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Samples from a high-dimensional first-order auto-regressive process generated by an independently and identically distributed random innovation sequence are observed by a sender which can communicate only finitely many bits per unit time to a receiver. The receiver seeks to form an estimate of the process value at every time instant in real-time. We consider a time-slotted communication model in a slow-sampling regime where multiple communication slots occur between two sampling instants. We propose a successive update scheme which uses communication between sampling instants to refine estimates of the latest sample and study the following question: Is it better to collect communication of multiple slots to send better refined estimates, making the receiver wait more for every refinement, or to be fast but loose and send new information in every communication opportunity? We show that the fast but loose successive update scheme with ideal spherical codes is universally optimal asymptotically for a large dimension. However, most practical quantization codes for fixed dimensions do not meet the ideal performance required for this optimality, and they typically will have a bias in the form of a fixed additive error. Interestingly, our analysis shows that the fast but loose scheme is not an optimal choice in the presence of such errors, and a judiciously chosen frequency of updates outperforms it.

Popular dispatching policies such as the join shortest queue (JSQ), join smallest work (JSW) and their power of two variants are used in load balancing systems where the instantaneous queue length or workload information at all queues or a subset of them can be queried. In situations where the dispatcher has an associated memory, one can minimize this query overhead by maintaining a list of idle servers to which jobs can be dispatched. Recent alternative approaches that do not require querying such information include the cancel on start and cancel on complete based replication policies. The downside of such policies however is that the servers must communicate the start or completion of each service to the dispatcher and must allow cancellation of redundant copies. In this work, we consider a load balancing environment where the dispatcher cannot query load information, does not have a memory, and cannot cancel any replica that it may have created. In such a rigid environment, we allow the dispatcher to possibly append a server side cancellation criteria to each job or its replica. A job or a replica is served only if it satisfies the predefined criteria at the time of service. We focus on a criteria that is based on the waiting time experienced by a job or its replica and analyze several variants of this policy based on the assumption of asymptotic independence of queues. The proposed policies are novel and perform remarkably well in spite of the rigid operating constraints.