Douglas G. Down

Server Assignment Policies for Maximizing the Steady-State Throughput of Finite Queueing Systems

For a system of finite queues, we study how servers should be assigned dynamically to stations in order to obtain optimal (or near-optimal) long-run average throughput. We assume that travel times between different service facilities are negligible, that each server can work on only one job at a time, and that several servers can work together on one job. We show that when the service rates depend only on either the server or the station (and not both), then all nonidling server assignment policies are optimal. Moreover, for a Markovian system with two stations in tandem and two servers, we show that the optimal policy assigns one server to each station unless that station is blocked or starved (in which case the server helps at the other station), and we specify the criterion used for assigning servers to stations. Finally, we propose a simple server assignment policy for tandem systems in which the number of stations equals the number of servers, and we present numerical results that show that our policy appears to yield near-optimal throughput under general conditions.

Dynamic Server Allocation for Queueing Networks with Flexible Servers

This paper is concerned with the design of dynamic server assignment policies that maximize the capacity of queueing networks with flexible servers. Flexibility here means that each server may be capable of performing service at several different classes in the network. We assume that the interarrival times and the service times are independent and identically distributed, and that routing is probabilistic. We also allow for server switching times, which we assume to be independent and identically distributed. We deduce the value of a tight upper bound on the achievable capacity by equating the capacity of the queueing network model with that of a limiting deterministic fluid model. The maximal capacity of the deterministic model is given by the solution to a linear programming problem that also provides optimal allocations of servers to classes. We construct particular server assignment policies, called generalized round-robin policies, that guarantee that the capacity of the queueing network will be arbitrarily close to the computed upper bound. The performance of such policies is studied using numerical examples.

Piecewise linear test functions for stability and instability of queueing networks

We develop the use of piecewise linear test functions for the analysis of stability of multiclass queueing networks and their associated fluid limit models. It is found that if an associated LP admits a positive solution, then a Lyapunov function exists. This implies that the fluid limit model is stable and hence that the network model is positive Harris recurrent with a finite polynomial moment. Also, it is found that if a particular LP admits a solution, then the network model is transient.

A Hybrid Scheduling Approach for Scalable Heterogeneous Hadoop Systems

The scalability of Cloud infrastructures has significantly increased their applicability. Hadoop, which works based on a MapReduce model, provides for efficient processing of Big Data. This solution is being used widely by most Cloud providers. Hadoop schedulers are critical elements for providing desired performance levels. A scheduler assigns MapReduce tasks to Hadoop resources. There is a considerable challenge to schedule the growing number of tasks and resources in a scalable manner. Moreover, the potential heterogeneous nature of deployed Hadoop systems tends to increase this challenge. This paper analyzes the performance of widely used Hadoop schedulers including FIFO and Fair sharing and compares them with the COSHH (Classification and Optimization based Scheduler for Heterogeneous Hadoop) scheduler, which has been developed by the authors. Based on our insights, a hybrid solution is introduced, which selects appropriate scheduling algorithms for scalable and heterogeneous Hadoop systems with respect to the number of incoming jobs and available resources.

Dynamic control of a single-server system with abandonments

In this paper, we discuss the dynamic server control in a two-class service system with abandonments. Two models are considered. In the first case, rewards are received upon service completion, and there are no abandonment costs (other than the lost opportunity to gain rewards). In the second, holding costs per customer per unit time are accrued, and each abandonment involves a fixed cost. Both cases are considered under the discounted or average reward/cost criterion. These are extensions of the classic scheduling question (without abandonments) where it is well known that simple priority rules hold.The contributions in this paper are twofold. First, we show that the classic c–μ rule does not hold in general. An added condition on the ordering of the abandonment rates is sufficient to recover the priority rule. Counterexamples show that this condition is not necessary, but when it is violated, significant loss can occur. In the reward case, we show that the decision involves an intuitive tradeoff between getting more rewards and avoiding idling. Secondly, we note that traditional solution techniques are not directly applicable. Since customers may leave in between services, an interchange argument cannot be applied. Since the abandonment rates are unbounded we cannot apply uniformization—and thus cannot use the usual discrete-time Markov decision process techniques. After formulating the problem as a continuous-time Markov decision process (CTMDP), we use sample path arguments in the reward case and a savvy use of truncation in the holding cost case to yield the results. As far as we know, this is the first time that either have been used in conjunction with the CTMDP to show structure in a queueing control problem. The insights made in each model are supported by a detailed numerical study.

