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| In [[queueing theory]], a discipline within the mathematical [[probability theory|theory of probability]], an '''M/G/1 queue''' is a queue model where arrivals are '''M'''arkovian (modulated by a [[Poisson process]]), service times have a '''G'''eneral [[probability distribution|distribution]] and there is a single server.<ref>{{cite book | page = 77 | title = Multi-armed Bandit Allocation Indices | first = John C. | last = Gittins | authorlink = John C. Gittins | publisher = John Wiley & Sons | year = 1989| isbn = 0471920592}}</ref> The model name is written in [[Kendall's notation]], and is an extension of the [[M/M/1 queue]], where service times must be [[exponential distribution|exponentially distributed]]. The classic application of the M/G/1 queue is to model performance of a fixed head [[hard disk]].<ref name="harrison" />
| | Malus pumila and China Peregrine flavour rig to tighten ties pursuit the found of the iPhone on the carrier's monolithic meshing live calendar month. Speechmaking to The [http://search.huffingtonpost.com/search?q=Paries+Street&s_it=header_form_v1 Paries Street] Journal, Orchard apple tree CEO Tim Captain Cook says he's "incredibly optimistic" approximately his company's next with PRC Mobile undermentioned the seemingly successful performance of Apple's devices concluded the preceding few weeks. |
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| ==Model definition==
| | According to Mainland China Mobile River chair Xi Guohua, "multi-millions" of iPhones suffer already been orderly by customers in the lead-up to the device's give up. "I think there are lots more things our companies can do together." The iPhone launch on the world's largest meshing is expected to offer Malus pumila with a gross revenue further run into tens of millions of devices. The Dry land manufacturer already has deals in come out with China's second- and third-largest carriers. |
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| A queue represented by a M/G/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 queue, including any being served. Transitions from state ''i'' to ''i'' + 1 represent the arrival of a new customer: the times between such arrivals have an [[exponential distribution]] with parameter λ. Transitions from state ''i'' to ''i'' − 1 represent a customer who has been served, finishing being served and departing: the length of time required for serving an individual customer has a general distribution function. The lengths of times between arrivals and of service periods are [[random variable]]s which are assumed to be [[statistically independent]].
| | "Today is a beginning," says Cook, "and I think there are lots more things our companies can do together in the future ... China Mobile already has a reach to many cities that Apple does not have a reach to." As for accurate gross revenue figures, both companies are staying quieten for instantly. The partnership is in all likelihood to follow below acute scrutiny ended the next few weeks; analysts accept wildly unlike forecasts for the deal, with gross revenue estimates for the succeeding class working betwixt 10 and 30 billion. |
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| ===Scheduling policies===
| | The market testament front to Tim Fake to have just about firm estimates on his expectations for the China Mobile River partnership during the company's lucre later on this month.<br><br>If you cherished this posting and you would like to acquire much more facts about [http://batuiti.com/shop/galaxy-s4/samsung-galaxy-s5-i9600-s4-i9500-case-mercury-flip-pu-leather-cover-case/ galaxy case] kindly check out the web site. |
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| Customers are typically served on a [[first-come, first-served]] basis, other popular scheduling policies include
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| * [[processor sharing]] where all jobs in the queue share the service capacity between them equally
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| * [[last-come, first served]] without preemption where a job in service cannot be interrupted
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| * last-come, first served with preemption where a job in service is interrupted by later arrivals, but work is conserved<ref name="hb30">{{cite doi|10.1017/CBO9781139226424.038}}</ref>
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| * generalized foreground-background (FB) scheduling also known as least-attained-service where the jobs which have received least processing time so far are served first and jobs which have received equal service time share service capacity using processor sharing<ref name="hb30" />
