We consider a system with N~parallel servers where incoming jobs are immediately replicated to, say, d~servers. Each of the N servers has its own queue and follows a FCFS discipline. As soon as the first job replica is completed, the remaining replicas are abandoned. We investigate the achievable stability region for a quite general workload model with different job types and heterogeneous servers, reflecting job-server affinity relations which may arise from data locality issues and soft compatibility constraints. Under the assumption that job types are known beforehand we show for New-Better-than-Used (NBU) distributed speed variations that no replication $(d=1)$ gives a strictly larger stability region than replication $(d>1)$. Strikingly, this does not depend on the underlying distribution of the intrinsic job sizes, but observing the job types is essential for this statement to hold. In case of non-observable job types we show that for New-Worse-than-Used (NWU) distributed speed variations full replication ($d=N$) gives a larger stability region than no replication $(d=1)$.
We examine a canonical scenario where several wireless data sources generate sporadic delay-sensitive messages that need to be transmitted to a common access point. The access point operates in a time-slotted fashion, and can instruct the various sources in each slot with what probability to transmit a message, if they have any. When several sources transmit simultaneously, the access point can detect a collision, but is unable to infer the identities of the sources involved. While the access point can use the channel activity observations to obtain estimates of the queue states at the various sources, it does not have any explicit queue length information otherwise. We explore the achievable delay performance in a regime where the number of sources n grows large while the relative load remains fixed. We establish that, under any medium access algorithm without queue state information, the average delay must be at least of the order of n slots when the load exceeds some threshold lambda* < 1. This demonstrates that bounded delay can only be achieved if a positive fraction of the system capacity is sacrificed. Furthermore, we introduce a scalable Two-Phase algorithm which achieves a delay upper bounded uniformly in n when the load is below e -1 , and a delay of the order of n slots when the load is between e -1 and 1. Additionally, this algorithm provides robustness against correlated source activity.
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