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Queueing models with multiple waiting lines
 Queueing Systems
, 2001
"... This paper discusses analytic solution methods for queueing models with multiple waiting lines. The methods are briefly illustrated, using key models like the 2 × 2 switch, the shortest queue and the cyclic polling system. AMS subject classification: 60K25, 90B22. ..."
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This paper discusses analytic solution methods for queueing models with multiple waiting lines. The methods are briefly illustrated, using key models like the 2 × 2 switch, the shortest queue and the cyclic polling system. AMS subject classification: 60K25, 90B22.
On the use of the power series algorithm for general Markov processes, with an application to a Petri net
 INFORMS J. on Computing 9
, 1997
"... The power series algorithm has been developed as a numerical procedure for solving queueing models. This paper shows that it can be used for each Markov process with a single recurrent class. This applies in particular to finite state processes, which is illustrated with the analysis of a bounded st ..."
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The power series algorithm has been developed as a numerical procedure for solving queueing models. This paper shows that it can be used for each Markov process with a single recurrent class. This applies in particular to finite state processes, which is illustrated with the analysis of a bounded stochastic Petri net model. Analytically obtaining performance measures of multidimensional queueing systems is often very difficult. Explicit solutions are only available for some very special models, like Jackson networks. Some specific twodimensional models can also be solved analytically, for example by showing that solving the problem is equivalent to solving a wellstudied complex analysis problem. See Boxma et al. [6] for an overview. The drawbacks of the analytical methods can be summarized as follows: the resulting problems are nontrivial to solve, we are confined to two dimensions, and small changes in the model usually lead to analytically intractable models. On the other hand, simply numerically solving the steady state equations usually does not work well either. Often the state space is countable, giving need to truncation of the
Queueing with Redundant Requests: First Exact Analysis
, 2015
"... Recent computer systems research has proposed using redundant requests to reduce latency. The idea is to run a single request on multiple servers and only wait for the first completion (discarding all remaining instances of the request). However no exact analysis of systems with redundancy exists. T ..."
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Recent computer systems research has proposed using redundant requests to reduce latency. The idea is to run a single request on multiple servers and only wait for the first completion (discarding all remaining instances of the request). However no exact analysis of systems with redundancy exists. This paper presents the first exact analysis of systems with redundancy. We allow for any number of classes of redundant requests, any number of classes of nonredundant requests, any degree of redundancy, and any number of heterogeneous servers. In all cases we derive the limiting distribution on the state of the system. In small (two or three server) systems, we derive simple forms for the distribution of response time of both the redundant classes and nonredundant classes, and we quantify the “gain ” to redundant classes and “pain ” to nonredundant classes caused by redundancy. We find some surprising results. First, in many cases the response time of the redundant class follows a simple Exponential distribution and that of the nonredundant class follows a Generalized Hyperexponential. Second, once a class is fully redundant, it becomes “immune ” to any pain caused by other classes becoming redundant. We also compare redundancy with other approaches for reducing latency, such as optimal probabilistic splitting of a class among servers (OptSplit) and JointheShortestQueue (JSQ) routing of a class. We find that redundancy outperforms JSQ and OptSplit with respect to overall response time, making it an attractive solution. 1
Computable Bounds in ForkJoin Queueing Systems
"... In a ForkJoin (FJ) queueing system an upstream fork station splits incoming jobs into N tasks to be further processed by N parallel servers, each with its own queue; the response time of one job is determined, at a downstream join station, by the maximum of the corresponding tasks ’ response times ..."
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In a ForkJoin (FJ) queueing system an upstream fork station splits incoming jobs into N tasks to be further processed by N parallel servers, each with its own queue; the response time of one job is determined, at a downstream join station, by the maximum of the corresponding tasks ’ response times. This queueing system is useful to the modelling of multiservice systems subject to synchronization constraints, such as MapReduce clusters or multipath routing. Despite their apparent simplicity, FJ systems are hard to analyze. This paper provides the first computable stochastic bounds on the waiting and response time distributions in FJ systems. We consider four practical scenarios by combining 1a) renewal and 1b) nonrenewal arrivals, and 2a) nonblocking and 2b) blocking servers. In the case of nonblocking servers we prove that delays scale as O(logN), a law which is known for first moments under renewal input only. In the case of blocking servers, we prove that the same factor of logN dictates the stability region of the system. Simulation results indicate that our bounds are tight, especially at high utilizations, in all four scenarios. A remarkable insight gained from our results is that, at moderate to high utilizations, multipath routing “makes sense ” from a queueing perspective for two paths only, i.e., response times drop the most when N = 2; the technical explanation is that the resequencing (delay) price starts to quickly dominate the tempting gain due to multipath transmissions.
International Conference on Computer Systems and Technologies CompSysTech’2005 Minimisation of the Average Response Time in a Cluster of Servers
"... Abstract: In this paper, we consider task assignment problem in a cluster of servers. We show that optimal static task assignment is tantamount to equalizing an appropriate cost functions associated with the servers. We also propose an improvement of dynamic Shortest Expected Delay (SED) task assign ..."
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Abstract: In this paper, we consider task assignment problem in a cluster of servers. We show that optimal static task assignment is tantamount to equalizing an appropriate cost functions associated with the servers. We also propose an improvement of dynamic Shortest Expected Delay (SED) task assignment policy.
STEADYSTATE SOLUTIONS OF MARKOV CHAINS
"... The paper is devoted on methods and algorithms for steadystate analysis of Markov chains. Basic, direct and iterative methods for steadystate analysis of Markov chains are concerned, where Gaussian Elimination method and Grassman method, as well as Power, Jacobis and GaussSeidels methods are impl ..."
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The paper is devoted on methods and algorithms for steadystate analysis of Markov chains. Basic, direct and iterative methods for steadystate analysis of Markov chains are concerned, where Gaussian Elimination method and Grassman method, as well as Power, Jacobis and GaussSeidels methods are implemented. Algorithms for computation of steadystate probability vector for finite Markov chains are developed. Comparison of numerical solutions to exact equilibrium solution for localbalance equation of DiscreteTime Markov Chain is given. Example and numerical results for feedback networks of Markovian queues are shown.
An Analytical Approach to the Performance Evaluation of MasterSlave Computational Models
, 1997
"... Many stochastic models and analysis techniques have been proposed in the literature during the last two decades for the performance evaluation of parallel and distributed systems. However few of them are directly applicable to practical systems which are generally too complex. In this paper we analy ..."
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Many stochastic models and analysis techniques have been proposed in the literature during the last two decades for the performance evaluation of parallel and distributed systems. However few of them are directly applicable to practical systems which are generally too complex. In this paper we analyze performances of the masterslave computational model, one of the most commonly used models in parallel and distributed computations. We propose a hybrid analytical approach by using techniques from the theories of both stochastic task graphs and queueing networks. We apply this method to the analysis of a computational chemistry application running on a Transputer based system. The proposed method turns out to be not only very efficient in time for large systems but also very accurate compared to measuring.