Dynamic load balancing in parallel queueing systems : stability and optimal control

We consider a system of parallel queues with dedicated arrival streams. At each decision epoch a decision-maker can move customers from one queue to another. The cost for moving customers consists of a fixed cost and a linear, variable cost dependent on the number of customers moved. There are also linear holding costs that may depend on the queue in which customers are stored. Under very mild assumptions, we develop stability (and instability) conditions for this system via a fluid model. Under the assumption of stability, we consider minimizing the long-run average cost. In the case of two-servers the optimal control policy is shown to prefer to store customers in the lowest cost queue. When the inter-arrival and service times are assumed to be exponential, we use a Markov decision process formulation to show that for a fixed number of customers in the system, there exists a level S such that whenever customers are moved from the high cost queue to the low cost queue, the number of customers moved brings the number of customers in the low cost queue to S. These results lead to the development of a heuristic for the model with more than two servers.

COSHH: A classification and optimization based scheduler for heterogeneous Hadoop systems

A Hadoop system provides execution and multiplexing of many tasks in a common datacenter. There is a rising demand for sharing Hadoop clusters amongst various users, which leads to increasing system heterogeneity. However, heterogeneity is a neglected issue in most Hadoop schedulers. In this work we design and implement a new Hadoop scheduling system, named COSHH, which considers heterogeneity at both the application and cluster levels. The main objective of COSHH is to improve the mean completion time of jobs. However, as it is concerned with other key Hadoop performance metrics, our proposed scheduler also achieves competitive performance under minimum share satisfaction, fairness and locality metrics with respect to other well-known Hadoop schedulers.

Compensating for Failures with Flexible Servers

We consider the problem of maximizing capacity in a queueing network with flexible servers, where the classes and servers are subject to failure. We assume that the interarrival and service times are independent and identically distributed, that routing is probabilistic, and that the failure state of the system can be described by a Markov process that is independent of the other system dynamics. We find that the maximal capacity is tightly bounded by the solution of a linear programming problem and that the solution of this problem can be used to construct timed, generalized round-robin policies that approach the maximal capacity arbitrarily closely. We then give a series of structural results for our policies, including identifying when server flexibility can completely compensate for failures and when the implementation of our policies can be simplified. We conclude with a numerical example that illustrates some of the developed insights.

On Optimal Policies for Energy-Aware Servers

As energy costs and energy used by server farms increase, so does the desire to implement energy-aware policies. Although under some metrics, optimal policies for single as well as multiple server systems are known, a number of metrics remain without sufficient knowledge of corresponding optimal policies. We describe and analyse a model to determine an optimal policy for on/off single server systems under a broad range of metrics that are based on expected response time, energy usage, and switching costs. We leverage this model in the determination of routing probabilities to show a range of nontrivial optimal routing probabilities and server configurations when energy concerns are a factor.

A state-dependent polling model with k -limited service

We consider a two-queue model with state-dependent setups, in which a single server alternately serves the two queues. The high-priority queue is served exhaustively, whereas the low-priority queue is served according to the k-limited strategy. A setup at a queue is incurred only if there are customers waiting at the polled queue. We obtain the transforms of the queue length and sojourn time distributions under the assumption of Poisson arrivals, generally distributed service times, and generally distributed setup times. The interest for this model is fueled by an application in the field of logistics. It is shown how the results of this analysis can be applied in the evaluation of a stochastic two-item single-capacity production system. From these results we can conclude that significant cost reductions are possible by bounding the production runs of the low-priority item, which indicates the potential of the k-limited service discipline as priority rule in production environments.