Nicky van Foreest | University of Groningen (original) (raw)

Papers by Nicky van Foreest

We consider two variants of a two-station tandem network with blocking. In both variants the firs... more We consider two variants of a two-station tandem network with blocking. In both variants the first server ceases to work when the queue length at the second station hits a 'blocking threshold'. In addition, in variant 2 the first server decreases its service rate when the second queue exceeds a 'slow-down threshold', which is smaller than the blocking level. In both variants the arrival process is Poisson and the service times at both stations are exponentially distributed. Note, however, that in case of slow-downs, server 1 works at a high rate, a slow rate, or not at all, depending on whether the second queue is below or above the slow-down threshold or at the blocking threshold, respectively. For variant 1, i.e., only blocking, we concentrate on the geometric decay rate of the number of jobs in the first buffer and prove that for increasing blocking thresholds the sequence of decay rates decreases monotonically and at least geometrically fast to max{ρ1, ρ2}, where ρi is the load at server i. The methods used in the proof also allow us to clarify the asymptotic queue length distribution at the second station. Then we generalize the analysis to variant 2, i.e., slow-down and blocking, and establish analogous results. a result also obtained in . However, in this paper we also show rigorously that the decay rate as a function of the blocking threshold decreases monotonically and at least geometrically fast to ρ.

Stochastic Models, 2005

We consider two variants of a two-station tandem network with blocking. In both variants the firs... more We consider two variants of a two-station tandem network with blocking. In both variants the first server ceases to work when the queue length at the second station hits a 'blocking threshold'. In addition, in variant 2 the first server decreases its service rate when the second queue exceeds a 'slow-down threshold', which is smaller than the blocking level. In both variants the arrival process is Poisson and the service times at both stations are exponentially distributed. Note, however, that in case of slow-downs, server 1 works at a high rate, a slow rate, or not at all, depending on whether the second queue is below or above the slow-down threshold or at the blocking threshold, respectively. For variant 1, i.e., only blocking, we concentrate on the geometric decay rate of the number of jobs in the first buffer and prove that for increasing blocking thresholds the sequence of decay rates decreases monotonically and at least geometrically fast to max{ρ1, ρ2}, where ρi is the load at server i. The methods used in the proof also allow us to clarify the asymptotic queue length distribution at the second station. Then we generalize the analysis to variant 2, i.e., slow-down and blocking, and establish analogous results. a result also obtained in . However, in this paper we also show rigorously that the decay rate as a function of the blocking threshold decreases monotonically and at least geometrically fast to ρ.

Operations Research Letters, 2005

We investigate a fluid buffer which is modulated by a stochastic background process, while the mo... more We investigate a fluid buffer which is modulated by a stochastic background process, while the momentary behavior of the background process depends on the current buffer level in a continuous way. Loosely speaking the feedback is such that the background process behaves 'as a Markov process' with generator ɴݵ at times when the buffer level is Ý, where the entries of ɴݵ are continuous functions of Ý. Moreover, the flow rates for the buffer may also depend continuously on the current buffer level. Such models are interesting in the context of closed-loop telecommunication networks, in which sources interact with network buffers, but may also be deployed in the study of certain production systems.

Performance Evaluation, 2007

In this paper we use stochastic Petri nets (SPNs) to study the interaction of multiple TCP source... more In this paper we use stochastic Petri nets (SPNs) to study the interaction of multiple TCP sources that share one or two buffers. No analytical nor numerical results have been presented for such cases yet. We use SPNs in an unconventional way: the tokens in the SPN do not represent the packets being sent in the network, but merely model fractions of buffer occupancy and the congestion window sizes. In this way, we use the SPNs to obtain a discretisation of a fluid model for TCP dynamics. Thus, we pair the modelling flexibility of SPNs with the modelling efficiency of fluid models. In doing so, our approach also avoids the (numerical) solution of partial differential equations; instead, just the steady-state solution of a (large) continuous-time Markov chain is required.We first consider two TCP sources sharing a single buffer and evaluate the consequences of two popular assumptions for the loss process in terms of fairness and link utilization. The results obtained with this model are in agreement with existing analytic models. A comparison with (more costly) simulations in ns2 shows that the real loss process is somewhere in between the two loss models.Secondly, we consider a network consisting of three sources and two buffers and study how the sources share the capacity of the links. This leads to an interesting conjecture on fairness in large TCP networks.

Journal of Computational and Applied Mathematics, 2009

We establish some representations for the smallest and largest zeros of orthogonal polynomials in... more We establish some representations for the smallest and largest zeros of orthogonal polynomials in terms of the parameters in the three-terms recurrence relation. As a corollary we obtain representations for the endpoints of the true interval of orthogonality. Implications of these results for the decay parameter of a birth–death process (with killing) are displayed.

