M/D/1 queue: Difference between revisions

In queueing theory, a discipline within the mathematical 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.^[1] Agner Krarup Erlang first published on this model in 1909, starting the subject of queueing theory.^[2]^[3]

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 customers 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 + 1.
Service times are deterministic time D (serving at rate μ = 1/D).
A single server serves customers one at a time from the front of the queue, according to a first-come, first-served discipline. When the service is complete the customer leaves the queue and the number of customers in the system reduces by one.
The buffer is of infinite size, so there is no limit on the number of customers it can contain.

Delay

Define ρ = λ/μ as the utilization; then the mean delay in an M/D/1 queue is^[4]

{\frac {1}{\mu }}\cdot {\frac {2-\rho }{2-2\rho }}.

Finite capacity

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.^[5]

References

^ Attention: This template ({{cite doi}}) is deprecated. To cite the publication identified by doi:10.1214/aoms/1177728975, please use {{cite journal}} (if it was published in a bona fide academic journal, otherwise {{cite report}} with |doi=10.1214/aoms/1177728975 instead.
^ Attention: This template ({{cite doi}}) is deprecated. To cite the publication identified by doi:10.1007/s11134-009-9147-4, please use {{cite journal}} (if it was published in a bona fide academic journal, otherwise {{cite report}} with |doi=10.1007/s11134-009-9147-4 instead.
^ "The theory of probabilities and telephone conversations" (PDF). Nyt Tidsskrift for Matematik B. 20: 33–39. 1909.
^ Cahn, Robert S. (1998). Wide Area Network Design:Concepts and Tools for Optimization. Morgan Kaufmann. p. 319. ISBN 1558604588.
^ Attention: This template ({{cite jstor}}) is deprecated. To cite the publication identified by jstor:3216008, please use {{cite journal}} with |jstor=3216008 instead.

v t e Queueing theory
Single queueing nodes	D/M/1 queue M/D/1 queue M/D/c queue M/M/1 queue Burke's theorem M/M/c queue M/M/∞ queue M/G/1 queue Pollaczek–Khinchine formula Matrix analytic method M/G/k queue G/M/1 queue G/G/1 queue Kingman's formula Lindley equation Fork–join queue Bulk queue
Arrival processes	Poisson point process Markovian arrival process Rational arrival process
Queueing networks	Jackson network Traffic equations Gordon–Newell theorem Mean value analysis Buzen's algorithm Kelly network G-network BCMP network
Service policies	FIFO LIFO Processor sharing Round-robin Shortest job next Shortest remaining time
Key concepts	Continuous-time Markov chain Kendall's notation Little's law Product-form solution Balance equation Quasireversibility Flow-equivalent server method Arrival theorem Decomposition method Beneš method
Limit theorems	Fluid limit Mean-field theory Heavy traffic approximation Reflected Brownian motion
Extensions	Fluid queue Layered queueing network Polling system Adversarial queueing network Loss network Retrial queue
Information systems	Data buffer Erlang (unit) Erlang distribution Flow control (data) Message queue Network congestion Network scheduler Pipeline (software) Quality of service Scheduling (computing) Teletraffic engineering
Category

