Note that c , c -FJ is the classical FJ system. Summarizing the important results from [ 16 , 17 , 18 ] ,. In [ 37 ] the authors study a multiserver FJ-system with Markovian arrival process MAP , phase type PH services, task vacations and intercommunication times modeled as PH random variables. An approximate solution based on stochastic decomposition is derived, and examples are presented comparing the approximation with simulation results under four dynamic scheduling policies for a few scenarios.

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We also note an interesting overview of FJ and related systems [ 39 ]. Specifically, in this survey paper, the author reviews the queueing systems related to FJ systems, the analysis of certain Markovian queues, methods to estimate the expected value of the maximum which play an important role in CC and DG models of different distributions, and approximations for the mean delay.

The MAP , introduced as a versatile Markovian point process by Neuts [ 28 ] and later simplified using less notations by Lucantioni, et. Note that the matrix D 0 governs transitions without arrivals and the matrix D 1 governs transitions inducing arrivals to the system. It should be pointed out that MAP is a rich class of point processes and includes many classical processes such as Poisson, PH-renewal processes, and Markov-modulated Poisson process.

For more details on MAP and their applications in stochastic modelling, we refer to [ 22 , 23 , 30 , 31 ]. Further, we refer the reader to [ 1 , 4 , 5 ] for a review and recent work on MAP. In simulations, however, we consider general service time distribution. In the sequel we need the following notations. Should there be a need to emphasize the dimension, we will do so like I m rather than I , and similarly for the others. For details and properties on Kronecker products and Kronecker sums we refer the reader to [ 11 , 24 , 38 ].

In this system, recall, an arriving job is split into c tasks and served by each of c servers until the earliest service completion of the task, when all other tasks of the job in service are removed from the system.

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The generator see e. The proof follows by applying the matrix-geometric results as seen in [ 29 ]. The necessary details may be found in [ 10 ]. It allows to evaluate the system performance for relatively large values of c. The results are displayed in Fig. Here, we consider a system in which an arriving job is split into c tasks and each task is sent to each of the c servers.

To obtain the two steady-state performance measures, we reuse a few notations from the FJ system. In this section we will look at the steady-state probability vector of the CTMC with generator given in 3. It is easy to verify see [ 29 ] the following theorem. In this case we split each job into c tasks only at beginning service epochs. The jobs sequentially enter into service in contrast to FJ system, where the jobs are dispatched to each of c servers immediately, and each server has its independent queue.

Thus, the service is offered according to FCFS basis. It is known see, e. However, the dimension of the PH-distribution of the k th order statistics grows exponentially with k and hence in this paper we will resort to simulation for the steady-state analysis of the model. We refer the reader to [ 36 ] for an extended discussion of these type of models. The purpose of this section is to discuss a few illustrative examples to bring out the qualitative nature of the models under study.

Towards this end, we consider five arrival processes and three service time distributions. The ratio of the standard deviation of the inter-arrival times of these five arrival processes with respect to ERLA are, respectively, 1, 1. The above MAP processes will be normalized so as to have a specific arrival rate.

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First, we want to point out that in [ 18 ] for the case of Poisson arrivals, it was shown that, as c increases,. We notice from Fig. Note that the results agree with observed in [ 18 ]. When considering Erlang services see Fig. Also, the crossing points for hyperexponential arrivals are less than the corresponding ones for Erlang arrivals. When looking hyperexponential services see Fig.

This is intuitively clear, since having a highly variable services such as hyperexponential will yield a smaller service time for FJ system compared to FEC system. While this has been observed in [ 18 ] for the case of Poisson arrivals and for log-concave and log-convex type services, we notice here the same phenomenon for other types of arrival processes.

To save space, we do not provide the figures here, since they do not provide additional insight. However, the state space of the CTMC governing the system under study grows exponentially with c , and hence we turn to simulation to study the systems for large values of c. To complete the experiments, we validated our simulation models with the analytical ones.

Towards this end, we consider the arrival processes and service time distributions defined in Section 3. We compare the analytical results for FJ and FEC systems with the simulated results, for the same set of parameter values. The simulation models were implemented using ARENA [ 19 ], a powerful software used in applied stochastic modeling and other areas.

We made five simulation runs with each run lasting 1,, units of model time. In Tables 1 and 2 , respectively, we display the error percentages i. We also denoted the scenarios with error percentages from 1. Having validated the simulated models with the analytical models for the FJ and FEC systems, we now simulate these two models when dealing with large values of c.

### Introduction to Matrix Analytic Methods in Stochastic Modeling

It is clear from these figures that. It decreases to a constant for the HEXS , and we stress the fast decrease for low values of c. In this case i. This indicates that when services have a coefficient of variation less than one and for positively correlated type of arrivals, one may need a larger number of servers as compared to other scenarios. In this section we concentrate on simulating c , k -SM system under various scenarios.

In addition to the exponential service time distribution EXPS mentioned earlier, we also consider a log-concave and a heavy-tailed one. The cumulative distribution functions CDF follow:. In the example below, we simulated the models using three simulation runs and each run is done to cover , jobs leaving the system.

The results of experiment are presented on Fig. As k increases to c , we see the expected latency for heavy-tailed services like Weibull appears to go from least value compared to other service distributions to the highest. This seems to be the case for higher c. This phenomenon can be explained as follows. In this paper we studied queuing models useful in the study of efficient redundancy techniques in CC and DG systems.

We combined both analytic and simulation approach in the study of such queuing models under the assumption that the arrivals occur according to a versatile Markovian point process. Through simulation and numerical methods we showed the effect of redundancy on the expected latency as well as on the expected cost.

The models studied in this paper can be extended in a number of ways. For example, heavy-tailed distributions for services [ 36 ] play a significant role in CC and DG areas.

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So, it will be of interest to explore this in the context of classical FJ system as well as extensions of FJ and SM systems. This will shed additional light similar to [ 18 ] but for more versatile arrival process. It is also worthy of consideration for further extensions both from theoretical and algorithmic points of view to compare the results with the model presented in [ 37 ]. It should be pointed that some preliminary results for multiserver queues with log-convex type services are available in [ 36 ].

The authors express their sincere thanks to the anonymous reviewers and the editor for their suggestions and paying attention to some key references, that improved the presentation of the paper.

Artalejo J. Baccelli F. Bladt M. Chakravarthy S. Krishnamoorthy et al. Chernov I. Furht B. Gardner K. Rama Murthy Garimella, Rumyantsev A. Graham A. Their intention is to show that the fundamental ideas extend beyond simple homogeneous QBD. Matrix analytic methods constitute a success story, illustrating the enrichment of a science, applied probability, by a technology, that of digital computers.

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Marcel Neuts has played a seminal role in these exciting developments, promoting numerical investigation as an essential part of the solution of probability models. He wrote in ,. When done properly, it conforms to the highest standard of scientific research. This had been long accepted among numerous scientific communities but was not at the time the prevalent view among applied probabilists. The excitement one feels when dealing with that subject stems from the synergy resulting from keeping algorithmic considerations in the forefront when solving stochastic problems.

The connection was already noted in by Kemeny and Snell who wrote,. Sign in Help View Cart. Manage this Book. Add to my favorites. Recommend to Library. Email to a friend. Digg This. Notify Me! E-mail Alerts. RSS Feeds.