# International Journal of Emerging Technologies in Computational and Applied Sciences (IJETCAS)

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2 Queuing model [5, 11] is used to approximate a real queuing situation or system, so the queuing behaviour can be analyzed mathematically. Queuing models allow a number of useful steady state performance measures to be determined. Concept of Queuing theory concept comes from Kendall s notation. Kendall s notation [11] is a standard notation for classifying queuing systems into different types. Kendall s notation mainly described by the notation A/B/C/D/E. A- Distribution of inter arrival times of customers; B- Distribution of service times; C- Number of servers; D- Maximum total number of customers which can be accommodated in system, i.e. system capacity; E- Queuing discipline. Let s take an example where- M/M/m/K/N- this would describe a queuing system with an exponential distribution for the inter arrival times of customers and the service times of customers, m servers, a maximum of K customers in the queuing system at once, and potential customers in the calling population. There are a lot of models available like M/M/1, M/M/2, etc. however we describe our model in M/M/2 structure in this paper to avoid calculation overhead. An M/M/2 queue [5] represents the queue length in a system having two servers, where arrivals are determined by a Poisson process and job service times have a exponential distribution rate. An M/M/2 queue is a stochastic process whose state space is the set {0, 1, 2, 3...} where the value corresponds to the number of customers in the system, including any currently in service. Mathematical formula of M/M/2 queuing model is shown in figure 1 and figure 2. Figure1: Diagram of M/M/1 Queuing Model Figure2: Mathematical formulation of M/M/1 Queuing Model For easy description as well as easy calculation we proposed our CSP model based on M/M/2 queuing model diagrammatically which is shown in figure 3. Figure 3: Smart Cloud Environment In the above CSP model n number of cloud user send request to the cloud service provider (CSP). CSP stored the request at the RQ and then it directly communicates with SJS inside CSP. SJS then apply any one Task Assignment Approach algorithms (STTA, OTTA) [13] and tries to assign each of these tasks or jobs to the resources as per the request comes from the CU. Now we briefly describe about the Task Assignment Approach (TAA) [13]. In this approach a process is consider to compose of multiple tasks and goal is to find optimal assignment policy for the tasks of individual process. Some preliminary assumptions we must consider in task assignment work as follows- IJETCAS ; 2013, IJETCAS All Rights Reserved Page 239

5 T2 T3 T4 T5 T6 Table4 shows a optimal assignment of the task to the two resources using the procedure of network flow graph and network flow algorithm. So first five tasks T1 to T5 must be assigned to resource and task T6 only assign to resource R6. This assignment is aimed to optimize the total execution costs. But here also we must need to consider the execution cost as well as inter-process communication cost which must be shown below- Optimal Assignment Execution cost, x = x 11 + x 21 + x 31 + x 41 + x 51 + x 62 = = 26 Optimal Assignment Communication cost, c = c 16 + c 26 + c 36 + c 46 + c 56 = = 12 Hence, Optimal Assignment Total cost = 38 So in case of OTAA overall assignment cost must be reduced to 38; which is too much smaller compare to STAA where total serial assignment costs are 58. Key benefits of Task Assignment Approach:- a. Minimizing Inter-process communication cost. b. Quick turnaround time of complete process. c. A high degree of parallelism. d. Efficient utilization of system resources. IV. Conclusion and Future Work Last two sections we have described two different types of Task Assignment Approach Models. Smart Job Scheduler is mainly responsible here to assign the tasks to the pertinent resources. If there are m number of tasks and p number of resources then there are mp numbers of possibility to assign the task to the resources. To remove this problem we mainly introduce here the Task Assignment Approach algorithms and remove the problem of task assignment to the resources. In the above section we also describe the Cloud Service Provider model where we applied the TAA algorithms in Smart Job Scheduler which is inside CSP and recover the problem of assignment of the requested tasks to the available resources within minimum execution time as well as minimizing inter-process communication time between the tasks. In future we will try to derive other Task Assignment Algorithms to reduce the overall assignment costs of CSP in Cloud environment. We also try to reduce the complexity of CSP model when the number of resources must be maximum. References [1] Service Performance and Analysis in Cloud Computing by Kaiqi Xiong, Harry Perros /09 \$ IEEE page [2] Virtual Infrastructure Management in Private and Hybrid Clouds by Borja Sotomayor, Rubén S. Montero and Ignacio M. Llorente, Ian Foster /09/\$ IEEE [3] Research on Distributed Architecture Based on SOA by Hongqi Li, Zhuang Wu /09 \$ IEEE [4] A Berkeley View of Cloud computing. M. Armbrust, A. Fox, R. Griffith, A. Joseph, R. Katz, A. Konwinski, G. Lee, D. Patterson, A. Rabkin, I. Stoica, M. Zaharia. Above the Clouds: Technical Report No. UCB/EECS , University of California at Berkley, USA, Feb. 10, 2009 [5] Hock, N.C., Queueing Modelling Fundamentals. JOHN WILEY&SONS, 1997 [6] An Approach to a Cloud Computing Network by Francesco Maria Aymerich, Gianni Fenu1, Simone Surcis /08/\$ IEEE 113 page [7] Cloud Computing and Services Platform Construction of Telecom Operator by Xu Lei, Xin Zhe, Ma Shaowu, Tang Xiongyan. Broadband Network & Multimedia Technology, IC-BNMT '09. 2nd IEEE International Conference on Digital Object Identifier, pp [8] Service Performance and Analysis in Cloud Computing, Kaiqi Xiong and Harry Perros 2009 Congress on Services I [9] An Optimistic Differentiated Service Job Scheduling System for Cloud Computing Service Users and Providers by Luqun Li 2009 Third International Conference on Multimedia and Ubiquitous Engineering page [10] Probability, Markov chains, queues, and simulation: the mathematical basis of performance modeling. Stewart, William J. (2009) Princeton University Press. p ISBN IJETCAS ; 2013, IJETCAS All Rights Reserved Page 242

6 [11] "Stochastic Processes Occurring in the Theory of Queues and their Analysis by the Method of the Imbedded Markov Chain". Kendall, David G. (September 1953). Annals of Mathematical Statistics 24 (3): doi: /aoms/ JSTOR [12] A Comparative Study into Distributed Load Balancing Algorithms for Cloud Computing by Martin Randles, David Lamb, A. Taleb- Bendiab 2010 IEEE 24th International Conference on Advanced Information Networking and Applications Workshops page [13] Distributed Operating System Concept and Design - Pradeep K. Sinha; PHI publication. Acknowledgements We want to thank all of our co-workers for their endless cooperation and are also grateful to our family for their support during the whole period of our work. We also want to thank all of our colleagues and to all who supported us in this regard. Finally, we sincerely thank to the reviewers and editors for their through and constructive suggestions to give proper shape to the manuscript. IJETCAS ; 2013, IJETCAS All Rights Reserved Page 243

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