No abstract

A simple queueing system, known as the fork-join queue, is considered with basic performance measure defined as the delay between the fork and join dates. Simple lower and upper bounds are derived for some of the statistics of this quantity. They are obtained, in both transient and steady-state regimes, by stochastically comparing the original system to other queueing systems with a structure simpler than the original system, yet with identical stability characteristics. In steady-state, under renewal assumptions, the computation reduces to standard GI/GI/1 calculations and the bounds constitute a first sizing-up of system performance. These bounds can also be used to show that for homogeneous fork-join queue system under assumptions, the moments of the system response time grow logarithmically in the number of parallel processors provided the service time distribution has rational Laplace–Stieltjes transform. The bounding arguments combine ideas from the theory of stochastic ordering with the notion of associated random variables, and are of independent interest to study various other queueing systems with synchronization constraints. The paper is an abridged version of a more complete report on the matter [6].

The theory of large deviations for jump Markov processes has been generally proved only when jump rates are bounded below, away from zero [4,9,13]. Yet various applications of interest do not satisfy this condition. We describe several classes of models where jump rates diminish to zero in a Lipschitz continuous way. Under appropriate conditions, we prove that the sample path large deviations principle continues to hold. Under our conditions, the rate function remains an integral over a local rate function, which retains its standard representation.

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We consider a system with a single queue and multiple server pools of heterogenous exponential servers. The system operates under a policy that always routes a job to the pool with longest cumulative idleness among pools with available servers, in an attempt to achieve fairness toward servers. It is easy to find examples of a system with a fixed number of servers, for which fairness is not achieved by this policy in any reasonable sense. Our main result shows that in the many-server regime of Halfin and Whitt, the policy does attain equalization of cumulative idleness, and that the equalization time, defined within any given precision level, remains bounded in the limit. An important feature of this policy is that it acts 'blindly', in that it requires no information on the service or arrival rates.