Heavy traffic approximation


In queueing theory, a discipline within the mathematical theory of probability, a heavy traffic approximation is 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 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
and the limit of this process is considered as n → ∞.
There are three classes of regime under which such approximations are generally considered.
  1. The number of servers is fixed and the traffic intensity is increased to 1. The queue length approximation is a reflected Brownian motion.
  2. 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.
  3. A quantity β is fixed where

    Results for a G/G/1 queue

Theorem 1. Consider a sequence of G/G/1 queues indexed by.
For queue

let denote the random inter-arrival time, denote the random service time;
let denote the traffic intensity with and ;
let denote the waiting time in queue for a customer in steady state;
Let and
Suppose that,, and. then
provided that:

for some, and are both less than some constant for all.

Heuristic argument

Let be the difference between the nth service time and the nth inter-arrival time;
Let be the waiting time in queue of the nth customer;
Then by definition:
After recursive calculation, we have:
Let, with are i.i.d;
Define and ;
Then we have
we get by taking limit over.
Thus the waiting time in queue of the nth customer is the supremum of a random walk with a negative drift.
Random walk can be approximated by a Brownian motion when the jump sizes approach 0 and the times between the jump approach 0.
We have and has independent and stationary increments. When the traffic intensity approaches 1 and tends to, we have after replaced with continuous value according to functional central limit theorem. Thus the waiting time in queue of the th customer can be approximated by the supremum of a Brownian motion with a negative drift.
Theorem 2. Let be a Brownian motion with drift and standard deviation starting at the origin, and let
if
otherwise

Conclusion

Thus, the heavy traffic limit theorem is heuristically argued. Formal proofs usually follow a different approach which involve characteristic functions.

Example

Consider an M/G/1 queue with arrival rate, the mean of the service time, and the variance of the service time. What is average waiting time in queue in the steady state?
The exact average waiting time in queue in steady state is given by:
The corresponding heavy traffic approximation:
The relative error of the heavy traffic approximation:
Thus when, we have :