is as follows:

==Application==

<math>W_N=(N-1-\frac{\sum_{k=0}^{N-1}b_k-N}{\rho b_{N-1}})T</math>

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 |access-date=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|pages=319}}</ref>

==References==

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{{DEFAULTSORT:M D 1 queue}}

[[Category:Stochastic processes]]

[[Category:Single queueing nodes]]