G/M/1 queue

In queueing theory, a discipline within the mathematical theory of probability, the G/M/1 queue represents the queue length in a system where interarrival times have a general (meaning arbitrary) distribution and service times for each job have an exponential distribution.^[1] The system is described in Kendall's notation where the G denotes a general distribution, M the exponential distribution for service times and the 1 that the model has a single server.

The arrivals of a G/M/1 queue are given by a renewal process. It is an extension of an M/M/1 queue, where this renewal process must specifically be a Poisson process (so that interarrival times have exponential distribution).

Models of this type can be solved by considering one of two M/G/1 queue dual systems, one proposed by Ramaswami and one by Bright.^[2]

Queue size at arrival times

Let $(X_{t}, t \geq 0)$ be a $G / M (μ) / 1$ queue with arrival times $(A_{n}, n \in ℕ)$ that have interarrival distribution A. Define the size of the queue immediately before the nth arrival by the process $U_{n} = X_{A_{n} -}$ . This is a discrete-time Markov chain with stochastic matrix:

$P = (\begin{matrix} 1 - a_{0} & a_{0} & 0 & 0 & 0 & \dots \\ 1 - (a_{0} + a_{1}) & a_{1} & a_{0} & 0 & 0 & \dots \\ 1 - (a_{0} + a_{1} + a_{2}) & a_{2} & a_{1} & a_{0} & 0 & \dots \\ 1 - (a_{0} + a_{1} + a_{2} + a_{3}) & a_{3} & a_{2} & a_{1} & a_{0} & \dots \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋱ \end{matrix})$

where $a_{v} = 𝔼 (\frac{(μ X)^{v} e^{- μ A}}{v!})$ .^[3]^{: 427–428}

The Markov chain $U_{n}$ has a stationary distribution if and only if the traffic intensity $ρ = (μ 𝔼 (A))^{- 1}$ is less than 1, in which case the unique such distribution is the geometric distribution with probability $η$ of failure, where $η$ is the smallest root of the equation $𝔼 (\exp (μ (η - 1) A))$ .^[3]^: 428

In this case, under the assumption that the queue is first-in first-out (FIFO), a customer's waiting time W is distributed by:^[3]^: 430

ℙ (W \leq x) = 1 - η \exp (- μ (1 - η) x) for x \geq 0

Busy period

The busy period can be computed by using a duality between the G/M/1 model and M/G/1 queue generated by the Christmas tree transformation.^[4]

Response time

The response time is the amount of time a job spends in the system from the instant of arrival to the time they leave the system. A consistent and asymptotically normal estimator for the mean response time, can be computed as the fixed point of an empirical Laplace transform.^[5]

References

↑ Adan, I.; Boxma, O.; Perry, D. (2005). "The G/M/1 queue revisited" (PDF). Mathematical Methods of Operations Research. 62 (3): 437. doi:10.1007/s00186-005-0032-6.
↑ Taylor, P. G.; Van Houdt, B. (2010). "On the dual relationship between Markov chains of GI/M/1 and M/G/1 type". Advances in Applied Probability. 42: 210. doi:10.1239/aap/1269611150.
1 2 3 Grimmett, G. R.; Stirzaker, D. R. (1992). Probability and Random Processes (second ed.). Oxford University Press. ISBN 0198572220.
↑ Perry, D.; Stadje, W.; Zacks, S. (2000). "Busy period analysis for M/G/1 and G/M/1 type queues with restricted accessibility". Operations Research Letters. 27 (4): 163. doi:10.1016/S0167-6377(00)00043-2.
↑ Chu, Y. K.; Ke, J. C. (2007). "Interval estimation of mean response time for a G/M/1 queueing system: Empirical Laplace function approach". Mathematical Methods in the Applied Sciences. 30 (6): 707. doi:10.1002/mma.806.

[adan2005-1] Adan, I.; Boxma, O.; Perry, D. (2005). "The G/M/1 queue revisited" (PDF). Mathematical Methods of Operations Research. 62 (3): 437. doi:10.1007/s00186-005-0032-6.

[2] Taylor, P. G.; Van Houdt, B. (2010). "On the dual relationship between Markov chains of GI/M/1 and M/G/1 type". Advances in Applied Probability. 42: 210. doi:10.1239/aap/1269611150.

[Grimmett&Stirzaker-3] 1 2 3 Grimmett, G. R.; Stirzaker, D. R. (1992). Probability and Random Processes (second ed.). Oxford University Press. ISBN 0198572220.

[4] Perry, D.; Stadje, W.; Zacks, S. (2000). "Busy period analysis for M/G/1 and G/M/1 type queues with restricted accessibility". Operations Research Letters. 27 (4): 163. doi:10.1016/S0167-6377(00)00043-2.

[5] Chu, Y. K.; Ke, J. C. (2007). "Interval estimation of mean response time for a G/M/1 queueing system: Empirical Laplace function approach". Mathematical Methods in the Applied Sciences. 30 (6): 707. doi:10.1002/mma.806.

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v t e Queueing theory
Single queueing nodes	D/M/1 queue M/D/1 queue M/D/c queue M/M/1 queue Burke's theorem M/M/c queue M/M/∞ queue M/G/1 queue Pollaczek–Khinchine formula Matrix analytic method M/G/k queue G/M/1 queue G/G/1 queue Kingman's formula Lindley equation Fork–join queue Bulk queue
Arrival processes	Poisson point process Markovian arrival process Rational arrival process
Queueing networks	Jackson network Traffic equations Gordon–Newell theorem Mean value analysis Buzen's algorithm Kelly network G-network BCMP network
Service policies	FIFO LIFO Processor sharing Round-robin Shortest job next Shortest remaining time
Key concepts	Continuous-time Markov chain Kendall's notation Little's law Product-form solution Balance equation Quasireversibility Flow-equivalent server method Arrival theorem Decomposition method Beneš method
Limit theorems	Fluid limit Mean-field theory Heavy traffic approximation Reflected Brownian motion
Extensions	Fluid queue Layered queueing network Polling system Adversarial queueing network Loss network Retrial queue
Information systems	Data buffer Erlang (unit) Erlang distribution Flow control (data) Message queue Network congestion Network scheduler Pipeline (software) Quality of service Scheduling (computing) Teletraffic engineering
Category