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ABSTRACT The arrival theorem is known to be valid for product form queueing networks that do not have capacity constraints at the queues. This paper investigates the validity of the arrival theorem for open and closed queueing networks... more
ABSTRACT The arrival theorem is known to be valid for product form queueing networks that do not have capacity constraints at the queues. This paper investigates the validity of the arrival theorem for open and closed queueing networks with finite queues under product form conditions, and derives a necessary and sufficient condition for the validity of the arrival theorem under product form conditions. It will be shown that a product form alone is not a guarantee for the validity of the arrival theorem. Further conditions are required to retain its validity. The paper provides an extensive set of examples and counterexamples, and can therefore also be considered as a survey to provide insight for the validity of the arrival theorem.
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Bloedtransfusies zijn onmisbaar en levensreddend, maar kunnen ook nadelige gevolgen voor de ontvanger met zich meebrengen. Risico’s kunnen geminimaliseerd worden door de best passende donors te selecteren. Elementaire statistiek in... more
Bloedtransfusies zijn onmisbaar en levensreddend, maar kunnen ook nadelige gevolgen voor de ontvanger met zich meebrengen. Risico’s kunnen geminimaliseerd worden door de best passende donors te selecteren. Elementaire statistiek in combinatie met OR biedt hiervoor hulp.
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For rare blood groups the recruitment of donor relatives, for example siblings, is expected to be effective, since the probability of a similar rare blood group is likely. However, the likelihood differs between blood groups and is not... more
For rare blood groups the recruitment of donor relatives, for example siblings, is expected to be effective, since the probability of a similar rare blood group is likely. However, the likelihood differs between blood groups and is not commonly available. This paper provides a unified mathematical formulation to calculate such likelihoods. From a mathematical and probabilistic point of view, it is shown that these likelihoods can be obtained from the computation of a stationary genotype distribution. This, in turn, can be brought down to a system of quadratic stochastic operators. A generic mathematical approach is presented which directly leads to a stationary genotype distribution for arbitrary blood groups. The approach enables an exact computation for the effectiveness of recruiting next of kin for blood donorship. Next to an illustration of computations for ‘standard’ ABO and Rhesus-D blood groups, it is particularly illustrated for the extended Rhesus blood group system. Also ...
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This note studies the comparison of finite-buffer and nonexponential batch arrival systems of the form Gx/M/c/c + N with the corresponding systems, with N replaced by N', where N' can be smaller, larger, or infinite. If N' = ∞... more
This note studies the comparison of finite-buffer and nonexponential batch arrival systems of the form Gx/M/c/c + N with the corresponding systems, with N replaced by N', where N' can be smaller, larger, or infinite. If N' = ∞ the service times can be arbitrarily distributed. Both comparison and error bounds are obtained for performance measures such as the throughput, the idle probability, and the active server distribution. The results are of practical interest to establish computational reductions, either by infinite-space approximation or by reduced finite truncations. Two different proof techniques will be employed: the sample path approach and the Markov reward approach. The comparison of these two techniques is of interest in itself, showing the advantage and disadvantage of each.
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Overflow mechanisms can be found in a variety of queueing models. This paper studies a simple and generic overflow system that allows the service times to be both job type and station dependent. This system does not exhibit a product... more
Overflow mechanisms can be found in a variety of queueing models. This paper studies a simple and generic overflow system that allows the service times to be both job type and station dependent. This system does not exhibit a product form. To justify simple product form computations, two product form modifications are given, as by a so-called call packing principle and by a stop protocol. The provided proofs are self-contained and straightforward for the exponential case and of merit by itself. Next, it is numerically studied whether and when, or under which conditions, the modifications lead to a reasonable approximation of the blocking probability, if not an ordering. The numerical results indicate that call packing provides a rather accurate approximation when the overflow station is not heavily utilized. Moreover, when overflowed jobs have an equal or faster service rate, the approximation is consistently found to be pessimistic, which can be useful for practical purposes. The s...
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ABSTRACT Do we have a product form? If so, how is it characterized? If not, how can product forms still be useful?
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ABSTRACT In general capacity constraints prohibit a product form limiting distribution for a queueing network. An exception to this rule is the case when jobs jump over a saturated node as if they were served at an infinite speed. This... more
ABSTRACT In general capacity constraints prohibit a product form limiting distribution for a queueing network. An exception to this rule is the case when jobs jump over a saturated node as if they were served at an infinite speed. This note generalizes the principle of “jumping-over” in that a job arriving at a node is rejected with a state-dependent probability. It is shown that also in this more general situation the steady state distribution exhibits a product form. This generalization requires a proof from a different point of view, which is of interest in itself.
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This paper presents a queueing network description of a cellular mobile communications network. These cellular networks are characterised by the call lengths, the fresh call arrival rate, and the handover rates, as well as the capacity... more
This paper presents a queueing network description of a cellular mobile communications network. These cellular networks are characterised by the call lengths, the fresh call arrival rate, and the handover rates, as well as the capacity restrictions on the number of calls because of limited bandwidth. In the resulting queueing network the routing probabilities depend on the age of the customers. This queueing network is amenable to analysis via known methods. In particular, insensitivity results are generalised to also include capacity restrictions under age-dependent routing and to include call holding time distributions of customers moving from one queue to another.
