MapReduce framework is widely used to parallelize batch jobs since it exploits a high degree of multi-tasking to process them. However, it has been observed that when the number of mappers increases, the map phase can take much longer than expected. This paper analytically shows that stochastic behavior of mapper nodes has a negative effect on the completion time of a MapReduce job, and continuously increasing the number of mappers without accurate scheduling can degrade the overall performance. We analytically capture the effects of stragglers (delayed mappers) on the performance. Based on an observed delayed exponential distribution (DED) of the response time of mappers, we then model the map phase by means of hardware, system, and application parameters. Mean sojourn time (MST), the time needed to sync the completed map tasks at one reducer, is mathematically formulated. Following that, we optimize MST by finding the task inter-arrival time to each mapper node. The optimal mapping problem leads to an equilibrium property investigated for different types of inter-arrival and service time distributions in a heterogeneous datacenter (i.e., a datacenter with different types of nodes). Our experimental results show the performance and important parameters of the different types of schedulers targeting MapReduce applications. We also show that, in the case of mixed deterministic and stochastic schedulers, there is an optimal scheduler that can always achieve the lowest MST.
So far, various components of image characteristics have been used for steganalysis, including the histogram characteristic function, adjacent colors distribution, and sample pair analysis. However, some certain steganography methods have been proposed that can thwart some analysis approaches through managing the embedding patterns. In this regard, the present paper is intended to introduce a new analytical method for detecting stego images, which is robust against some of the embedding patterns designed specifically to foil steganalysis attempts. The proposed approach is based on the analysis of the eigenvalues of the cover correlation matrix used for the purpose of the study. Image cloud partitioning, vertical correlation function computation, constellation of the correlated data, and eigenvalues examination are the major challenging stages of this analysis method. The proposed method uses the LSB plane of images in spatial domain, extendable to transform domain, to detect low embedding rates-a major concern in the area of the LSB steganography. The simulation results based on deviation detection and rate estimation methods indicated that the proposed approach outperforms some well-known LSB steganalysis methods, specifically at low embedding rates.