# M/G/k queue

In queueing theory, a discipline within the mathematical theory of probability, an M/G/k queue is a queue model where arrivals are Markovian (modulated by a Poisson process), service times have a 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 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.[1]

## Model definition

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."[2]

Various approximations for the average queue size,[3] stationary distribution[4][5] and approximation by a reflected Brownian motion[6][7] 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..[8][9] 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.

## Average delay/waiting time

There are numerous approximations for the average delay a job experiences.[5][7][10][11][12][13] The first such was given in 1959 using a factor to adjust the mean waiting time in an M/M/c queue[14][15] This result is sometimes known as Kingman's law of congestion.[16]

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

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."[17]

However, it is known that no approximation using only the first two moments can be accurate in all cases.[14]

A Markov–Krein characterization has been shown to produce tight bounds on the mean waiting time.[18]

## Inter-departure times

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.[19]

## Two servers

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[20] or the Laplace transform of the distribution when the service time distribution is a mixture of exponential distributions.[21] The Laplace transform of queue length[22] and waiting time distributions[23] can be computed when the waiting time distribution has a rational Laplace transform.

## References

1. ^ Kingman, J. F. C. (2009). "The first Erlang century—and the next". Queueing Systems. 63 (1–4): 3–4. doi:10.1007/s11134-009-9147-4. S2CID 38588726.
2. ^ Tijms, H. C.; Van Hoorn, M. H.; Federgruen, A. (1981). "Approximations for the Steady-State Probabilities in the M/G/c Queue". Advances in Applied Probability. 13 (1): 186–206. doi:10.2307/1426474. JSTOR 1426474.
3. ^ Ma, B. N. W.; Mark, J. W. (1995). "Approximation of the Mean Queue Length of an M/G/c Queueing System". Operations Research. 43 (1): 158–165. doi:10.1287/opre.43.1.158. JSTOR 171768.
4. ^ Breuer, L. (2008). "Continuity of the M/G/c queue". Queueing Systems. 58 (4): 321–331. doi:10.1007/s11134-008-9073-x. S2CID 2345317.
5. ^ a b Hokstad, Per (1978). "Approximations for the M/G/m Queue". Operations Research. INFORMS. 26 (3): 510–523. doi:10.1287/opre.26.3.510. JSTOR 169760.
6. ^ Kimura, T. (1983). "Diffusion Approximation for an M/G/m Queue". Operations Research. 31 (2): 304–321. doi:10.1287/opre.31.2.304. JSTOR 170802.
7. ^ a b Yao, D. D. (1985). "Refining the Diffusion Approximation for the M/G/m Queue". Operations Research. 33 (6): 1266–1277. doi:10.1287/opre.33.6.1266. JSTOR 170637.
8. ^ Khazaei, H.; Misic, J.; Misic, V. B. (2012). "Performance Analysis of Cloud Computing Centers Using M/G/m/m+r Queuing Systems". IEEE Transactions on Parallel and Distributed Systems. 23 (5): 936. doi:10.1109/TPDS.2011.199. S2CID 16934438.
9. ^ Khazaei, H.; Misic, J.; Misic, V. B. (2011). "Modelling of Cloud Computing Centers Using M/G/m Queues". 2011 31st International Conference on Distributed Computing Systems Workshops. p. 87. doi:10.1109/ICDCSW.2011.13. ISBN 978-1-4577-0384-3. S2CID 16067523.
10. ^ Hokstad, Per (1980). "The Steady-State Solution of the M/K2/m Queue". Advances in Applied Probability. Applied Probability Trust. 12 (3): 799–823. doi:10.2307/1426432. JSTOR 1426432.
11. ^ Köllerström, Julian (1974). "Heavy Traffic Theory for Queues with Several Servers. I". Journal of Applied Probability. Applied Probability Trust. 11 (3): 544–552. doi:10.1017/s0021900200096327. JSTOR 3212698.
12. ^ Nozaki, S. A.; Ross, S. M. (1978). "Approximations in Finite-Capacity Multi-Server Queues with Poisson Arrivals". Journal of Applied Probability. 15 (4): 826–834. doi:10.2307/3213437. JSTOR 3213437.
13. ^ Boxma, O. J.; Cohen, J. W.; Huffels, N. (1979). "Approximations of the Mean Waiting Time in an M/G/s Queueing System". Operations Research. INFORMS. 27 (6): 1115–1127. doi:10.1287/opre.27.6.1115. JSTOR 172087.
14. ^ a b Gupta, V.; Harchol-Balter, M.; Dai, J. G.; Zwart, B. (2009). "On the inapproximability of M/G/K: Why two moments of job size distribution are not enough". Queueing Systems. 64: 5–48. doi:10.1007/s11134-009-9133-x. S2CID 16755599.
15. ^ Lee, A. M.; Longton, P. A. (1959). "Queueing Processes Associated with Airline Passenger Check-in". Journal of the Operational Research Society. 10: 56–71. doi:10.1057/jors.1959.5.
16. ^ Gans, N.; Koole, G.; Mandelbaum, A. (2003). "Telephone Call Centers: Tutorial, Review, and Research Prospects" (PDF). Manufacturing & Service Operations Management. 5 (2): 79. doi:10.1287/msom.5.2.79.16071.
17. ^ Whitt, W. (2009). "Approximations for the GI/G/m Queue" (PDF). Production and Operations Management. 2 (2): 114–161. doi:10.1111/j.1937-5956.1993.tb00094.x.
18. ^ Gupta, V.; Osogami, T. (2011). "On Markov–Krein characterization of the mean waiting time in M/G/K and other queueing systems". Queueing Systems. 68 (3–4): 339. doi:10.1007/s11134-011-9248-8. S2CID 35061112.
19. ^ Veeger, C.; Kerner, Y.; Etman, P.; Adan, I. (2011). "Conditional inter-departure times from the M/G/s queue". Queueing Systems. 68 (3–4): 353. doi:10.1007/s11134-011-9240-3. S2CID 19382087.
20. ^ Knessl, C.; Matkowsky, B. J.; Schuss, Z.; Tier, C. (1990). "An Integral Equation Approach to the M/G/2 Queue". Operations Research. 38 (3): 506. doi:10.1287/opre.38.3.506. JSTOR 171363.
21. ^ Cohen, J. W. (1982). "On the M/G/2 queueing model". Stochastic Processes and Their Applications. 12 (3): 231–248. doi:10.1016/0304-4149(82)90046-1.
22. ^ Hokstad, Per (1979). "On the Steady-State Solution of the M/G/2 Queue". Advances in Applied Probability. Applied Probability Trust. 11 (1): 240–255. doi:10.2307/1426776. JSTOR 1426776.
23. ^ Boxma, O. J.; Deng, Q.; Zwart, A. P. (2002). "Waiting-Time Asymptotics for the M/G/2 Queue with Heterogeneous Servers". Queueing Systems. 40: 5–31. doi:10.1023/A:1017913826973. S2CID 2513624.