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Product Form Solution to Production Systems With Job Type Restricted Machines
, 1999
"... In this paper we present a class of queueing models with multiple job types, multiple machines and machine dependent processing times. Each machine is restricted in such a way that it can only handle jobs from a specific, machine dependent, set of job types. If a job arrives at the system and it ..."
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In this paper we present a class of queueing models with multiple job types, multiple machines and machine dependent processing times. Each machine is restricted in such a way that it can only handle jobs from a specific, machine dependent, set of job types. If a job arrives at the system and it can be handled by two or more idle machines, then a probability distribution determines to which machine the job is sent. These probability distributions are control parameters for this model and the value of the probabilities influences the performance of the model. Using partial balance equations we show that only for appropriately chosen values of these probabilities, this model has a product form solution. 1 Introduction In research considerable attention has been devoted to product form stationary distributions for queueing models. Most models that are known to have a product form distribution are reversible or quasi reversible ([Muntz, 1972],[Kelly, 1979], [Walrand & Varaiya, 198...
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.
Order Independent Loss Networks
"... . The Order Independent (OI) queues are a class of quasireversible queues where the total departure rate from any queue state is independent of the order of the customers in the queue. The OI class is extended to include networks of queues which can be used to model systems with complex loss mechan ..."
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. The Order Independent (OI) queues are a class of quasireversible queues where the total departure rate from any queue state is independent of the order of the customers in the queue. The OI class is extended to include networks of queues which can be used to model systems with complex loss mechanisms. We present several examples where OI loss networks are used to to model circuitswitched networks where some blocked calls are queued for connection in a finite buffer  if the buffer is full the call is lost. 1 INTRODUCTION A class of quasireversible queues was recently investigated [2] where the essential property leading to the quasireversibility of these queues is the fact that the total departure rate of customers from any queue state is independent of the order of the customers in the queue. We call these queues order independent (OI) queues. OI queues are quasireversible and can therefore be connected in networks with productform stationary distribution. The OI class include...
Sum of product forms solutions to MSCCC queues with job type dependent processing times
"... Queueing models with simultaneous resource possession can be used to model production systems at which the production process occupies two or more resources (machines, operators, product carriers etc.) at the same time. A special class of these queueing models is the class of MSCCC queues, for which ..."
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Queueing models with simultaneous resource possession can be used to model production systems at which the production process occupies two or more resources (machines, operators, product carriers etc.) at the same time. A special class of these queueing models is the class of MSCCC queues, for which the stationary distribution has a product form. This was shown by Berezner et al. whose result depends on one special characteristic of MSCCC queues, being the processing times are job type independent exponentially distributed. However in many production situations processing times are job type dependent. Therefore we examined MSCCC queues with job type dependent exponentially distributed processing times. We determined the equilibrium probabilities of two special models using a detailed state description, for which a solution using an aggregated state description is known. Comparing these two solutions we gained more insight in the structure of the solution to more general models for whi...
Product Form Solutions to Production Systems With Simultaneous Resource Possession
, 1999
"... Queueing models with simultaneous resource possession can be used to model production systems, in which several resources are needed simultaneously to process a job. With these models, performance characteristics of the production systems can be calculated. This is done by recognising the relevan ..."
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Queueing models with simultaneous resource possession can be used to model production systems, in which several resources are needed simultaneously to process a job. With these models, performance characteristics of the production systems can be calculated. This is done by recognising the relevant Markov chain and calculating the equilibrium probabilities. However these Markov chains become highdimensional and allow very large jumps, which makes it extremely hard to nd the exact solution to the equilibrium equations. In this paper we study three relatively simple models with simultaneous resource possession in order gain insight into the solution of these models in general. We analyse the equilibrium equations of the relevant Markov chains and examine whether they allow a product form solution. We show that for these three models the behaviour on the horizontal boundary of the corresponding random walk is crucial for the existence of a product form solution. Two of the thr...