M/G/k queue: Difference between revisions

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{{Use American English|date = January 2019}}

{{Short description|Queue model}}

In [[queueing theory]], a discipline within the mathematical [[probability theory|theory of probability]], an '''M/G/k queue''' is a queue model where arrivals are '''M'''arkovian[[Markov property|Markovian]] (modulated by a [[Poisson process]]), service times have a '''G'''eneral [[probability distribution|general distribution]] and there are ''k'' servers. The model name is written in [[Kendall's notation]], and is an extension of the [[M/M/c queue]], where service times must be [[exponential distribution|exponentially distributed]] and of the [[M/G/1 queue]] with a single server. Most performance metrics for this queueing system are not known and remain an [[open problem]].<ref>{{Cite journal | last1 = Kingman | first1 = J. F. C. | author-link1 = John Kingman | title = The first Erlang century—and the next | journal = [[Queueing Systems]] | volume = 63 | pages = 3–4 | year = 2009 | issue = 1–4 | doi = 10.1007/s11134-009-9147-4| s2cid = 38588726 }}</ref>

==Model definition==

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==Average delay/waiting time==

There are numerous approximations for the average delay a job experiences.<ref name="cite jstor|169760"/><ref name="yao" /><ref>{{cite journal | last1 = Hokstad | first1 = Per | year = 1980 | title = The Steady-State Solution of the M/K₂/m Queue | journal = Advances in Applied Probability | volume = 12 | issue = 3 | pages = 799–823 | publisher = Applied Probability Trust | doi = 10.2307/1426432 | jstor = 1426432| s2cid = 124883099 }}</ref><ref>{{cite journal | last1 = Köllerström | first1 = Julian | year = 1974 | title = Heavy Traffic Theory for Queues with Several Servers. I | journal = Journal of Applied Probability | volume = 11 | issue = 3 | pages = 544–552 | publisher = Applied Probability Trust | jstor = 3212698 | doi=10.1017/s0021900200096327}}</ref><ref>{{Cite journal | last1 = Nozaki | first1 = S. A. | last2 = Ross | first2 = S. M. | title = Approximations in Finite-Capacity Multi-Server Queues with Poisson Arrivals | journal = Journal of Applied Probability | volume = 15 | issue = 4 | pages = 826–834 | doi = 10.2307/3213437 | year = 1978 | jstor = 3213437 | s2cid = 32476285 }}</ref><ref>{{cite journal | last1 = Boxma | first1 = O. J. | author-link1 = Onno Boxma | last2 =Cohen | first2 = J. W. | author-link2 = Wim Cohen | first3 = N. | last3 = Huffels | year = 1979 | title = Approximations of the Mean Waiting Time in an M/G/s Queueing System | journal = [[Operations Research (journal)|Operations Research]] | volume = 27 | issue = 6 | pages = 1115–1127 | publisher = INFORMS | jstor = 172087 | doi=10.1287/opre.27.6.1115}}</ref> The first such was given in 1959 using a factor to adjust the mean waiting time in an [[M/M/c queue]]<ref name="gbdz" /><ref>{{Cite journal | last1 = Lee | first1 = A. M. | last2 = Longton | first2 = P. A. | doi = 10.1057/jors.1959.5 | title = Queueing Processes Associated with Airline Passenger Check-in | journal = [[Journal of the Operational Research Society]]| volume = 10 | pages = 56–71 | year = 1959 }}</ref> This result is sometimes known as Kingman's law of congestion.<ref>{{Cite journal | last1 = Gans | first1 = N. | last2 = Koole | first2 = G. | last3 = Mandelbaum | first3 = A. | doi = 10.1287/msom.5.2.79.16071 | title = Telephone Call Centers: Tutorial, Review, and Research Prospects | journal = [[Manufacturing & Service Operations Management]] | volume = 5 | issue = 2 | pages = 79 | year = 2003 | url = http://ie.technion.ac.il/Labs/Serveng/files/CCReview.pdf | doi-access = free | archive-date = 2015-12-08 | access-date = 2015-08-29 | archive-url = https://web.archive.org/web/20151208113728/http://ie.technion.ac.il/Labs/Serveng/files/CCReview.pdf | url-status = dead }}</ref>

:<math>E[W^{\text{M/G/}k}] = \frac{C^2+1}{2} \mathbb E [ W^{\text{M/M/}c}]</math>

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where ''C'' is the [[coefficient of variation]] of the service time distribution. [[Ward Whitt]] described this approximation as “usually an excellent approximation, even given extra information about the service-time distribution."<ref>{{Cite journal | last1 = Whitt | first1 = W. | author-link1 = Ward Whitt| title = Approximations for the GI/G/m Queue| doi = 10.1111/j.1937-5956.1993.tb00094.x | journal = [[Production and Operations Management]]| volume = 2 | issue = 2 | pages = 114–161 | year = 2009 | url = http://www.columbia.edu/~ww2040/ApproxGIGm1993.pdf}}</ref>