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Tandem queue configurations naturally arise in multistage stochastic systems such as assembly lines in manufacturing or multiphase transmissions in telecommunications. As finite capacity or storage constraints (buffers) are usually... more
Tandem queue configurations naturally arise in multistage stochastic systems such as assembly lines in manufacturing or multiphase transmissions in telecommunications. As finite capacity or storage constraints (buffers) are usually involved, the celebrated closed product form expression is generally not applicable. In this paper, a new bounding methodology for nonproduct form systems is applied to finite single-server exponential tandem queues. The methodology is based on modifying the original system into product form systems that provide bounds for some performance measure of interest. The product form modifications given for this finite tandem queue propose a computationally attractive and intuitively obvious lower and upper bound for the call congestion and throughput. Numerical results indicate that the bounds are reasonable indicators of the order of magnitude. This can be useful for quick engineering purposes as will be illustrated by an optimal design example. A formal proof...
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As an extension of the discrete-time case, this note investigates the variance of the total cumulative reward for continuous-time Markov reward chains with finite state spaces. The results correspond to discrete-time results. In... more
As an extension of the discrete-time case, this note investigates the variance of the total cumulative reward for continuous-time Markov reward chains with finite state spaces. The results correspond to discrete-time results. In particular, the variance growth rate is shown to be asymptotically linear in time. Expressions are provided to compute this growth rate.
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... M. 72 Claessens et al. European Journal of Operational Research 110 (1998) 474489 479 The circulation of rolling stock affects the num ber of compositions (ie identical trains) needed to operate the lines. A composition is the name... more
... M. 72 Claessens et al. European Journal of Operational Research 110 (1998) 474489 479 The circulation of rolling stock affects the num ber of compositions (ie identical trains) needed to operate the lines. A composition is the name for a train as a whole. ...
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A general condition is provided from which an error bound can be concluded for approximations of queueing networks which are based on modifications of the transition and state space structure. This condition relies upon Markov reward... more
A general condition is provided from which an error bound can be concluded for approximations of queueing networks which are based on modifications of the transition and state space structure. This condition relies upon Markov reward theory and can be verified inductively in concrete situations. The results are illustrated by estimating the accuracy of a simple throughput bound for a closed queueing network with alternate routing and a large finite source input. An explicit error bound for this example is derived which is of order M—1, where M is the number of sources.
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Practical motivation some fundamental tools station balance - a down-to-Earth approach station balance - a more formal approach station and cluster balance for networks with limited clusters communications networks - source balance and... more
Practical motivation some fundamental tools station balance - a down-to-Earth approach station balance - a more formal approach station and cluster balance for networks with limited clusters communications networks - source balance and insensitivity.
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ABSTRACT Blood banks produce and store blood products in order to fulfil the uncertain demand at hospitals. Platelet pools are the most expensive and most perishable blood product having a shelf life of only four to six days. Production... more
ABSTRACT Blood banks produce and store blood products in order to fulfil the uncertain demand at hospitals. Platelet pools are the most expensive and most perishable blood product having a shelf life of only four to six days. Production volumes need to be chosen carefully in order to reduce outdating while keeping the occurrence of shortages low. We investigate the structure of the optimal production policy by solving a down sized periodic Markov Decision Problem. The optimal production volumes appear to depend on the number of pools on stock and their ages. Simulation results for the optimal MDP-policy suggest two rules: the 1D and 2D rule. Both rules perform quite well. The 2D rule performs nearly optimal even if one acknowledges the distinction of multiple and limited compatible blood groups and the uncertainty in the supply by donors.
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Continuous-time Markov reward structures over a finite time interval are studied. Conditions are provided to conclude error bounds or comparison results when studying systems under modified data assumptions such as for sensitivity,... more
Continuous-time Markov reward structures over a finite time interval are studied. Conditions are provided to conclude error bounds or comparison results when studying systems under modified data assumptions such as for sensitivity, computitional or bounding purposes. A reliability network is studied as an application. An explicit sensitivity error bound on the effect of breakdown and repair rates is obtained.
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A discrete-time communication framework is studied under an approximate socalled "stop protocol" related to the lost-calls-held protocol. An explicit insensitive product form expression as recently obtained under a less... more
A discrete-time communication framework is studied under an approximate socalled "stop protocol" related to the lost-calls-held protocol. An explicit insensitive product form expression as recently obtained under a less realistic recirculated protocol is shown to remain valid. This extension seems of both practical and theoretical interest. Applications include: • Circuit switching structures • MAN-allocation schemes • CSMA and BTMA • Rude-CSMA
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Queueing networks are used widely as modelling and evaluation tools in manufacturing, telecommunications, computer networking, and related areas. Much of the research effort has been devoted to so-called Jackson networks, that is,... more
Queueing networks are used widely as modelling and evaluation tools in manufacturing, telecommunications, computer networking, and related areas. Much of the research effort has been devoted to so-called Jackson networks, that is, networks with Poisson arrivals, exponential service times and routing independent of the state of the system and the history of the customer. The steady-state distribution of Jackson networks can be expressed in a so-called product form. This computationally attractive form will be shown to be directly related to the principle of balance per station. This principle will be used to provide practical insights concerning the following questions
ABSTRACT We consider a class of models for multicomponent systems in which components can break down and be repaired in a dependent manner and where breakdown and repair times can be arbitrarily distributed. The problem of calculating the... more
ABSTRACT We consider a class of models for multicomponent systems in which components can break down and be repaired in a dependent manner and where breakdown and repair times can be arbitrarily distributed. The problem of calculating the equilibrium distribution and, from this, the expected performability for these models is intractable unless certain assumptions are made about breakdowns and repairs. In this paper we show that the performability of multicomponent systems that do not satisfy these rules can be bounded by tractable modifications. Our results are proved by stochastic comparability arguments and a Markov reward technique, which is of interest in itself as it enables one to prove that the equilibrium distribution of one process can be bounded by that of another even when the sample paths of the process are not. This is illustrated by an example.