Many-server scaling of the N-system under FCFS–ALIS

Abstract

The N-system with independent Poisson arrivals and exponential serverdependent service times under the first come first served and assign to the longest idle server policy has an explicit steady-state distribution. We scale the arrival rate and the number of servers simultaneously, and obtain the fluid and central limit approximation for the steady state. This is the first step toward exploring the many-server scaling limit behavior of general parallel service systems.

Design heuristic for parallel many server systems

“…A proof of the conjecture, in general form or under more limiting assumptions, is the first task. A preliminary attempt to verify the conjecture for many server N-system in the Poisson-exponential case is presented in [32]. It would also be interesting to evaluate the efficacy of FCFS-ALIS policy in comparison with other policies.…”

“…It is our conjecture that when λ and n are large, the matching rates of the general parallel service system under FCFS-ALIS are approximated by those of the FCFS infinite bipartite matching model: this is the basis for our current paper. A first step towards verifying the conjectured many server behavior is made in [32] by studying the limiting behavior of the "N" system. Even for this very simple system, the derivations of the limits are quite laborious, emphasizing the difficulty in verifying the conjecture for the general system.…”

Matching Queues with Reneging: a Product Form Solution

“…Supply and demand units are matched, in an FCFS manner, according to the compatibility graph depicted in Figure 1. Our model is a variation of the widely studied N-system [Green, 1985, Adan et al, 2009, Visschers et al, 2012, Adan et al, 2018, Zhan and Weiss, 2018. There are two types of supply, flexible, indexed by 1, and inflexible, indexed by 2.…”