Given a family @math of graphs and a positive integer @math , a graph @math is called vertex @math -fault-tolerant with respect to @math , denoted by @math -FT @math , if @math contains some @math as a subgraph, for every @math with @math . Vertex-fault-tolerance has been introduced by Hayes [A graph model for fault-tolerant computing systems, IEEE Transactions on Computers, C-25 (1976), pp. 875-884.], and has been studied in view of potential applications in the design of interconnection networks operating correctly in the presence of faults. We define the Fault-Tolerant Complete Matching (FTCM) Problem in bipartite graphs of order @math : to design a bipartite @math , with @math , @math , @math , that has a FTCM, and the tuple @math , where @math and @math are the maximum degree in @math and @math , respectively, is lexicographically minimum. @math has a FTCM if deleting at most @math vertices from @math creates @math that has a complete matching, i.e., a matching of size @math . We show that if @math is integer, solutions of the FTCM Problem can be found among @math -regular bipartite graphs of order @math , with @math , and @math . If @math then all @math -regular bipartite graphs of order @math have a FTCM, and for @math , it is not the case. We characterize the values of @math , @math , @math , and @math that admit an @math -regular bipartite graph of order @math , with @math , and give a simple construction that creates such a graph with a FTCM whenever possible. Our techniques are based on Hall's marriage theorem, elementary number theory, linear Diophantine equations, properties of integer functions and congruences, and equations involving them.
Nevertheless, the relation of our model with the process flexibility literature is still remote, since the process flexibility community has so far focused on systems where a server can process different kinds of compatible requests within the same time period. Moreover, only recently unbalanced systems, i.e., with @math , have started to be considered (see, @cite_21 and @cite_4 for examples).
@cite_21 @cite_4
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In round-by-round models of distributed computing processes run in a sequence of (synchronous or asynchronous) rounds. The advantage of the round-by-round approach is that invariants established in the first round are preserved in later rounds. An elegant asynchronous round-by-round shared memory model, is the iterated snapshots model (IS). Instead of the snapshots model where processes share an array m[ċ] that can be accessed any number of times, indexed by process ID, where Pi writes to m[i] and can take a snapshot of the entire array, we have processes share a two-dimensional array m[ċ, ċ], indexed by iteration number and by process ID, where Pi in iteration r writes once to m[r, i] and takes one snapshot of row r, m[r, ċ]. The IS model lends itself more easily to combinatorial analysis. However, to show that whenever a task is impossible in the IS model the task is impossible in the snapshots model, a simulation is needed. Such a simulation was presented by Borowsky and Gafni in PODC97; namely, it was shown how to take a wait-free protocol for the snapshots model, and transform it into a protocol for the IS model, solving the same task. In this paper we present a new simulation from the snapshots model to the IS model, and show that it can be extended to work with models stronger that wait-free. The main contribution is to show that the simulation can work with models that have access to certain communication objects, called 01-tasks. This extends the result of Gafni, Rajsbaum and Herlihy in DISC'2006 stating that renaming is strictly weaker than set agreement from the IS model to the usual non-iterated wait-free read write shared memory model. We also show that our simulation works with t-resilient models and the more general dependent process failure model of Junqueira and Marzullo. This version of the simulation extends previous results by Herlihy and Rajsbaum in PODC'2010 and DISC'2010 about the topological connectivity of a protocol complex in an iterated dependent process failure model, to the corresponding non-iterated model. One of the most effective ways of minimizing supply demand mismatch costs, with little increase in operational costs, is to deploy valuable resources in a flexible and timely manner to meet the realized demand. This notion of flexible processes has significantly changed operations in many manufacturing and service companies. For example, a flexible production system is now commonly used by automobile manufacturers, and a workforce cross-training system is now a common practice in many service industries. However, there is a trade-off between the level of flexibility available in the system and the associated complexity and operating costs. The challenge is to have the “right” level of flexibility to capture the bulk of the benefits from a fully flexible system, while controlling the increase in implementation costs. This paper reviews developments in process flexibility over the past decade. In particular, we focus on the phenomenon, often observed in practice, that a slight increase in process flexibility can lead to a significant improvement in system performance. This review explores the issues from three perspectives: design, evaluation, and applications. We also discuss how the concept of process flexibility has been deployed in several manufacturing and service systems. Recent computer systems research has proposed using redundant requests to reduce latency. The idea is to replicate a request so that it joins the queue at multiple servers. The request is considered complete as soon as any one of its copies completes. Redundancy allows us to overcome server-side variability–the fact that a server might be temporarily slow due to factors such as background load, network interrupts, and garbage collection to reduce response time. In the past few years, queueing theorists have begun to study redundancy, first via approximations, and, more recently, via exact analysis. Unfortunately, for analytical tractability, most existing theoretical analysis has assumed an Independent Runtimes (IR) model, wherein the replicas of a job each experience independent runtimes (service times) at different servers. The IR model is unrealistic and has led to theoretical results that can be at odds with computer systems implementation results. This paper introduces a much more realistic model of redundancy. Our model decouples the inherent job size ( @math ) from the server-side slowdown ( @math ), where we track both @math and @math for each job. Analysis within the @math model is, of course, much more difficult. Nevertheless, we design a dispatching policy, Redundant-to-Idle-Queue, which is both analytically tractable within the @math model and has provably excellent performance. Abstract In this paper, we present a unifying analysis for redundancy systems with cancel-on-start ( c . o . s . ) and cancel-on-complete ( c . o . c . ) with exponentially distributed service requirements. With c . o . s . ( c . o . c . ) all redundant copies are removed as soon as one of the copies starts (completes) service. As a consequence, c . o . s . does not waste any computing resources, as opposed to c . o . c . We show that the c . o . s . model is equivalent to a queueing system with multi-type jobs and servers, which was analyzed in , (2012), and show that c . o . c . (under the assumption of i.i.d. copies) can be analyzed by a generalization of , (2012) where state-dependent departure rates are permitted. This allows us to show that the stationary distribution for both the c . o . c . and c . o . s . models has a product form. We give a detailed first-time analysis for c . o . s and derive a closed form expression for important metrics like mean number of jobs in the system, and probability of waiting. We also note that the c . o . s . model is equivalent to Join-Shortest-Work queue with power of d (JSW( d )). In the latter, an incoming job is dispatched to the server with smallest workload among d randomly chosen ones. Thus, all our results apply mutatis-mutandis to JSW( d ). Comparing the performance of c . o . s . with that of c . o . c . with i.i.d. copies gives the unexpected conclusion (since c . o . s . does not waste any resources) that c . o . s . is worse in terms of mean number of jobs. As part of ancillary results, we illustrate that this is primarily due to the assumption of i.i.d. copies in case of c . o . c . (together with exponentially distributed requirements) and that such assumptions might lead to conclusions that are qualitatively different from that observed in practice.