However, it is known that no approximation using only the first two moments can be accurate in all cases.<ref name="gbdz">{{Cite journal | last1 = Gupta | first1 = V. | last2 = Harchol-Balter | first2 = M. |author2-link=Mor Harchol-Balter| last3 = Dai | first3 = J. G. | last4 = Zwart | first4 = B. | title = On the inapproximability of M/G/K: Why two moments of job size distribution are not enough | doi = 10.1007/s11134-009-9133-x | journal = [[Queueing Systems]]| volume = 64 | pages = 5–48 | year = 2009 | s2cid = 16755599 | urlciteseerx = http://repository10.cmu1.edu/cgi/viewcontent1.cgi?article=1867&context=compsci151.5844 }}</ref>

A [[Markov–Krein]] characterization has been shown to produce tight bounds on the mean waiting time.<ref>{{Cite journal | last1 = Gupta | first1 = V. | last2 = Osogami | first2 = T. | doi = 10.1007/s11134-011-9248-8 | title = On Markov–Krein characterization of the mean waiting time in M/G/K and other queueing systems | journal = Queueing Systems | volume = 68 | issue = 3–4 | pages = 339 | year = 2011 | s2cid = 35061112 }}</ref>

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For an M/G/2 queue (the model with two servers) the problem of determining marginal probabilities can be reduced to solving a pair of [[integral equation]]s<ref>{{Cite journal | last1 = Knessl | first1 = C. | last2 = Matkowsky | first2 = B. J. | last3 = Schuss | first3 = Z. | last4 = Tier | first4 = C. | title = An Integral Equation Approach to the M/G/2 Queue | doi = 10.1287/opre.38.3.506 | journal = [[Operations Research (journal)|Operations Research]]| volume = 38 | issue = 3 | pages = 506 | year = 1990 | jstor = 171363}}</ref> or the Laplace transform of the distribution when the service time distribution is a mixture of exponential distributions.<ref>{{Cite journal | last1 = Cohen | first1 = J. W. | author-link1 = Wim Cohen| title = On the M/G/2 queueing model | doi = 10.1016/0304-4149(82)90046-1 | journal = Stochastic Processes and Their Applications | volume = 12 | issue = 3 | pages = 231–248 | year = 1982 | doi-access = free }}</ref> The Laplace transform of queue length<ref>{{cite journal | last1 = Hokstad | first1 = Per | year = 1979 | title = On the Steady-State Solution of the M/G/2 Queue | journal = Advances in Applied Probability | volume = 11 | issue = 1 | pages = 240–255 | publisher = Applied Probability Trust | doi = 10.2307/1426776 | jstor = 1426776| s2cid = 125014523 }}</ref> and waiting time distributions<ref>{{Cite journal | last1 = Boxma | first1 = O. J. | author-link1 = Onno Boxma| last2 = Deng | first2 = Q. | last3 = Zwart | first3 = A. P. | journal = [[Queueing Systems]]| volume = 40 | pages = 5–31 | year = 2002 | doi = 10.1023/A:1017913826973 | title = Waiting-Time Asymptotics for the M/G/2 Queue with Heterogeneous Servers| s2cid = 2513624 }}</ref> can be computed when the waiting time distribution has a rational Laplace transform.

== Software ==

=== Ciw ===

Ciw is a Python package for simulating queueing networks.<ref>{{Cite web |title=Welcome to Ciw’s documentation! — Ciw 3.1.0 documentation |url=https://ciw.readthedocs.io/en/latest/ |access-date=2023-12-19 |website=ciw.readthedocs.io}}</ref> Since G in M/G/k can be any distribution with non-negative support, we will use a [[gamma distribution]] for the sake of example. A gamma distribution is the result of a sum of independent [[Exponential distribution|exponentially-distributed]] [[random variable]], and thus for this example we have an interpretation that the servicing is implicitly a multi-step process where each step has an exponentially-distributed completion time.

A M/G/k queue can be implemented and simulated using Ciw in the following way.<syntaxhighlight lang="python3" line="1">

import ciw

ciw.seed(2018)

ARRIVAL_TIME = 1

SERVICE_SHAPE = 6 / 10

SERVICE_SCALE = 12 / 10

NUM_SERVERS = 2

HORIZON = 365

network = ciw.create_network(

arrival_distributions = [ciw.dists.Exponential(ARRIVAL_TIME)],

service_distributions = [ciw.dists.Gamma(shape=SERVICE_SHAPE, scale=SERVICE_SCALE)],

number_of_servers = [NUM_SERVERS]

)

simulation = ciw.Simulation(network)

simulation.simulate_until_max_time(HORIZON)

records = simulation.get_all_records()

</syntaxhighlight>

==References==