# A computational model for MapReduce job flow

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2 In this paper we propose a formal approach to model the MapReduce framework using model checking and tempora l logics to verify some relevant properties as reliability, load balancing, lack of deadlock of the MapReduce job flow. To the best of our knowledge only two works have combined MapReduce and model checking with a different aim from ours: in [2] MapReduce is adopted to compute distributed CTL algorithms and in [6] MapReduce is modeled using CSP formalism. The remaining of the paper is organized as follows. In Section 2 we recall basics of model checking and temporal logics, the formalism used to define and simulate our model. Section 3 provides a brief overview of the main features of MapReduce. Section 4 proposes our formal model of job flow in Mapreduce computation while Section 5 proposes an analytical model in the Uppaal model checker language with properties to be checked. Conclusion and future works are drawn in the last section. 2 Model Checking and Temporal Logics The logical language we use for the model checking task is the Computation Tree Logic (CTL), a propositional, branching, temporal logic [5]. The syntax of the formulae can be defined, using Backus-Naur form, as follows (where p is an atomic proposition): φ, ψ ::= p φ ψ φ ψ φ φ ψ φ ψ EF φ EXφ EGφ E(φUψ) AGφ AF φ AXφ A(φUψ). An atomic proposition is the formula true or a ground atom CTL formulae can also contain path quantifiers followed by temporal operators. The path quantifier E specifies some path from the current state, while the path quantifier A specifies all paths from the current state. The temporal operators are X, the next-state operator; U, the Until operator; G, the Globally operator; and F the Future operator. The symbols X, U, G, F cannot occur without being preceded by the quantifiers E and A. The semantics of the language is defined through a Kripke structure as the triple (S,, L) where S is a collection of states, is a binary relation on S S, stating that the system can move from state to state. Associated with each state s, the interpretation function L provides the set of atomic propositions L(s) that are true at that particular state [3]. The semantics of boolean connectives is as usual. The semantics for temporal connectives is as follows: Xφ specifies that φ holds in the next state along the path. φuψ specifies that φ holds on every state along the path until ψ is true. Gφ specifies that φ holds on every state along the path. F φ specifies that there is at least one state along the path in which φ is true. The semantics of formulae is defined as follows: EXφ: φ holds in some next state; EF φ: a path exists such that φ holds in some Future state ; EGφ: a path exists such that φ holds Globally along the path; E(φUψ): a path exists such that φ Untilψ holds on it; AXφ: φ holds in every next state; AF φ: for All paths there will be some Future state where φ holds; AGφ: for All paths the property φ holds Globally; A(φUψ): All paths satisfy φ Until ψ. The model checking problem is the following: Given a model M, an initial state s and a CTL formula φ, check whether M, s = φ. M = φ ehen the formula must be checked for every state of M.

3 3 MapReduce Overview and proposed model MapReduce is a software framework for solving large-scale computing problems over large data-sets and data-intensive computing. It has grown to be the programming model for current distributed systems, i.e. cloud computing. It also forms the basis of the data-center software stack [4]. MapReduce framework was developed at Google Research as a parallel programming model with an associated implementation. The framework is highly scalable and location independent. It is used for the generation of data for Google s production web search service, for sorting, for data-intensive applications, for optimizing parallel jobs performance in data-intensive clusters. The most relevant feature of MapReduce processing is that computation runs on a large cluster of commodity machines [1]; while the main feature with respect to other existing parallel computational models is the sequential and parallel computation interleaving. The MapReduce model is made up of the Map and Reduce functions, which are borrowed from functional languages such as Lisp [1]. Users computations are written as Map and Reduce functions. The Map function processes a key/value pair to generate a set of intermediate key/value pairs. The Reduce function merges all intermediate values associated with the same intermediate key. Intermediate functions of Shuffle and Sorting are useful to split and sort the data chunks to be given in input to the Reduce function. We now define the computational model for MapReduce framework. We model jobs and tasks in MapReduce using the flow description as shown in Figure 1. Definition 1 (MapReduce Graph (MRG)). A MapReduce Graph (MRG) is a Direct Acyclic Graph G = {N, E}, where nodes N in the computation graph are the tasks of computation N = M S SR R (M = map, S = shuffle, SR = sort, R = reduce) and edges e E are such that: 1. E (M S) (S SR) (SR R), i.e. edges connect map with shuffle, shuffle with sort and sort with reduce tasks ; 2. e M S breaks input into tokens; e S SR sorts input tokens by type; e SR R gives sort tokens to reducer. 3. Reduce sends input data for cluster allocation to the file system Definition 2 (MapReduce Task). A MapReduce task t is a token of computation such that t (M S SR R). Definition 3 (MapReduce Job). A MapReduce Job is the sequence t 1 t 2... t n of MapReduce tasks where t 1 j = (M i,.., M i+p ) t 2 j = S k t 3 j = SR k t 4 j = R k with M i M, i = 1... n, S j S, SR j SR, R j R, j = 1... m.

5 Fig. 2. Uppaal simulation model M i. In the simulation model, mapper c ount is the counter of the total number of mappers and mapper s ched is the variable that counts the scheduled mappers. The assigned value, m is the number of mappers. EF (M.Out of order) AG(mapper count = m mapper sched = m) From the simulation results we checked that the model ensures load balancing and fault tolerance. 5 Conclusion and future work We proposed a formal model of MapReduce framework for data-intensive and computing-intensive environment. The model introduces the definition of MapReduce graph and MapReduce job and task. At this stage of the work we implemented and simulated the model with Uppaal model checker to verify basics properties of its computation as fault tolerance, load balancing and lack of deadlock. We are currently modeling other relevant features as scalability, data locality and extending the model with advanced job management activities such as job folding and job chaining. We acknowledge support of project A Knowledge based Holistic Integrated Research Approach (KHIRA - PON ).

6 References 1. Jeffrey Dean and Sanjay Ghemawat. Mapreduce: Simplified data processing on large clusters, OSDI04: Sixth symposium on operating system design and implementation, san francisco, ca, december, S. Dill, R. Kumar, K. McCurley, S. Rajagopalan, D. Sivakumar, ad A. Tomkins, Self-similarity in the Web, Proc VLDB, Feng Guo, Guang Wei, Mengmeng Deng, and Wanlin Shi. Ctl model checking algorithm using mapreduce. In Emerging Technologies for Information Systems, Computing, and Management, pages Springer, M.R.A. Huth and M.D. Ryan. Logic in Computer Science. Cambridge University Press, Krishna Kant. Data center evolution: A tutorial on state of the art, issues, and challenges. Computer Networks, 53(17): , Edmund M.Clarke, Orna Grumberg, and Doron A. Peled. Model Checking. Cambridge, Massachusetts, USA: MIT press., Fan Yang, Wen Su, Huibiao Zhu, and Qin Li. Formalizing mapreduce with csp. In Engineering of Computer Based Systems (ECBS), th IEEE International Conference and Workshops on, pages IEEE, 2010.

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