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| * [[shortest job first]] without preemption (SJF) where the job with the smallest size receives service and cannot be interrupted until service completes
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| * preemptive shortest job first where at any moment in time the job with the smallest original size is served<ref>{{cite doi|10.1017/CBO9781139226424.040}}</ref>
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| * [[shortest remaining processing time]] (SRPT) where the next job to serve is that with the smallest remaining processing requirement<ref>{{cite doi|10.1017/CBO9781139226424.041}}</ref>
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| Service policies are often evaluated by comparing mean sojourn times in the queue. If service times that jobs require are known on arrival then the optimal scheduling policy is SRPT.<ref name="gautam">{{cite book | first = Natarajan | last = Gautam | title = Analysis of Queues: Methods and Applications | publisher = CRC Press | year = 2012 | isbn = 9781439806586}}</ref>{{rp|296}}
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| Policies can also be evaluated using a measure of fairness.<ref>{{cite doi|10.1145/885651.781057}}</ref>
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| ==Queue length==
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| ===Pollaczek–Khinchine method===
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| The [[probability generating function]] of the [[stationary process|stationary]] queue length distribution is given by the [[Pollaczek–Khinchine transform equation]]<ref name="harrison">{{cite book|first=Peter|last=Harrison|authorlink=Peter G. Harrison|first2=Naresh M. |last2=Patel|title=Performance Modelling of Communication Networks and Computer Architectures|publisher=Addison–Wesley|year=1992}}</ref>
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| :<math>\pi(z) = \frac{(1-z)(1-\rho)g(\lambda(1-z))}{g(\lambda(1-z))-z}</math>
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| where g(''s'') is the Laplace transform of the service time probability density function.<ref>{{cite doi|10.1088/0967-1846/3/1/003}}</ref> In the case of an [[M/M/1 queue]] where service times are exponentially distributed with parameter ''μ'', g(''s'') = ''μ''/(''μ'' + ''s'').
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| This can be solved for individual state probabilities either using by direct computation or using the [[method of supplementary variables]]. The [[Pollaczek–Khinchine formula]] gives the mean queue length and mean waiting time in the system.<ref>{{cite doi|10.1007/BF01194620}}</ref><ref>{{cite journal|last=Khintchine|first=A. Y|authorlink=Aleksandr Khinchin|year=1932|title=Mathematical theory of a stationary queue|journal=[[Matematicheskii Sbornik]]|volume=39|number=4|pages=73–84|url=http://mi.mathnet.ru/rus/msb/v39/i4/p73|accessdate=2011-07-14}}</ref>
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| ===Matrix analytic method===
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| {{Main|Matrix analytic method}}
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| Consider the [[embedded Markov chain]] of the M/G/1 queue, where the time points selected are immediately after the moment of departure. The customer being served (if there is one) has received zero seconds of service. Between departures, there can be 0, 1, 2, 3,… arrivals. So from state ''i'' the chain can move to state ''i'' – 1, ''i'', ''i'' + 1, ''i'' + 2, ….<ref>{{cite book|title=Probability, Markov chains, queues, and simulation|first=William J.|last=Stewart|publisher=Princeton University Press|year=2009|page=510|isbn=0-691-14062-6}}</ref> The embedded [[Markov chain]] has [[transition matrix]]
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| :<math>P = \begin{pmatrix}
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| a_0 & a_1 & a_2 & a_3 & a_4 & \cdots \\
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| a_0 & a_1 & a_2 & a_3 & a_4 & \cdots \\
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| 0 & a_0 & a_1 & a_2 & a_3 & \cdots \\
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| 0 & 0 & a_0 & a_1 & a_2 & \cdots \\
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| 0 & 0 & 0 & a_0 & a_1 & \cdots \\
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| \vdots & \vdots & \vdots & \vdots & \vdots & \ddots
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| \end{pmatrix}</math>
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| where
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| :<math>a_v = \int_0^\infty e^{-\lambda u} \frac{(\lambda u)^v}{v!} \text{d}F(u) ~\text{ for } v \geq 0</math>
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| and ''F''(''u'') is the service time distribution and λ the Poisson arrival rate of jobs to the queue.
