Gated Infinite Server Queues in Light Traffic

TL;DR

This paper analyzes gated infinite server queues in light traffic, deriving the distribution of stage length and customer count, with convergence results for the series representations in the stationary regime.

Contribution

It provides new analytical results for the distribution of stage length and customer numbers in gated $M/G/ ext{infinity}$ and $GI/M/ ext{infinity}$ queues, including convergence proofs.

Findings

Series representation of stage length density derived

Convergence of solutions established in light traffic

Results extend to $GI/M/\infty$ queues

Abstract

We consider $M/G/\infty$ queues with gated service and obtain results on the distribution of the stage length and the number of customers served in a stage when the system is stationary. The stage length density is expressed as an infinite series of terms, involving the solution of an infinite system of linear equations. The convergence of a sequence of solutions arising from truncations of the infinite system is established in the light traffic case. Analogous results are established for a similar $GI/M/\infty$ gated system.