We establish some representations for the smallest and largest zeros of orthogonal polynomials in... more We establish some representations for the smallest and largest zeros of orthogonal polynomials in terms of the parameters in the three-terms recurrence relation. As a corollary we obtain representations for the endpoints of the true interval of orthogonality. Implications of these results for the decay parameter of a birth death process (with killing) are displayed.

Journal of Computational and Applied Mathematics, 2009

We establish some representations for the smallest and largest zeros of orthogonal polynomials in... more We establish some representations for the smallest and largest zeros of orthogonal polynomials in terms of the parameters in the three-terms recurrence relation. As a corollary we obtain representations for the endpoints of the true interval of orthogonality. Implications of these results for the decay parameter of a birth-death process (with killing) are displayed.

Probability in The Engineering and Informational Sciences, 2010

... M X /G Y /1/K + B BULK QUEUE REMCO GERMS AND NICKY VAN FOREEST Faculty of Economics and Busin... more ... M X /G Y /1/K + B BULK QUEUE REMCO GERMS AND NICKY VAN FOREEST Faculty of Economics and Business University of Groningen 9700 AV Groningen,The Netherlands E-mail: r.germs@rug.nl ... Chang, Choi, and Kim [2] gave computational and analytical Page 3. ...

International Journal of Production Research, 2012

In many make-to-order production situations with batch setup times, customer orders are grouped i... more In many make-to-order production situations with batch setup times, customer orders are grouped into family-dependent batches to limit the loss of capacity due to setups. These batches, however, cannot be too large, since the make-to-order character requires that orders have to be produced in time. This trade-off between setup time efficiency and due-date adherence creates a challenging scheduling problem referred to in the literature as the Customised Stochastic Lot Scheduling Problem. Typically, suppliers reduce the complexity of the production problem by quoting lead times that are equal for all customer families. This choice, however, is in many cases too restrictive. In this paper, we show quantitatively by means of Markov decision processes (MDPs) that using family-dependent lead times can result in a significant gain in profit as compared with using standard lead times. We develop a simple heuristic acceptance/scheduling policy, and demonstrate that this heuristic performs very well compared with the optimal policy of the MDP for a wide range of parameters.

Fuel and Energy Abstracts, 2011

This papers considers admission control and scheduling of customer orders in a production system ... more This papers considers admission control and scheduling of customer orders in a production system that produces different items on a single machine. Customer orders drive the production and belong to product families, and have family dependent due-date, size, and reward. When production changes from one family to another a setup time is incurred. Moreover, if an order cannot be accepted, it is considered lost upon arrival. The problem is to find a policy that accepts/rejects and schedules orders such that long run profit is maximized. This problem finds its motivation in batch industries in which suppliers have to realize high machine utilization while delivery times should be short and reliable and the production environment is subject to long setup times.We model the joint admission control/scheduling problem as a Markov decision process (MDP) to gain insight into the optimal control of the production system and use the MDP to benchmark the performance of a simple heuristic acceptance/scheduling policy. Numerical results show that the heuristic performs very well compared with the optimal policy for a wide range of parameter settings, including product family asymmetries in arrival rate, order size, and order reward.► We consider a stochastic lot scheduling problem with setup times between orders of the different families and orders can be rejected upon arrival. ► A stochastic dynamic programming formulation allows to compute an optimal acceptance and scheduling policy. ► This optimal policy is used to benchmark a heuristic simple threshold policy. ► The heuristic is shown to perform very well under a wide range of parameters settings.

International Journal of Production Research, 2010

The single-product, stationary inventory problem with set-up cost is one of the classical problem... more The single-product, stationary inventory problem with set-up cost is one of the classical problems in stochastic operations research. Theories have been developed to cope with finite production capacity in periodic review systems, and it has been proved that optimal policies for these cases are not of the (modified) (s, S)-type in general, but more complex. In this paper we consider a production system such that the production rate is constrained, rather than the amount as is common in periodic review models. Thus, in our case the production rate is positive and finite when the system is on and zero when off, while a cost is incurred to switching on or off. We prove that a long-run optimal stationary policy exists for this single-item continuous review inventory problem with non-zero switching cost and finite production rate, and that this optimal policy has an (s, S)-structure. We also provide an efficient numerical procedure to compute the parameters of the optimal policy. Another, and perhaps more precise, way to include a capacity constraint is to constrain the production rate, rather then the production amount per review period. In this paper we follow this idea, and determine the structure of the optimal policy for the single-item inventory problem with set-up cost, backlogging, and finite production rate. Orders arrive according to a Poisson process and the i.i.d. demands follow some (rather) arbitrary distribution. The inventory is replenished at a constant rate only when production is on. It will be proved that a long-run optimal stationary policy exists and is described by two parameters: s and S. As soon as the inventory gets below s, production is switched on, while as soon as the inventory hits S, production is switched off. We also provide a dynamic-programming based numerical method to efficiently characterize the optimal policy. In passing we mention that in (s, S)-policy literature, see e.g. and references therein, an important objective of knowing that the optimal policy is of the (s, S)-type is to use this to find the optimal values for s and S. For our approach, however, it is not necessary to know the structure of the optimal policy; it is a corollary that the optimal stationary policy has an (s, S)-structure and can be found simply by iteration (bisection).

