Inqueueing theory, a discipline within the mathematicaltheory of probability, anM/G/1 queue is a queue model where arrivals areMarkovian (modulated by aPoisson process), service times have aGeneraldistribution and there is a single server.^[1] The model name is written inKendall's notation, and is an extension of theM/M/1 queue, where service times must beexponentially distributed. The classic application of the M/G/1 queue is to model performance of a fixed headhard disk.^[2]

A queue represented by a M/G/1 queue is a stochastic process whosestate 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 statei toi + 1 represent the arrival of a new customer: the times between such arrivals have anexponential distribution with parameter λ. Transitions from statei toi − 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 arerandom variables which are assumed to bestatistically independent.

Customers are typically served on afirst-come, first-served basis, other popular scheduling policies include

• processor sharing where all jobs in the queue share the service capacity between them equally
• last-come, first served without preemption where a job in service cannot be interrupted
• last-come, first served with preemption where a job in service is interrupted by later arrivals, but work is conserved^[3]
• 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^[3]
• shortest job first without preemption (SJF) where the job with the smallest size receives service and cannot be interrupted until service completes
• preemptive shortest job first where at any moment in time the job with the smallest original size is served^[4]
• shortest remaining processing time (SRPT) where the next job to serve is that with the smallest remaining processing requirement^[5]

Service policies are often evaluated by comparing themean sojourn time in the queue. If service times that jobs require are known on arrival then the optimal scheduling policy is SRPT.^[6]: 296

Policies can also be evaluated using a measure of fairness.^[7]

Theprobability generating function of thestationary queue length distribution is given by thePollaczek–Khinchine transform equation^[2]

${\displaystyle \pi (z)=\frac{(1-z)(1-\rho )g(\lambda (1-z))}{g(\lambda (1-z))-z}}$

where g(s) is the Laplace transform of the service time probability density function.^[8] In the case of anM/M/1 queue where service times are exponentially distributed with parameterμ, g(s) = μ/(μ + s).

This can be solved for individual state probabilities either using by direct computation or using themethod of supplementary variables. ThePollaczek–Khinchine formula gives the mean queue length and mean waiting time in the system.^[9][10]

Recently, thePollaczek–Khinchine formula has been extended to the case of infinite service moments, thanks to the use of Robinson's Non-Standard Analysis.^[11]

Consider theembedded 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 statei the chain can move to statei – 1,i,i + 1,i + 2, ....^[12] The embeddedMarkov chain hastransition matrix

where

${\displaystyle a_v={\int }_0^{\mathrm{\infty }}{e}^{-\lambda u}\frac{(\lambda u{)}^v}{v!}\text{d}F(u)\text{ for }v\ge 0}$

andF(u) is the service time distribution and λ the Poisson arrival rate of jobs to the queue.

Markov chains with generator matrices or block matrices of this form are called M/G/1 type Markov chains,^[13] a term coined byMarcel F. Neuts.^[14][15]

An M/G/1 queue has a stationary distribution if and only if the traffic intensity${\displaystyle \rho =\lambda \mathbb{E}(G)}$ is less than 1, in which case the unique such distribution hasprobability-generating function:^[16]

${\displaystyle G(s)=\frac{(1-\rho )(1-s)}{1-s/M_S(\lambda (s-1))}}$

where${\displaystyle M_S}$ is themoment-generating function of a general service time. The stationary distribution of an M/G/1 type Markov model can be computed using thematrix analytic method.^[17]

The busy period is the time spent in states$1,2,3,...$ between visits to the state$0$. Theexpected length of a busy period is${\displaystyle \frac{1}{\mu -\lambda }}$ where${\displaystyle \frac{1}{\mu }}$ is theexpected length of service time and$\lambda$ the rate of the Poisson process governing arrivals.^[18] Let${\displaystyle \varphi (s)}$ denote theLaplace transform of the busy periodprobability density function (so${\displaystyle \varphi (s)}$ is also theLaplace–Stieltjes transform of the busy period cumulative distribution function). Then${\displaystyle \varphi (s)}$ can be shown to obey the Kendall functional equation^{[19][20]: 92}

${\displaystyle \varphi (s)=g[s+\lambda -\lambda \varphi (s)]}$

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 theM/M/1 queue), but for any$s$ the value of$\varphi (s)$ can be calculated and by iteration with upper and lower bounds the distribution function numerically computed.^[21]

Writing S^*(s) for theLaplace–Stieltjes transform of the sojourn time distribution,^[22] is given by the Pollaczek–Khinchine transform as

${\displaystyle S^{\ast }(s)=\frac{(1-\rho )sg(s)}{s-\lambda (1-g(s))}}$

where g(s) is the Laplace–Stieltjes transform of service time probability density function.

Writing W^*(s) for theLaplace–Stieltjes transform of the waiting time distribution,^[22] is given by the Pollaczek–Khinchine transform as

${\displaystyle W^{\ast }(s)=\frac{(1-\rho )s}{s-\lambda (1-g(s))}}$

where g(s) is the Laplace–Stieltjes transform of service time probability density function.

As the arrivals are determined by a Poisson process, thearrival theorem holds.

Many metrics for theM/G/k queue withk servers remain an open problem, though some approximations and bounds are known.

• M/M/1 queue
• M/M/c queue

• Single queueing nodes

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