M/G/k queue

A queue represented by a M/G/k 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 the departure of a customer who has just finished being served: 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 variables which are assumed to be statistically independent.

Tijms et al. believe it is "not likely that computationally tractable methods can be developed to compute the exact numerical values of the steady-state probability in the M/G/k queue."{{Cite journal | last1 = Tijms | first1 = H. C. | last2 = Van Hoorn | first2 = M. H. | last3 = Federgruen | first3 = A. | title = Approximations for the Steady-State Probabilities in the M/G/c Queue | journal = Advances in Applied Probability | volume = 13 | issue = 1 | pages = 186–206 | doi = 10.2307/1426474 | jstor = 1426474| year = 1981 | s2cid = 222335724 }}

Various approximations for the average queue size,{{Cite journal | last1 = Ma | first1 = B. N. W. | last2 = Mark | first2 = J. W. | doi = 10.1287/opre.43.1.158 | title = Approximation of the Mean Queue Length of an M/G/c Queueing System | journal = Operations Research| volume = 43 | pages = 158–165 | year = 1995 | issue = 1 | jstor = 171768}} stationary distribution{{Cite journal | last1 = Breuer | first1 = L. | title = Continuity of the M/G/c queue | doi = 10.1007/s11134-008-9073-x | journal = Queueing Systems| volume = 58 | issue = 4 | pages = 321–331 | year = 2008 | s2cid = 2345317 }}{{cite journal | last1 = Hokstad | first1 = Per | year = 1978 | title = Approximations for the M/G/m Queue | journal = Operations Research | volume = 26 | issue = 3 | pages = 510–523 | publisher = INFORMS | jstor = 169760 | doi = 10.1287/opre.26.3.510}} and approximation by a reflected Brownian motion{{Cite journal | last1 = Kimura | first1 = T. | title = Diffusion Approximation for an M/G/m Queue | doi = 10.1287/opre.31.2.304 | journal = Operations Research| volume = 31 | issue = 2 | pages = 304–321 | year = 1983 | jstor = 170802}}{{Cite journal | last1 = Yao | first1 = D. D. | title = Refining the Diffusion Approximation for the M/G/m Queue | doi = 10.1287/opre.33.6.1266 | journal = Operations Research| volume = 33 | issue = 6 | pages = 1266–1277 | year = 1985 | jstor = 170637}} have been offered by different authors. Recently a new approximate approach based on Laplace transform for steady state probabilities has been proposed by Hamzeh Khazaei et al..{{Cite journal | last1 = Khazaei | first1 = H. | last2 = Misic | first2 = J. | last3 = Misic | first3 = V. B. | doi = 10.1109/TPDS.2011.199 | title = Performance Analysis of Cloud Computing Centers Using M/G/m/m+r Queuing Systems | journal = IEEE Transactions on Parallel and Distributed Systems | volume = 23 | issue = 5 | pages = 936 | year = 2012 | s2cid = 16934438 }}{{Cite book | last1 = Khazaei | first1 = H. | last2 = Misic | first2 = J. | last3 = Misic | first3 = V. B. | doi = 10.1109/ICDCSW.2011.13 | chapter = Modelling of Cloud Computing Centers Using M/G/m Queues | title = 2011 31st International Conference on Distributed Computing Systems Workshops | pages = 87 | year = 2011 | isbn = 978-1-4577-0384-3 | s2cid = 16067523 }} This new approach is yet accurate enough in cases of large number of servers and when the distribution of service time has a Coefficient of variation more than one.

There are numerous approximations for the average delay a job experiences.{{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 }}{{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}}{{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 }}{{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 | volume = 27 | issue = 6 | pages = 1115–1127 | publisher = INFORMS | jstor = 172087 | doi=10.1287/opre.27.6.1115}} The first such was given in 1959 using a factor to adjust the mean waiting time in an M/M/c queue{{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 }} This result is sometimes known as Kingman's law of congestion.{{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 }}

:$E[W^\{\backslash text\{M/G/\}k\}]\; =\; \backslash frac\{C^2+1\}\{2\}\; \backslash mathbb\; E\; [\; W^\{\backslash text\{M/M/\}c\}]$

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."{{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}}

However, it is known that no approximation using only the first two moments can be accurate in all cases.{{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 | citeseerx = 10.1.1.151.5844 }}

A Markov–Krein characterization has been shown to produce tight bounds on the mean waiting time.{{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 }}

It is conjectured that the times between departures, given a departure leaves n customers in a queue, has a mean which as n tends to infinity is different from the intuitive 1/μ result.{{Cite journal | last1 = Veeger | first1 = C. | last2 = Kerner | first2 = Y. | last3 = Etman | first3 = P. | last4 = Adan | first4 = I. | title = Conditional inter-departure times from the M/G/s queue | doi = 10.1007/s11134-011-9240-3 | journal = Queueing Systems| volume = 68 | issue = 3–4 | pages = 353 | year = 2011 | s2cid = 19382087 }}

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 equations{{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| volume = 38 | issue = 3 | pages = 506 | year = 1990 | jstor = 171363}} or the Laplace transform of the distribution when the service time distribution is a mixture of exponential distributions.{{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 }} The Laplace transform of queue length{{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 }} and waiting time distributions{{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 }} can be computed when the waiting time distribution has a rational Laplace transform.

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{{Queueing theory}}

{{Stochastic processes}}

{{DEFAULTSORT:M G k queue}}

Category:Single queueing nodes

Category:Unsolved problems in mathematics