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.

The model has transition rate matrix

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

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)

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]

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]

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

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

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

where W is a standard Brownian motion.


  1. Harrison, Peter; Patel, Naresh M. (1992). Performance Modelling of Communication Networks and Computer Architectures. Addison–Wesley. p. 173.
  2. 1 2 Morrison, J. A.; Shepp, L. A.; Van Wyk, C. J. (1987). "A Queueing Analysis of Hashing with Lazy Deletion" (PDF). SIAM Journal on Computing. 16 (6): 1155. doi:10.1137/0216073.
  3. 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.
  4. 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.
  5. Kulkarni, Vidyadhar G. (1995). Modeling and analysis of stochastic systems (First ed.). Chapman & Hall. ISBN 0412049910.
  6. Kleinrock, Leonard (1975). Queueing Systems Volume 1: Theory. pp. 101–103, 404. ISBN 0471491101.
  7. 1 2 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. doi:10.2307/3215073. ISSN 0021-9002. JSTOR 3215073 via JSTOR. (registration required (help)).
  8. Guillemin, Fabrice; Simonian, Alain (1995). "Transient Characteristics of an M/M/∞ System". Advances in Applied Probability. Applied Probability Trust. 27 (3): 862–88. doi:10.2307/1428137. ISSN 0001-8678. JSTOR 1428137 via JSTOR. (registration required (help)).
  9. 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.
  10. Newell, G. F. (1966). "The M/G/∞ Queue". SIAM Journal on Applied Mathematics. Society for Industrial and Applied Mathematics. 14 (1): 86–8. doi:10.1137/0114007. ISSN 0036-1399. JSTOR 2946178 via JSTOR. (registration required (help)).
  11. 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.
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