We consider two variants of a two-station tandem network with blocking. In both variants the firs... more We consider two variants of a two-station tandem network with blocking. In both variants the first server ceases to work when the queue length at the second station hits a 'blocking threshold'. In addition, in variant 2 the first server decreases its service rate when the second queue exceeds a 'slow-down threshold', which is smaller than the blocking level. In both variants the arrival process is Poisson and the service times at both stations are exponentially distributed. Note, however, that in case of slow-downs, server 1 works at a high rate, a slow rate, or not at all, depending on whether the second queue is below or above the slow-down threshold or at the blocking threshold, respectively. For variant 1, i.e., only blocking, we concentrate on the geometric decay rate of the number of jobs in the first buffer and prove that for increasing blocking thresholds the sequence of decay rates decreases monotonically and at least geometrically fast to max{ρ1, ρ2}, where ρi is the load at server i. The methods used in the proof also allow us to clarify the asymptotic queue length distribution at the second station. Then we generalize the analysis to variant 2, i.e., slow-down and blocking, and establish analogous results. a result also obtained in . However, in this paper we also show rigorously that the decay rate as a function of the blocking threshold decreases monotonically and at least geometrically fast to ρ.

Stochastic Models, 2005

We consider two variants of a two-station tandem network with blocking. In both variants the firs... more We consider two variants of a two-station tandem network with blocking. In both variants the first server ceases to work when the queue length at the second station hits a 'blocking threshold'. In addition, in variant 2 the first server decreases its service rate when the second queue exceeds a 'slow-down threshold', which is smaller than the blocking level. In both variants the arrival process is Poisson and the service times at both stations are exponentially distributed. Note, however, that in case of slow-downs, server 1 works at a high rate, a slow rate, or not at all, depending on whether the second queue is below or above the slow-down threshold or at the blocking threshold, respectively. For variant 1, i.e., only blocking, we concentrate on the geometric decay rate of the number of jobs in the first buffer and prove that for increasing blocking thresholds the sequence of decay rates decreases monotonically and at least geometrically fast to max{ρ1, ρ2}, where ρi is the load at server i. The methods used in the proof also allow us to clarify the asymptotic queue length distribution at the second station. Then we generalize the analysis to variant 2, i.e., slow-down and blocking, and establish analogous results. a result also obtained in . However, in this paper we also show rigorously that the decay rate as a function of the blocking threshold decreases monotonically and at least geometrically fast to ρ.

Operations Research Letters, 2005

We investigate a fluid buffer which is modulated by a stochastic background process, while the mo... more We investigate a fluid buffer which is modulated by a stochastic background process, while the momentary behavior of the background process depends on the current buffer level in a continuous way. Loosely speaking the feedback is such that the background process behaves 'as a Markov process' with generator ɴݵ at times when the buffer level is Ý, where the entries of ɴݵ are continuous functions of Ý. Moreover, the flow rates for the buffer may also depend continuously on the current buffer level. Such models are interesting in the context of closed-loop telecommunication networks, in which sources interact with network buffers, but may also be deployed in the study of certain production systems.

Performance Evaluation, 2007

In this paper we use stochastic Petri nets (SPNs) to study the interaction of multiple TCP source... more In this paper we use stochastic Petri nets (SPNs) to study the interaction of multiple TCP sources that share one or two buffers. No analytical nor numerical results have been presented for such cases yet. We use SPNs in an unconventional way: the tokens in the SPN do not represent the packets being sent in the network, but merely model fractions of buffer occupancy and the congestion window sizes. In this way, we use the SPNs to obtain a discretisation of a fluid model for TCP dynamics. Thus, we pair the modelling flexibility of SPNs with the modelling efficiency of fluid models. In doing so, our approach also avoids the (numerical) solution of partial differential equations; instead, just the steady-state solution of a (large) continuous-time Markov chain is required.We first consider two TCP sources sharing a single buffer and evaluate the consequences of two popular assumptions for the loss process in terms of fairness and link utilization. The results obtained with this model are in agreement with existing analytic models. A comparison with (more costly) simulations in ns2 shows that the real loss process is somewhere in between the two loss models.Secondly, we consider a network consisting of three sources and two buffers and study how the sources share the capacity of the links. This leads to an interesting conjecture on fairness in large TCP networks.