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| Markov chains with generator matrices or block matrices of this form are called M/G/1 type Markov chains,<ref>{{cite book|title=Matrix-geometric solutions in stochastic models: an algorithmic approach (Johns Hopkins Studies in Mathematical Sciences)|first=Marcel F.|last=Neuts|page=2|year=1981|isbn=0-486-68342-7|publisher=[[Johns Hopkins University Press]]}}</ref> a term coined by M. F. Neuts.<ref>{{cite journal | first = M. F . | last= Neuts | title= Structured Stochastic Matrices of M/G/1 Type and Their Applications| publisher = Marcel Dekk.|location= New York| year = 1989}}</ref><ref>{{cite doi|10.1080/15326349808807483}}</ref> The stationary distribution of an M/G/1 type Markov model can be computed using the [[matrix analytic method]].<ref>{{cite doi|10.1093/acprof:oso/9780198527688.001.0001}}</ref>
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| ==Busy period==
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| The busy period is the time spent in states 1, 2, 3,… between visits to the state 0. The [[expected value|expected]] length of a busy period is 1/(μ−λ) where 1/μ is the [[expected value|expected]] length of service time and λ the rate of the Poisson process governing arrivals.<ref>{{cite journal|title=Hitting time in an M/G/1 queue|last1=Ross|first1=Sheldon M.|last2=Seshadri|first2=Sridhar|journal=Journal of Applied Probability |year=1999|pages=934–940 |jstor=3215453 |url=http://pages.stern.nyu.edu/~sseshadr/journalArticles/hitting_time_in_mg1_queue.pdf}}</ref> The busy period [[probability density function]] <math>\phi(s)</math> can be shown to obey the Kendall functional equation<ref>{{cite doi|10.1016/0167-6377(95)00049-6}}</ref><ref>{{cite doi|10.1017/CBO9781139173087.004}}</ref>{{rp|91}}
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| ::<math>\phi(s) = g[s+\lambda - \lambda \phi(s)]</math>
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| where as above g is the [[Laplace–Stieltjes transform]] of the service time distribution function. This relationship can only be solved exactly in special cases (such as the [[M/M/1 queue]]), but for any ''s'' the value of ϕ(''s'') can be calculated and by iteration with upper and lower bounds the distribution function numerically computed.<ref>{{cite doi|10.1016/0167-6377(92)90085-H}}</ref>
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| ==Waiting/response time==
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| Writing W<sup>*</sup>(''s'') for the [[Laplace–Stieltjes transform]] transform of the waiting time distribution,<ref name="diagle">{{cite doi|10.1007/0-387-22859-4_5}}</ref> is given by the Pollaczek–Khinchine transform as
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| :<math>W^\ast(s) = \frac{(1-\rho)s g(s)}{s-\lambda(1-g(s))}</math>
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| where g(''s'') is the Laplace–Stieltjes transform of serivice time probability density function.
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| ==Arrival theorem==
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| As the arrivals are determined by a Poisson process, the [[arrival theorem]] holds.
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| ==Multiple servers==
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| {{Main|M/G/k queue}}
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| Many metrics for the [[M/G/k queue]] with ''k'' servers remain an open problem, though some approximations and bounds are known.
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| ==References==
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| {{Reflist}}
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| {{Queueing theory}}
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| {{Stochastic processes}}
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| {{DEFAULTSORT:M G 1 queue}}
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| [[Category:Single queueing nodes]]
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| [[Category:Stochastic processes]]
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According to Mainland China Mobile River chair Xi Guohua, "multi-millions" of iPhones suffer already been orderly by customers in the lead-up to the device's give up. "I think there are lots more things our companies can do together." The iPhone launch on the world's largest meshing is expected to offer Malus pumila with a gross revenue further run into tens of millions of devices. The Dry land manufacturer already has deals in come out with China's second- and third-largest carriers.
"Today is a beginning," says Cook, "and I think there are lots more things our companies can do together in the future ... China Mobile already has a reach to many cities that Apple does not have a reach to." As for accurate gross revenue figures, both companies are staying quieten for instantly. The partnership is in all likelihood to follow below acute scrutiny ended the next few weeks; analysts accept wildly unlike forecasts for the deal, with gross revenue estimates for the succeeding class working betwixt 10 and 30 billion.
The market testament front to Tim Fake to have just about firm estimates on his expectations for the China Mobile River partnership during the company's lucre later on this month.
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