Dudley triangle

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. 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.

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

$Q=\left(\begin{array}{cc}-\lambda & \lambda \\ \mu & -(\mu +\lambda )& \lambda \\ & 2\mu & -(2\mu +\lambda )& \lambda \\ & & 3\mu & -(3\mu +\lambda )& \lambda \\ & & & & \ddots \end{array}\right).$

The state space diagram for this chain is as below.

File:Mminfinity-statespace.svg

Transient solution

The transient distribution can be written using moment generating functions and formulas for transient means and variances computed by solving differential equations. 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 asPrimarily based on the most recent URA personal property value index (PPPI) flash estimates, we know that the PPPI, which represents the overall real property price development, has dipped in 2013Q4. That is the first dip the market has seen within the final 2 years.

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

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)

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

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.

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.

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

$\frac{\mu }{\lambda }\sum _{i>c}\frac{c!}{i!}{\left(\frac{\lambda }{\mu }\right)}^{i-c}$

and the Laplace transform can be expressed in terms of Kummer's function.

Stationary analysis

The stationary probability mass function is a Poisson distribution

${\pi }_k=\frac{(\lambda /\mu {)}^k{e}^{-\lambda /\mu }}{k!}k\ge 0$

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

Heavy traffic

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. 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

$d{X}_t=-Xdt+\sqrt{2}d{W}_t$

where W is a standard Brownian motion.

References

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