Block-Structured Supermarket Models

Supermarket models are a class of parallel queueing networks with an adaptive control scheme that play a key role in the study of resource management of, such as, computer networks, manufacturing systems and transportation networks. When the arrival processes are non-Poisson and the service times are non-exponential, analysis of such a supermarket model is always limited, interesting, and challenging. This paper describes a supermarket model with non-Poisson inputs: Markovian Arrival Processes (MAPs) and with non-exponential service times: Phase-type (PH) distributions, and provides a generalized matrix-analytic method which is first combined with the operator semigroup and the mean-field limit. When discussing such a more general supermarket model, this paper makes some new results and advances as follows: (1) Providing a detailed probability analysis for setting up an infinite-dimensional system of differential vector equations satisfied by the expected fraction vector, where "the invariance of environment factors" is given as an important result. (2) Introducing the phase-type structure to the operator semigroup and to the mean-field limit, and a Lipschitz condition can be obtained by means of a unified matrix-differential algorithm. (3) The matrix-analytic method is used to compute the fixed point which leads to performance computation of this system. Finally, we use some numerical examples to illustrate how the performance measures of this supermarket model depend on the non-Poisson inputs and on the non-exponential service times. Thus the results of this paper give new highlight on understanding influence of non-Poisson inputs and of non-exponential service times on performance measures of more general supermarket models.Comment: 65 pages; 7 figure

A Matrix-Analytic Solution for Randomized Load Balancing Models with Phase-Type Service Times

In this paper, we provide a matrix-analytic solution for randomized load balancing models (also known as \emph{supermarket models}) with phase-type (PH) service times. Generalizing the service times to the phase-type distribution makes the analysis of the supermarket models more difficult and challenging than that of the exponential service time case which has been extensively discussed in the literature. We first describe the supermarket model as a system of differential vector equations, and provide a doubly exponential solution to the fixed point of the system of differential vector equations. Then we analyze the exponential convergence of the current location of the supermarket model to its fixed point. Finally, we present numerical examples to illustrate our approach and show its effectiveness in analyzing the randomized load balancing schemes with non-exponential service requirements.Comment: 24 page

Stochastic Modeling of Hybrid Cache Systems

In recent years, there is an increasing demand of big memory systems so to perform large scale data analytics. Since DRAM memories are expensive, some researchers are suggesting to use other memory systems such as non-volatile memory (NVM) technology to build large-memory computing systems. However, whether the NVM technology can be a viable alternative (either economically and technically) to DRAM remains an open question. To answer this question, it is important to consider how to design a memory system from a "system perspective", that is, incorporating different performance characteristics and price ratios from hybrid memory devices. This paper presents an analytical model of a "hybrid page cache system" so to understand the diverse design space and performance impact of a hybrid cache system. We consider (1) various architectural choices, (2) design strategies, and (3) configuration of different memory devices. Using this model, we provide guidelines on how to design hybrid page cache to reach a good trade-off between high system throughput (in I/O per sec or IOPS) and fast cache reactivity which is defined by the time to fill the cache. We also show how one can configure the DRAM capacity and NVM capacity under a fixed budget. We pick PCM as an example for NVM and conduct numerical analysis. Our analysis indicates that incorporating PCM in a page cache system significantly improves the system performance, and it also shows larger benefit to allocate more PCM in page cache in some cases. Besides, for the common setting of performance-price ratio of PCM, "flat architecture" offers as a better choice, but "layered architecture" outperforms if PCM write performance can be significantly improved in the future.Comment: 14 pages; mascots 201

On Modeling Clustering Indexes of BT-Like Systems

Abstract—In this paper, we explore the “clustering ” phenomenon in BT-like systems. A high clustering implies peers have a high tendency to exchange information with peers of the similar bandwidth type. We first show the clustering does exist in BT-like systems. Although high clustering is desirable for file sharing application, it may not be appropriate for multimedia streaming applications. We provide analytical models to calculate the clustering index and illustrate how one can control the clustering index for different P2P applications. I

Stochastic Analysis on RAID Reliability for Solid-State Drives

Solid-state drives (SSDs) have been widely deployed in desktops and data centers. However, SSDs suffer from bit errors, and the bit error rate is time dependent since it increases as an SSD wears down. Traditional storage systems mainly use parity-based RAID to provide reliability guarantees by striping redundancy across multiple devices, but the effectiveness of RAID in SSDs remains debatable as parity updates aggravate the wearing and bit error rates of SSDs. In particular, an open problem is that how different parity distributions over multiple devices, such as the even distribution suggested by conventional wisdom, or uneven distributions proposed in recent RAID schemes for SSDs, may influence the reliability of an SSD RAID array. To address this fundamental problem, we propose the first analytical model to quantify the reliability dynamics of an SSD RAID array. Specifically, we develop a "non-homogeneous" continuous time Markov chain model, and derive the transient reliability solution. We validate our model via trace-driven simulations and conduct numerical analysis to provide insights into the reliability dynamics of SSD RAID arrays under different parity distributions and subject to different bit error rates and array configurations. Designers can use our model to decide the appropriate parity distribution based on their reliability requirements.Comment: 12 page