1. RAIDs Performance Prediction based on Fuzzy Queue Theory Carlos Campos Bracho ECE 510 Project Prof. Dr. Duncan Elliot

2. Outline Goal Introduction. RAIDs Performance Analysis based on Queue Theory (QT). RAIDs Performance Analysis based on Fuzzy Queue Theory (FQT). Performance Evaluation of different RAIDs using FQT ( Pending). Conclusion.

3. Goal To explore how Fuzzy Queing Theory can be used to predict the performance of RAIDs disks. To investigate how both RAIDs level and the workload influence service time, disk utilization and response time.

4. Introduction Roughly speaking, in the study of processor as well as memory performance, we deal basically with two fundamental techniques: measurement and simulation. With I/O systems, given the probabilistic nature of I/O events, the study of I/O performance is basically carry out by an analytical model known as Queuing Theory.

5. Introduction Within the context of Queuing Theory, there are some parameters (such as the interarrival and services times), which are required to have a certain probability distributions. However, when a real queuing system (i.e. RAIDs disks) is in the planning stage, these parameters are described most of the times by linguistic terms ( such as fast, moderate or slow) or are estimated by experts.

6. RAIDs Performance Analysis based on Queue Theory (QT). Benefits: Using a (Fuzzy) Queue model will allow us to predict important RAIDs performance parameters such as the average time a job spends in the RAIDs queue, the average number of jobs in the RAIDs queue, the probability of having a given number of jobs in the RAIDs queue, among others.

7. RAIDs Performance Analysis based on Queue Theory (QT). Basic assumptions:  A RAID disk is modeled as a single-open queue system.  There is a Job-flow balance in the RAID disk.  One-step RAID disk’s behavior.  The average job-arrival rate and the average rate are independent of the RAID disk’s state.

8. RAIDs Performance Analysis based on Queue Theory (QT). Notation:  s : RAIDs Average time required to service a job.   = 1/s : RAIDs average service rate time.  : average arrival rate.   = /  : Traffic intensity at RAIDs.  r : RAIDs Average response time.  w : Average time a job spends waiting at RAIDs.  q : Average number of jobs at RAIDs.  n : Number of jobs at RAIDs.  U : Utilization of RAIDs.  umber of jobs at RAIDs.  a : The number of arrivals that occur within the fixed observation interval T.  d : Number of departures during observation interval T.

9. RAIDs Performance Analysis based on Queue Theory (QT). More notation ( w.r.t stochastic queue model) :  It is completely specified with six parameters ( Kendall notation), which are ( A/ S/ c/ B/ N/ D): A: Arrival Process: This stochastic process describes when jobs arrive at the queue. What is actually more useful, though, is knowing the times between arrivals of jobs (interarrival times). S: Service Process: This stochastic process describes the distribution of times required to service a job when it leaves the queue and enters one of the servers. c: Number of servers ( disks in our case) B: Refer to the total number of jobs that can be in the system, including both those in the queue and those being served. Most real systems have finite maximum queue sizes due to their having limited amounts of buffer memory. N: Refer to the total number of jobs that can enter in the system D: It specifies the order in which jobs are removed from the queue and passed to a server.

10. RAID 0,1,4 Modelling A RAID 0 containing N disks will be modeled as N separate M/M/1 (Fuzzy) queues. A RAID 1, considering a read-only workload will be modeled as an M/M/#mirrors (Fuzzy) queues. In addition, a write-only workload will be modeled as an M/M/1 (Fuzzy) queue. A RAID 4 containing N disks, considering a read- only workload will be modeled as M/M/ N-1 (Fuzzy) queues. In addition, a write-only workload will be modeled as an M/M/1 (Fuzzy) queue.

11. Basic RAIDs Model: A single-queue with N disks (servers) model. A single-queue model of a system consists of one or more servers (i.e. RAIDs disks) that process jobs entering the system. A single queue temporarily stores (buffers) jobs that must wait to be processed while jobs that arrived earlier are being processed.

12. Basic RAIDs Model: single-server (disk) (M/ M/ 1) system.. The state of a system described by a birth-death process is the number of jobs in the system, n. A birth occurs when a job enters the system and increases n by 1. The departure of a job from the system is called a death, and causes n to decrease by 1.

13. Basic RAIDs Model: single-server (M/ M/ 1) system.. A discrete-state process which follow the last assumptions is known as Markov Process.

14. Basic RAIDs Model: single-server (M/ M/ 1) Fuzzy system.. For the case, in which the interarrival process as well as the service time is unknown, a Fuzzy Markov Process will be used. Under this context, the transitions probabilities will be considered as intervals not just a single real numbers.

15. Conclusions Queuing Theory provide us with a framework for predicting the RAIDs performance under the assumption mentioned before. However, in many real-applications, these assumptions are not completely satisfactory.

16. Conclusions Fuzzy Queuing Theory provide us with a more realistic model for predicting the RAIDs performance under conditions of uncertainty ( in this case for not knowing in advance the behavior of the interarrival time as well as the service time), and without the assumption of poisson distribution.

17. References Ching Kao. Parametric programming to the analysis of fuzzy queues. Fuzzy Sets & Systems,1999. J. Buckley. Fuzzy Probability and Statistics. Springer 2006. H&P. Computer Architecture.Morgan & Kauffman. 2003.

18. Questions ????