On the Performance Evaluation of Distributed Join-Idle-Queue Load Balancing

Distributed Join-Idle-Queue load balancing is known to achieve vanishing waiting times in the large-scale limit provided that the number of dispatchers remains fixed, while the number of servers tends to infinity. When the number of dispatchers
$m$
scales to infinity together with the number of servers
$n$
, “such that
$r=n/m$
remains fixed, the large-scale performance of Join-Idle-Queue load balancing is less clear as waiting times no longer vanish. In this paper we first discuss some existing mean field models for distributed Join-Idle-Queue load balancing with
$r=n/r\mathrm{r}$
fixed and explain why the well-known model introduced in [1] is not exact in the large-scale limit. The inexactness is caused by mixing two variants of distributed Join-Idle-Queue load balancing: a variant with and one without token withdrawals. Next we introduce mean field models for Join-Idle-Queue load balancing with token withdrawals, where an idle server places a token at a dispatcher with the shortest among
$d$
randomly chosen dispatchers. The introduced mean field models imply that in case of phase type distributed service times and a total job arrival rate of
$\lambda n < \gamma$
, the response time of a job corresponds to that in a standard M/PH/1 queue with load
$\lambda q_{(}$
. The value of
$q0$
can be determined numerically and depends on
$\lambda, \uparrow$
. and
$d$
J
but not on the job size distribution (apart from its mean). This simple behavior is due to the token withdrawals and is lost if such withdrawals do not take place. We present simulation experiments that suggest that the unique fixed point of the introduced mean field models provides exact results in the large-scale limit.