The MDS Queue: Analysing the Latency Performance of Erasure Codes

In order to scale economically, data centers are increasingly evolving their data storage methods from the use of simple data replication to the use of more powerful erasure codes, which provide the same level of reliability as replication but at a significantly lower storage cost. In particular, it is well known that Maximum-Distance-Separable (MDS) codes, such as Reed-Solomon codes, provide the maximum storage efficiency. While the use of codes for providing improved reliability in archival storage systems, where the data is less frequently accessed (or so-called "cold data"), is well understood, the role of codes in the storage of more frequently accessed and active "hot data", where latency is the key metric, is less clear. In this paper, we study data storage systems based on MDS codes through the lens of queueing theory, and term this the "MDS queue." We analytically characterize the (average) latency performance of MDS queues, for which we present insightful scheduling policies that form upper and lower bounds to performance, and are observed to be quite tight. Extensive simulations are also provided and used to validate our theoretical analysis. We also employ the framework of the MDS queue to analyse different methods of performing so-called degraded reads (reading of partial data) in distributed data storage

Mixed Polling with Rerouting and Applications

Queueing systems with a single server in which customers wait to be served at a finite number of distinct locations (buffers/queues) are called discrete polling systems. Polling systems in which arrivals of users occur anywhere in a continuum are called continuous polling systems. Often one encounters a combination of the two systems: the users can either arrive in a continuum or wait in a finite set (i.e. wait at a finite number of queues). We call these systems mixed polling systems. Also, in some applications, customers are rerouted to a new location (for another service) after their service is completed. In this work, we study mixed polling systems with rerouting. We obtain their steady state performance by discretization using the known pseudo conservation laws of discrete polling systems. Their stationary expected workload is obtained as a limit of the stationary expected workload of a discrete system. The main tools for our analysis are: a) the fixed point analysis of infinite dimensional operators and; b) the convergence of Riemann sums to an integral. We analyze two applications using our results on mixed polling systems and discuss the optimal system design. We consider a local area network, in which a moving ferry facilitates communication (data transfer) using a wireless link. We also consider a distributed waste collection system and derive the optimal collection point. In both examples, the service requests can arrive anywhere in a subset of the two dimensional plane. Namely, some users arrive in a continuous set while others wait for their service in a finite set. The only polling systems that can model these applications are mixed systems with rerouting as introduced in this manuscript.Comment: to appear in Performance Evaluatio

Super-Exponential Solution in Markovian Supermarket Models: Framework and Challenge

Marcel F. Neuts opened a key door in numerical computation of stochastic models by means of phase-type (PH) distributions and Markovian arrival processes (MAPs). To celebrate his 75th birthday, this paper reports a more general framework of Markovian supermarket models, including a system of differential equations for the fraction measure and a system of nonlinear equations for the fixed point. To understand this framework heuristically, this paper gives a detailed analysis for three important supermarket examples: M/G/1 type, GI/M/1 type and multiple choices, explains how to derive the system of differential equations by means of density-dependent jump Markov processes, and shows that the fixed point may be simply super-exponential through solving the system of nonlinear equations. Note that supermarket models are a class of complicated queueing systems and their analysis can not apply popular queueing theory, it is necessary in the study of supermarket models to summarize such a more general framework which enables us to focus on important research issues. On this line, this paper develops matrix-analytical methods of Markovian supermarket models. We hope this will be able to open a new avenue in performance evaluation of supermarket models by means of matrix-analytical methods.Comment: Randomized load balancing, supermarket model, matrix-analytic method, super-exponential solution, density-dependent jump Markov process, Batch Markovian Arrival Process (BMAP), phase-type (PH) distribution, fixed poin