We consider a switch with uniform traffic operating under the MaxWeight scheduling algorithm . This traffic pattern is interesting to study in the heavy-traffic regime since the queue lengths exhibit a multi-dimensional state-space collapse  . We use a Lyapunov-type drift technique to characterize the heavy-traffic behavior of the expectation of the sum queue lengths in steady-state . Specifically, in the case of Bernoulli arrivals ,  we show that the heavy-traffic scaled queue length is \left({2n}}\right)\left( n-3{2}+1{2n} . Our result implies that the MaxWeight algorithm has optimal queue-length scaling behavior in the \right) heavy-traffic regime with respect to the size of a switch with a uniform traffic pattern. This settles the heavy-traffic version of an open conjecture  .
We consider a switch operating under the MaxWeight scheduling algorithm , under any traffic pattern such that all the ports are loaded. This system is interesting to study since the queue lengths exhibit a multi-dimensional state-space collapse in the heavy-traffic regime . We use a Lyapunov-type drift technique to characterize the heavy-traffic behavior of the expectation of the sum queue lengths in steady-state , under the assumption that all ports are saturated and all queues receive non-zero traffic. Under these conditions, we show that the heavy-traffic scaled queue length is given by\left(1-\frac{1{2n}}\right)||\sigma||^2, where \sigma is the vector of the standard deviations of arrivals to each port in the heavy-traffic limit. In the special case of uniform Bernoulli arrivals, the corresponding formula is given by\left( n-3{2}+1{2n} \right). The result shows that the heavy-traffic scaled queue length has optimal scaling with respect to n, thus settling one version of an open conjecture ; in fact, it is shown that the heavy-traffic queue length is at most within a factor of two from the optimal. We then consider certain asymptotic regimes where the load of the system scales simultaneously with the number of ports. We show that the MaxWeight algorithm has optimal queue length scaling behavior provided that the arrival rate approaches capacity sufficiently fast .
D
with uniform traffic
R
. This traffic pattern
, under any traffic pattern such that all the ports are loaded. This system
D
in the heavy-traffic regime
A
in the heavy-traffic regime
D
. Specifically, in the case of Bernoulli arrivals
A
under the assumption that all ports are saturated and all queues receive non-zero traffic. Under these conditions,
A
given by
A
1-\frac{1
A
||\sigma||^2, where \sigma is the vector of the standard deviations of arrivals to each port in the heavy-traffic limit. In the special case of uniform Bernoulli arrivals, the corresponding formula is given by
D
. Our result implies that the MaxWeight algorithm has optimal queue-length scaling behavior in the
A
. The result shows that the
R
regime
scaled queue length has optimal scaling
R
the size of a switch with a uniform traffic pattern. This settles the heavy-traffic
n, thus settling one
A
; in fact, it is shown that the heavy-traffic queue length is at most within a factor of two from the optimal. We then consider certain asymptotic regimes where the load of the system scales simultaneously with the number of ports. We show that the MaxWeight algorithm has optimal queue length scaling behavior provided that the arrival rate approaches capacity sufficiently fast