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Computing distributions and moments in polling models by numerical transform inversion
 Eval
, 1996
"... We show that probability distributions and moments of performance measures in many polling models can be effectively computed by numerically inverting transforms (generating functions and Laplace transforms). We develop new efficient iterative algorithms for computing the transform values and then u ..."
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We show that probability distributions and moments of performance measures in many polling models can be effectively computed by numerically inverting transforms (generating functions and Laplace transforms). We develop new efficient iterative algorithms for computing the transform values and then use our recently developed variant of the Fourierseries method to perform the inversion. We also show how to use this approach to compute moments and asymptotic parameters of the distributions. We compute a twoterm asymptotic expansion of the tail probabilities, which turns out to be remarkably accurate for small tail probabilities. The tail probabilities are especially helpful in understanding the performance of different polling disciplines. For instance, it is known that the exhaustive discipline produces smaller mean steadystate waiting times than the gated discipline, but we show that the reverse tends to be true for small tail probabilities. The algorithms apply to describe the transient behavior of stationary or nonstationary models as well as the steadystate behavior of stationary models. We demonstrate effectiveness by analyzing the computational complexity and by doing several numerical examples for the gated and exhaustive service disciplines, with both zero and nonzero switchover times. We also show that our approach applies to other polling models. Our main focus is on computing exact tail probabilities and asymptotic approximations to them, which seems not to have been done before. However, even for mean waiting times, our algorithm is faster than previous algorithms for large models. The computational complexity of our algorithm is O(N α) for computing performance measures at one queue and O(N 1 + α) for computing performance measures at all queues, where N is the number of queues and α is typically between 0.6 and 0.8.
Iterative approximation of klimited polling systems
, 2006
"... The present paper deals with the problem of calculating queue length distributions in a polling model with (exhaustive) klimited service under the assumption of general arrival, service and setup distributions. The interest for this model is fueled by an application in the field of logistics. Knowl ..."
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Cited by 9 (0 self)
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The present paper deals with the problem of calculating queue length distributions in a polling model with (exhaustive) klimited service under the assumption of general arrival, service and setup distributions. The interest for this model is fueled by an application in the field of logistics. Knowledge of the queue length distributions is needed to operate the system properly. The multiqueue polling system is decomposed into singlequeue vacation systems with klimited service and statedependent vacations, for which the vacation distributions are computed in an iterative approximate manner. These vacation models are analyzed via matrixanalytic techniques. The accuracy of the approximation scheme is verified by means of an extensive simulation study. The developed approximation turns out be accurate, robust and computationally efficient.
ResponseTime Approximations For MultiServer, MultiClass Production Systems With Significant Setups
, 1995
"... A multiserver polling model is a queueing model where many order classes share a set of identical servers and a setup time is incurred whenever a server changes class. This paper develops approximations for the waiting time distribution in a multiserver polling model with cyclic servetoexhaustio ..."
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A multiserver polling model is a queueing model where many order classes share a set of identical servers and a setup time is incurred whenever a server changes class. This paper develops approximations for the waiting time distribution in a multiserver polling model with cyclic servetoexhaustion service. These approximations are derived using previously established heavytraffic results and the assumption of significant setup times. They improve as the total setup over all classes increases. In a manufacturing environment, setups are only performed if there are orders waiting. We introduce the concept of positivequeue setups for the case where setups depend on the presence of orders, in contrast to the traditional polling model with emptyqueue setups. The approximation is derived both for systems with emptyqueue setups and systems with positivequeue setups. This study of polling models was motivated by a production scheduling problem at Raychem Corporation. The approximation f...
the Dutch Ministry of Economic Affairs.
"... Abstract The present paper deals with the problem of calculating queue length distributions in a polling model with (exhaustive) klimited service under the assumption of general arrival, service and setup distributions. The interest for this model is fueled by an application in the field of logis ..."
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Abstract The present paper deals with the problem of calculating queue length distributions in a polling model with (exhaustive) klimited service under the assumption of general arrival, service and setup distributions. The interest for this model is fueled by an application in the field of logistics. Knowledge of the queue length distributions is needed to operate the system properly. The multiqueue polling system is decomposed into singlequeue vacation systems with klimited service and statedependent vacations, for which the vacation distributions are computed in an iterative approximate manner. These vacation models are analyzed via matrixanalytic techniques. The accuracy of the approximation scheme is verified by means of an extensive simulation study. The developed approximation turns out to be accurate, robust and computationally efficient.
Iterative approximation of limited polling systems
, 2006
"... The present paper deals with the problem of calculating queue length distributions in a polling model with (exhaustive) limited service under the assumption of general arrival, service and setup distributions. The interest for this model is fueled by an application in the field of logistics. Knowled ..."
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The present paper deals with the problem of calculating queue length distributions in a polling model with (exhaustive) limited service under the assumption of general arrival, service and setup distributions. The interest for this model is fueled by an application in the field of logistics. Knowledge of the queue length distributions is needed to operate the system properly. The multiqueue polling system is decomposed into singlequeue vacation systems with limited service and statedependent vacations, for which the vacation distributions are computed in an iterative approximate manner. These vacation models are analyzed via matrixanalytic techniques. The accuracy of the approximation scheme is verified by means of an extensive simulation study. The developed approximation turns out be accurate, robust and computationally efficient.
Analysis of MultiClass Queueing Systems with Finite Buffers and Setup Times Using Decomposition Methods
"... In this paper, we consider two variants of a multiclass queueing system with a single server, finite buffers, and setup times. In the first variant, a setup occurs every time the server turns to a different class of customers, even if the queue of the scanned customer class is empty. In the second ..."
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In this paper, we consider two variants of a multiclass queueing system with a single server, finite buffers, and setup times. In the first variant, a setup occurs every time the server turns to a different class of customers, even if the queue of the scanned customer class is empty. In the second variant, the server skips empty queues and a setup is performed only at queues that contain at least one customer. We propose decomposition methods for the analysis of the steadystate behavior of both system variants. We also show that multiproduct kanban systems with setup times and lost sales are possible applications of the discussed models and we give numerical results that indicate the accuracy of the proposed approximation methods. Keywords:
The Stochastic Economic Lot Scheduling Problem: A Survey
, 2005
"... We consider the production of multiple standardized products on a single machine with limited capacity and setup times under random demands and random production times, i.e., the socalled stochastic economic lot scheduling problem (SELSP). The main task for the production manager in this setting i ..."
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We consider the production of multiple standardized products on a single machine with limited capacity and setup times under random demands and random production times, i.e., the socalled stochastic economic lot scheduling problem (SELSP). The main task for the production manager in this setting is the construction of a production plan for the machine that minimizes the total costs, i.e., the sum of holding, backlogging and setup costs. Based on the critical elements of such a production plan, we give a classification and extensive overview of the research on the SELSP together with an indication of open research areas.