Stability, memory, and messaging tradeoffs in heterogeneous service systems

TL;DR

This paper investigates the minimal memory and messaging resources needed for a dispatcher to maintain stability in a heterogeneous distributed service system with unknown server processing rates.

Contribution

It introduces a maximally stable dispatching policy using minimal message exchange and logarithmic memory, and establishes lower bounds on resource requirements for stability.

Findings

A policy with arbitrarily small message rate and logarithmic memory achieves maximal stability.

Policies with sub-quadratic message exchange and sub-logarithmic memory cannot guarantee maximal stability.

Logarithmic memory is both necessary and sufficient for stability under limited messaging.

Abstract

We consider a heterogeneous distributed service system, consisting of $n$ servers with unknown and possibly different processing rates. Jobs with unit mean and independent processing times arrive as a renewal process of rate $\lambda n$, with $0<\lambda<1$, to the system. Incoming jobs are immediately dispatched to one of several queues associated with the $n$ servers. We assume that the dispatching decisions are made by a central dispatcher endowed with a finite memory, and with the ability to exchange messages with the servers. We study the fundamental resource requirements (memory bits and message exchange rate) in order for a dispatching policy to be {\bf maximally stable}, i.e., stable whenever the processing rates are such that the arrival rate is less than the total available processing rate. First, for the case of Poisson arrivals and exponential service times, we present a policy that is maximally stable while using a positive (but arbitrarily small) message rate, and $\log_2(n)$ bits of memory. Second, we show that within a certain broad class of policies, a dispatching policy that exchanges $o\big(n^2\big)$ messages per unit of time, and with $o(\log(n))$ bits of memory, cannot be maximally stable. Thus, as long as the message rate is not too excessive, a logarithmic memory is necessary and sufficient for maximal stability.