M/M/∞ queue

In queueing theory, a discipline within the mathematical theory of probability, the M/M/∞ queue is a multi-server queueing model where every arrival experiences immediate service and does not wait.^[1] In Kendall's notation it describes a system where arrivals are governed by a Poisson process, there are infinitely many servers, so jobs do not need to wait for a server. Each job has an exponentially distributed service time. It is a limit of the M/M/c queue model where the number of servers c becomes very large.

The model can be used to model bound lazy deletion performance.^[2]

Model definition

An M/M/∞ queue is a stochastic process whose state space is the set {0,1,2,3,...} where the value corresponds to the number of customers currently being served.

Arrivals occur at rate λ according to a Poisson process and move the process from state i to i + 1.
Service times have an exponential distribution with parameter μ and there are always sufficient servers such that every arriving job is served immediately. Transitions from state i to i − 1 are at rate iμ

The model has transition rate matrix

Q=\begin{pmatrix} -\lambda & \lambda \\ \mu & -(\mu+\lambda) & \lambda \\ &2\mu & -(2\mu+\lambda) & \lambda \\ &&3\mu & -(3\mu+\lambda) & \lambda \\ &&&&\ddots \end{pmatrix}.

The state space diagram for this chain is as below.

Transient solution

The transient distribution can be written using moment generating functions^[3] and formulas for transient means and variances computed by solving differential equations.^[4] Assuming the system starts in state 0 at time 0, then the probability the system is in state j at time t can be written as^[5]^:356

p_{0j}(t) = \exp \left( -\frac{\lambda}{\mu}(1-e^{-\mu t}) \right) \frac{\left(\frac{\lambda}{\mu}(1-e^{-\mu t})\right)^j}{j!} \text{ for } j \geq 0

from which the mean queue length at time t can be computed (writing N(t) for the number of customers in the system at time t given the system is empty at time zero)

\mathbb E(N(t) | N(0)=0) = \frac{\lambda}{\mu} (1-e^{-\mu t}) \text{ for } t \geq 0.

Response time

The response time for each arriving job is a single exponential distribution with parameter μ. The average response time is therefore 1/μ.^[6]

Maximum queue length

Given the system is in equilibrium at time 0, we can compute the cumulative distribution function of the process maximum over a finite time horizon T in terms of Charlier polynomials.^[2]

Congestion period

The congestion period is the length of time the process spends above a fixed level c, starting timing from the instant the process transitions to state c + 1. This period has mean value^[7]

\frac{1}{\lambda}\sum_{i>c} \frac{c!}{i!}\left( \frac{\lambda}{\mu} \right)^{i-c}

and the Laplace transform can be expressed in terms of Kummer's function.^[8]

Stationary analysis

The stationary probability mass function is a Poisson distribution^[9]

\pi_k = \frac{(\lambda/\mu)^k e^{-\lambda/\mu}}{k!} \quad k \geq 0

so the mean number of jobs in the system is λ/μ.

The stationary distribution of the M/G/∞ queue is the same as that of the M/M/∞ queue.^[10]

Heavy traffic

Main article: heavy traffic approximation

Writing N_t for the number of customers in the system at time t as ρ → ∞ the scaled process

X_t = \frac{N_t - \lambda/\mu}{\sqrt{\lambda/\mu}}

converges to an Ornstein–Uhlenbeck process with normal distribution and correlation parameter 1, defined by the Itō calculus as^[7]^[11]

dX_t = -Xdt+\sqrt{2}dW_t

where W is a standard Brownian motion.

References

↑ Harrison, Peter; Patel, Naresh M. (1992). Performance Modelling of Communication Networks and Computer Architectures. Addison–Wesley. p. 173.
↑ 2.0 2.1 Morrison, J. A.; Shepp, L. A.; Van Wyk, C. J. (1987). "A Queueing Analysis of Hashing with Lazy Deletion". SIAM Journal on Computing 16 (6): 1155. doi:10.1137/0216073.
↑ El-Sherbiny, A. A. (2010). "Transient Solution to an infinite Server Queue with Varying Arrival and Departure Rate". Journal of Mathematics and Statistics 6: 1–5. doi:10.3844/jmssp.2010.1.3.
↑ Ellis, Peter M. (2010). "The Time-Dependent Mean and Variance of the Non-Stationary Markovian Infinite Server System". Journal of Mathematics and Statistics 6: 68–71. doi:10.3844/jmssp.2010.68.71.
↑ Kulkarni, Vidyadhar G. (1995). Modeling and analysis of stochastic systems (First edition ed.). Chapman & Hall. ISBN 0412049910.
↑ Kleinrock, Leonard (1975). Queueing Systems Volume 1: Theory. pp. 101–103, 404. ISBN 0471491101.
↑ 7.0 7.1 Guillemin, Fabrice M.; Mazumdar, Ravi R.; Simonian, Alain D. (1996). "On Heavy Traffic Approximations for Transient Characteristics of M/M/∞ Queues". Journal of Applied Probability (Applied Probability Trust) 33 (2): 490–506. JSTOR 3215073.
↑ Guillemin, Fabrice; Simonian, Alain (1995). "Transient Characteristics of an M/M/∞ System". Advances in Applied Probability (Applied Probability Trust) 27 (3): 862–888. doi:10.2307/1428137. JSTOR Probability Trust Applied Probability Trust.
↑ Bolch, Gunter; Greiner, Stefan; de Meer, Hermann; Trivedi, Kishor Shridharbhai (2006). Queueing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications. John Wiley & Sons. p. 249. ISBN 0471791563.
↑ Newell, G. F. (1966). "The M/G/∞ Queue". SIAM Journal on Applied Mathematics 14: 86. doi:10.1137/0114007. JSTOR 2946178.
↑ Knessl, C.; Yang, Y. P. (2001). "Asymptotic Expansions for the Congestion Period for the M/M/∞ Queue". Queueing Systems 39 (2/3): 213. doi:10.1023/A:1012752719211.

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 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

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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

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