Heavy traffic approximation

In queueing theory, a discipline within the mathematical theory of probability, a heavy traffic approximation (sometimes called heavy traffic limit theorem or diffusion approximation) involves the matching of a queueing model with a diffusion process under some limiting conditions on the model's parameters. The first such result was published by John Kingman, who showed that when the utilisation parameter of an M/M/1 queue is near 1, a scaled version of the queue length process can be accurately approximated by a reflected Brownian motion.

Heavy traffic condition

Heavy traffic approximations are typically stated for the process X(t) describing the number of customers in the system at time t. They are arrived at by considering the model under the limiting values of some model parameters and therefore for the result to be finite the model must be rescaled by a factor n, denoted

X

^

n

(

t

)

=

X

(

n

t

)

−

E

(

X

(

n

t

)

)

n

{\displaystyle {\hat {X}}_{n}(t)={\frac {X(nt)-\mathbb {E} (X(nt))}{\sqrt {n}}}}

and the limit of this process is considered as n → ∞.

There are three classes of regime under which such approximations are generally considered.

The number of servers is fixed and the traffic intensity (utilization) is increased to 1 (from below). The queue length approximation is a reflected Brownian motion.

Traffic intensity is fixed and the number of servers and arrival rate are increased to infinity. Here the queue length limit converges to the normal distribution.

A quantity β is fixed where

β

=

(

1

−

ρ

)

s

{\displaystyle \beta =(1-\rho ){\sqrt {s}}}

with ρ representing the traffic intensity and s the number of servers.