An algorithm for improved delay-scaling in input-queued switches

Abstract

Abstract We consider an

$$n\times n$$

n × n

input-queued switch with uniform Bernoulli traffic and study the delay (or equivalently, the queue length) in the regime where the size of the switch n and the load (denoted by

$$\rho $$

ρ

) simultaneously become large. We devise an algorithm with expected total queue length equal to

$$O((n^{5/4}(1-\rho )^{-1})\log \max (1/\rho ,n))$$

O (

(

n

5 / 4

( 1 - ρ )

- 1

)

log max

( 1 / ρ , n )

)

for large n and

$$\rho $$

ρ

such that

$$(1-\rho )^{-1} \ge n^{3/4}$$

( 1 - ρ )

- 1

≥

n

3 / 4

. This result improves the previous best queue length bound in the regime

$$n^{3/4}< (1-\rho )^{-1} < n^{7/4}$$

n

3 / 4

<

( 1 - ρ )

- 1

<

n

7 / 4

. Under same conditions, the algorithm has an amortized time complexity

$$O(n+(1-\rho )^2 n^{7/2} / \log \max (1/\rho ,n))$$

O ( n +

( 1 - ρ )

2

n

7 / 2

/ log max

( 1 / ρ , n )

)

. The time complexity becomes O(n) when

$$(1-\rho )^{-1} \ge n^{5/4}.$$

( 1 - ρ )

- 1

≥

n

5 / 4

.