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Register a new userScheduling in a random environment: stability and asymptotic optimality
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Added by
I.M. Verloop
Publication date
2011
fields
Informatics Engineering
and research's language is
English
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We investigate the scheduling of a common resource between several concurrent users when the feasible transmission rate of each user varies randomly over time. Time is slotted and users arrive and depart upon service completion. This may model for example the flow-level behavior of end-users in a narrowband HDR wireless channel (CDMA 1xEV-DO). As performance criteria we consider the stability of the system and the mean delay experienced by the users. Given the complexity of the problem we investigate the fluid-scaled system, which allows to obtain important results and insights for the original system: (1) We characterize for a large class of scheduling policies the stability conditions and identify a set of maximum stable policies, giving in each time slot preference to users being in their best possible channel condition. We find in particular that many opportunistic scheduling policies like Score-Based, Proportionally Best or Potential Improvement are stable under the maximum stability conditions, whereas the opportunistic scheduler Relative-Best or the cmu-rule are not. (2) We show that choosing the right tie-breaking rule is crucial for the performance (e.g. average delay) as perceived by a user. We prove that a policy is asymptotically optimal if it is maximum stable and the tie-breaking rule gives priority to the user with the highest departure probability. We will refer to such tie-breaking rule as myopic. (3) We derive the growth rates of the number of users in the system in overload settings under various policies, which give additional insights on the performance. (4) We conclude that simple priority-index policies with the myopic tie-breaking rule, are stable and asymptotically optimal. All our findings are validated with extensive numerical experiments.
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Dynamic affinity scheduling has been an open problem for nearly three decades. The problem is to dynamically schedule multi-type tasks to multi-skilled servers such that the resulting queueing system is both stable in the capacity region (throughput optimality) and the mean delay of tasks is minimized at high loads near the boundary of the capacity region (heavy-traffic optimality). As for applications, data-intensive analytics like MapReduce, Hadoop, and Dryad fit into this setting, where the set of servers is heterogeneous for different task types, so the pair of task type and server determines the processing rate of the task. The load balancing algorithm used in such frameworks is an example of affinity scheduling which is desired to be both robust and delay optimal at high loads when hot-spots occur. Fluid model planning, the MaxWeight algorithm, and the generalized $cmu$-rule are among the first algorithms proposed for affinity scheduling that have theoretical guarantees on being optimal in different senses, which will be discussed in the related work section. All these algorithms are not practical for use in data center applications because of their non-realistic assumptions. The join-the-shortest-queue-MaxWeight (JSQ-MaxWeight), JSQ-Priority, and weighted-workload algorithms are examples of load balancing policies for systems with two and three levels of data locality with a rack structure. In this work, we propose the Generalized-Balanced-Pandas algorithm (GB-PANDAS) for a system with multiple levels of data locality and prove its throughput optimality. We prove this result under an arbitrary distribution for service times, whereas most previous theoretical work assumes geometric distribution for service times. The extensive simulation results show that the GB-PANDAS algorithm alleviates the mean delay and has a better performance than the JSQ-MaxWeight algorithm by twofold
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Dynamic affinity load balancing of multi-type tasks on multi-skilled servers, when the service rate of each task type on each of the servers is known and can possibly be different from each other, is an open problem for over three decades. The goal is to do task assignment on servers in a real time manner so that the system becomes stable, which means that the queue lengths do not diverge to infinity in steady state (throughput optimality), and the mean task completion time is minimized (delay optimality). The fluid model planning, Max-Weight, and c-$mu$-rule algorithms have theoretical guarantees on optimality in some aspects for the affinity problem, but they consider a complicated queueing structure and either require the task arrival rates, the service rates of tasks on servers, or both. In many cases that are discussed in the introduction section, both task arrival rates and service rates of different task types on different servers are unknown. In this work, the Blind GB-PANDAS algorithm is proposed which is completely blind to task arrival rates and service rates. Blind GB-PANDAS uses an exploration-exploitation approach for load balancing. We prove that Blind GB-PANDAS is throughput optimal under arbitrary and unknown distributions for service times of different task types on different servers and unknown task arrival rates. Blind GB-PANDAS desires to route an incoming task to the server with the minimum weighted-workload, but since the service rates are unknown, such routing of incoming tasks is not guaranteed which makes the throughput optimality analysis more complicated than the case where service rates are known. Our extensive experimental results reveal that Blind GB-PANDAS significantly outperforms existing methods in terms of mean task completion time at high loads.
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