On Large Loss Models with Splittable Jobs

Sunday, October 27, 2024, 11:10 -- 11:40

In this talk I will introduce a new class of models that are of relevance in the context of parallelization in multiserver systems. In particular we consider adaptive multi-rate multi-server jobs where each job can be run on a variable number of servers up to a maximum number. This leads to a new class of loss moders with a variable number of servers.

The job allocation policy follows a greedy approach, meaning that when an arriving job en- counters $j$ idle servers, it occupies $i = min(d, j)$ servers, with a mean holding time of $ \tfrac{1}{i}$ (assuming no overhead due to job splitting). Given a flow of jobs with rate $ n \lambda$, we analyze the behavior of the system for large $n$. Focusing on the occupation vector, which describes the number of jobs split across $i$ servers for $ i = 1, \dotsc , d$, we demonstrate that as $n$ tends to infinity, a state space collapse occurs, where all jobs are distributed over $d$ servers. Consequently, the system asymptotically behaves like a blocking system, in which jobs are either split over $d$ servers or rejected.

This model is difficult to analyze as there is no monotonicity to obtain any sample-path coupling argument. So instead we use an idea of coupling the generator of a mean-field fluid limit to underlying Markov process related to the server occupancy and then exploit Stein’s technique.

Joint work with S. Ghanbarian (Waterloo), F. Guillemin (Orange), and A. Mukhopadhyay (Warwick).