Two Parallel Queues Created by Arrivals with Two Demands  I

Flatto, Leopold; Hahn, Sebastian

doi:10.1137/0144074

Cited by 294 publications

(222 citation statements)

References 3 publications

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“…According to Lemma 4, for a large value of M and l > M, we have ε l < [log(l)] −1 . Then, from (5), (6), and Lemma 4, we have…”

Section: Lemmamentioning

confidence: 99%

Exact Monte Carlo simulation for fork-join networks

Dai

2011

Adv. Appl. Probab.

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In a fork-join network each incoming job is split into K tasks and the K tasks are simultaneously assigned to K parallel service stations for processing. For the distributions of response times and queue lengths of fork-join networks, no explicit formulae are available. Existing methods provide only analytic approximations for the response time and the queue length distributions. The accuracy of such approximations may be difficult to justify for some complicated fork-join networks. In this paper we propose a perfect simulation method based on coupling from the past to generate exact realisations from the equilibrium of fork-join networks. Using the simulated realisations, Monte Carlo estimates for the distributions of response times and queue lengths of fork-join networks are obtained. Comparisons of Monte Carlo estimates and theoretical approximations are also provided. The efficiency of the sampling algorithm is shown theoretically and via simulation.

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“…According to Lemma 4, for a large value of M and l > M, we have ε l < [log(l)] −1 . Then, from (5), (6), and Lemma 4, we have…”

Section: Lemmamentioning

confidence: 99%

Exact Monte Carlo simulation for fork-join networks

Dai

2011

Adv. Appl. Probab.

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“…Each processor has exponentially distributed service times with rates α and β, respectively. The system considered here is a generalization of a two-server system considered by Flatto and Hahn [27] and Flatto [26]. The PGF f (x, y) of the two-dimensional distribution characterizing the system yields FE (2) with…”

Section: Fe 6: Two Parallel Processors With Coupled Inputsmentioning

confidence: 99%

“…Now, we present the equation, which arises [27] from a double queue model, illustrated in Fig. 3, where the arriving customers simultaneously place two demands handled independently by two servers.…”

Section: Fe 3: Inventory Control Of Database Systemsmentioning

confidence: 99%

On the structure and solutions of functional equations arising from queueing models

2017

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Abstract. It is a survey on functional equations of a certain type, for functions in two complex variables, which often arise in queueing models. They share a common pattern despite their apparently different forms. In particular, they invariably characterize the probability generating function of the bivariate distribution characterizing a two-queue system and their forms depend on the composition of the underlying system. Unfortunately, there is no general methodology of solving them, but rather various ad-hoc techniques depending on the nature of a particular equation; most of the techniques involve advanced complex analysis tools. Also, the known solutions to particular cases of this type of equations are in general of quite involved forms and therefore it is very difficult to apply them practically. So, it is clear that the issues connected with finding useful descriptions of solutions to these equations create a huge area of research with numerous open problems. The aim of this article is to stimulate a methodical study of this area. To this end we provide a survey of the queueing literature with such two-place functional equations. We also present several observations obtained while preparing it. We hope that in this way we will make it easier to take some steps forward on the road towards a (more or less) general solving theory for this interesting class of equations.Mathematics Subject Classification. 05A15, 30D05, 30E25, 39B32, 60K25, 65Q20.

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“…As stated in [22], because the arrival processes to the servers are correlated, the model is difficult to decompose. The exact analytical solution is only known for the particular case of two homogeneous servers [23,24]. For a larger number of servers, approximation and heuristic methods have been proposed.…”

Section: Introductionmentioning

confidence: 99%

A model of periodic acknowledgement

Brandwajn

2003

Performance Evaluation

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We study a problem abstracted from modeling a multicast protocol. In our model, messages generated by a single source are simultaneously forwarded to a set of receivers where they are independently processed. We assume a state-dependent message arrival rate and memoryless service time distributions. The receivers may process messages at different average rates. Messages processed by all receivers are periodically acknowledged and cleared from the system. Due to finite buffer space, the total number of non-acknowledged messages in the system is limited. Our focus in this paper is on the number of messages cleared at acknowledgement time.The problem under consideration bears resemblance to a fork/join process with heterogeneous servers, used in the study of multiprocessing computer systems. Our model includes the additional features of finite buffer space and delayed periodic departure of completed jobs. Even without these features, the resulting type of queuing model has no known closed-form solution in the general case of more than two servers. Because the arrival processes to the servers are correlated, the model is difficult to decompose. We propose a relatively simple decomposition technique and a fixed-point iteration scheme. In our approach, we consider each receiver (server) in isolation, and we account for the influence of other receivers through the probability that a given number of messages can be cleared at acknowledgement time. We derive elementary differential equations for the number of messages processed by a receiver. These equations involve the conditional probability of the number of messages not yet processed given the number of messages waiting to be cleared. We compute an approximate solution using the conditional probability obtained from a model with exponentially distributed acknowledgement periods. Our numerical results for the average number of messages cleared at acknowledgment time are typically within a few percent of simulation midpoints.

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Two Parallel Queues Created by Arrivals with Two Demands I

Cited by 294 publications

References 3 publications

Exact Monte Carlo simulation for fork-join networks

Exact Monte Carlo simulation for fork-join networks

On the structure and solutions of functional equations arising from queueing models

A model of periodic acknowledgement

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