v t e Stochastic processes
Discrete time	Bernoulli process Branching process Chinese restaurant process Galton–Watson process Independent and identically distributed random variables Markov chain Moran process Random walk Loop-erased Self-avoiding Biased Maximal entropy
Continuous time	Additive process Bessel process Birth–death process pure birth Brownian motion Bridge Excursion Fractional Geometric Meander Cauchy process Contact process Continuous-time random walk Cox process Diffusion process Dyson Brownian motion Empirical process Feller process Fleming–Viot process Gamma process Geometric process Hawkes process Hunt process Interacting particle systems Itô diffusion Itô process Jump diffusion Jump process Lévy process Local time Markov additive process McKean–Vlasov process Ornstein–Uhlenbeck process Poisson process Compound Non-homogeneous Schramm–Loewner evolution Semimartingale Sigma-martingale Stable process Superprocess Telegraph process Variance gamma process Wiener process Wiener sausage
Both	Branching process Gaussian process Hidden Markov model (HMM) Markov process Martingale Differences Local Sub- Super- Random dynamical system Regenerative process Renewal process Stochastic chains with memory of variable length White noise
Fields and other	Dirichlet process Gaussian random field Gibbs measure Hopfield model Ising model Potts model Boolean network Markov random field Percolation Pitman–Yor process Point process Cox Poisson Random field Random graph
Time series models	Autoregressive conditional heteroskedasticity (ARCH) model Autoregressive integrated moving average (ARIMA) model Autoregressive (AR) model Autoregressive–moving-average (ARMA) model Generalized autoregressive conditional heteroskedasticity (GARCH) model Moving-average (MA) model
Financial models	Binomial options pricing model Black–Derman–Toy Black–Karasinski Black–Scholes Chan–Karolyi–Longstaff–Sanders (CKLS) Chen Constant elasticity of variance (CEV) Cox–Ingersoll–Ross (CIR) Garman–Kohlhagen Heath–Jarrow–Morton (HJM) Heston Ho–Lee Hull–White Korn-Kreer-Lenssen LIBOR market Rendleman–Bartter SABR volatility Vašíček Wilkie
Actuarial models	Bühlmann Cramér–Lundberg Risk process Sparre–Anderson
Queueing models	Bulk Fluid Generalized queueing network M/G/1 M/M/1 M/M/c
Properties	Càdlàg paths Continuous Continuous paths Ergodic Exchangeable Feller-continuous Gauss–Markov Markov Mixing Piecewise-deterministic Predictable Progressively measurable Self-similar Stationary Time-reversible
Limit theorems	Central limit theorem Donsker's theorem Doob's martingale convergence theorems Ergodic theorem Fisher–Tippett–Gnedenko theorem Large deviation principle Law of large numbers (weak/strong) Law of the iterated logarithm Maximal ergodic theorem Sanov's theorem Zero–one laws (Blumenthal, Borel–Cantelli, Engelbert–Schmidt, Hewitt–Savage, Kolmogorov, Lévy)
Inequalities	Burkholder–Davis–Gundy Doob's martingale Doob's upcrossing Kunita–Watanabe Marcinkiewicz–Zygmund
Tools	Cameron–Martin formula Convergence of random variables Doléans-Dade exponential Doob decomposition theorem Doob–Meyer decomposition theorem Doob's optional stopping theorem Dynkin's formula Feynman–Kac formula Filtration Girsanov theorem Infinitesimal generator Itô integral Itô's lemma Karhunen–Loève theorem Kolmogorov continuity theorem Kolmogorov extension theorem Lévy–Prokhorov metric Malliavin calculus Martingale representation theorem Optional stopping theorem Prokhorov's theorem Quadratic variation Reflection principle Skorokhod integral Skorokhod's representation theorem Skorokhod space Snell envelope Stochastic differential equation Tanaka Stopping time Stratonovich integral Uniform integrability Usual hypotheses Wiener space Classical Abstract
Disciplines	Actuarial mathematics Control theory Econometrics Ergodic theory Extreme value theory (EVT) Large deviations theory Mathematical finance Mathematical statistics Probability theory Queueing theory Renewal theory Ruin theory Signal processing Statistics Stochastic analysis Time series analysis Machine learning
List of topics Category

@@ Line 14: / Line 14: @@
 Define ''ρ''&nbsp;=&nbsp;''λ''/''μ'' as the utilization; then the mean delay 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}{\mu}\cdot\frac{2-\rho}{2-2\rho}.</math>
-==Multiple servers==
-Write ''c'' for the number of servers.
-===Waiting time distribution===
-Erlang showed that when ''ρ''&nbsp;=&nbsp;''λ''&nbsp;''D''/''c''&nbsp;<&nbsp;1, the waiting time distribution has distribution F(''y'') given by<ref name="franx">{{cite doi|10.1016/S0167-6377(01)00108-0}}</ref>
-::<math>F(y) = \int_0^\infty F(x+y-D)\frac{\lambda^c x^{c-1}}{(c-1)!} e^{-\lambda x} \text{d} x, \quad y \geq 0 \quad c \in \mathbb N.</math>
-Crommelin showed that, writing ''P''<sub>''n''</sub> for the stationary probability of a system with ''n'' or fewer customers,
-<ref>{{cite journal | first = C.D. | last = Crommelin | title = Delay probability formulas when the holding times are constant | journal = P.O. Electr. Engr. J.| volume = 25| year=1932| pages= 41–50}}</ref>
-::<math>\mathbb P(W \leq x) = \sum_{n=0}^{c-1} P_n \sum_{k=1}^m \frac{(-\lambda(x-kD))^{(k+1)c-1-n}}{((K+1)c-1-n)!}e^{\lambda(x-kD)}, \quad mD \leq x <(m+1)D.</math>
 ==Finite capacity==

Revision as of 10:52, 14 June 2013

Model definition

Delay

Finite capacity

Transient solution

References