Journal of Computational and Applied Mathematics, 2009

We establish some representations for the smallest and largest zeros of orthogonal polynomials in... more We establish some representations for the smallest and largest zeros of orthogonal polynomials in terms of the parameters in the three-terms recurrence relation. As a corollary we obtain representations for the endpoints of the true interval of orthogonality. Implications of these results for the decay parameter of a birth–death process (with killing) are displayed.

We establish some representations for the smallest and largest zeros of orthogonal polynomials in... more We establish some representations for the smallest and largest zeros of orthogonal polynomials in terms of the parameters in the three-terms recurrence relation. As a corollary we obtain representations for the endpoints of the true interval of orthogonality. Implications of these results for the decay parameter of a birth death process (with killing) are displayed.

Journal of Computational and Applied Mathematics, 2009

We establish some representations for the smallest and largest zeros of orthogonal polynomials in... more We establish some representations for the smallest and largest zeros of orthogonal polynomials in terms of the parameters in the three-terms recurrence relation. As a corollary we obtain representations for the endpoints of the true interval of orthogonality. Implications of these results for the decay parameter of a birth-death process (with killing) are displayed.

Probability in The Engineering and Informational Sciences, 2010

... M X /G Y /1/K + B BULK QUEUE REMCO GERMS AND NICKY VAN FOREEST Faculty of Economics and Busin... more ... M X /G Y /1/K + B BULK QUEUE REMCO GERMS AND NICKY VAN FOREEST Faculty of Economics and Business University of Groningen 9700 AV Groningen,The Netherlands E-mail: r.germs@rug.nl ... Chang, Choi, and Kim [2] gave computational and analytical Page 3. ...

International Journal of Production Research, 2012

In many make-to-order production situations with batch setup times, customer orders are grouped i... more In many make-to-order production situations with batch setup times, customer orders are grouped into family-dependent batches to limit the loss of capacity due to setups. These batches, however, cannot be too large, since the make-to-order character requires that orders have to be produced in time. This trade-off between setup time efficiency and due-date adherence creates a challenging scheduling problem referred to in the literature as the Customised Stochastic Lot Scheduling Problem. Typically, suppliers reduce the complexity of the production problem by quoting lead times that are equal for all customer families. This choice, however, is in many cases too restrictive. In this paper, we show quantitatively by means of Markov decision processes (MDPs) that using family-dependent lead times can result in a significant gain in profit as compared with using standard lead times. We develop a simple heuristic acceptance/scheduling policy, and demonstrate that this heuristic performs very well compared with the optimal policy of the MDP for a wide range of parameters.

Fuel and Energy Abstracts, 2011

This papers considers admission control and scheduling of customer orders in a production system ... more This papers considers admission control and scheduling of customer orders in a production system that produces different items on a single machine. Customer orders drive the production and belong to product families, and have family dependent due-date, size, and reward. When production changes from one family to another a setup time is incurred. Moreover, if an order cannot be accepted, it is considered lost upon arrival. The problem is to find a policy that accepts/rejects and schedules orders such that long run profit is maximized. This problem finds its motivation in batch industries in which suppliers have to realize high machine utilization while delivery times should be short and reliable and the production environment is subject to long setup times.We model the joint admission control/scheduling problem as a Markov decision process (MDP) to gain insight into the optimal control of the production system and use the MDP to benchmark the performance of a simple heuristic acceptance/scheduling policy. Numerical results show that the heuristic performs very well compared with the optimal policy for a wide range of parameter settings, including product family asymmetries in arrival rate, order size, and order reward.► We consider a stochastic lot scheduling problem with setup times between orders of the different families and orders can be rejected upon arrival. ► A stochastic dynamic programming formulation allows to compute an optimal acceptance and scheduling policy. ► This optimal policy is used to benchmark a heuristic simple threshold policy. ► The heuristic is shown to perform very well under a wide range of parameters settings.

International Journal of Production Research, 2010

The single-product, stationary inventory problem with set-up cost is one of the classical problem... more The single-product, stationary inventory problem with set-up cost is one of the classical problems in stochastic operations research. Theories have been developed to cope with finite production capacity in periodic review systems, and it has been proved that optimal policies for these cases are not of the (modified) (s, S)-type in general, but more complex. In this paper we consider a production system such that the production rate is constrained, rather than the amount as is common in periodic review models. Thus, in our case the production rate is positive and finite when the system is on and zero when off, while a cost is incurred to switching on or off. We prove that a long-run optimal stationary policy exists for this single-item continuous review inventory problem with non-zero switching cost and finite production rate, and that this optimal policy has an (s, S)-structure. We also provide an efficient numerical procedure to compute the parameters of the optimal policy. Another, and perhaps more precise, way to include a capacity constraint is to constrain the production rate, rather then the production amount per review period. In this paper we follow this idea, and determine the structure of the optimal policy for the single-item inventory problem with set-up cost, backlogging, and finite production rate. Orders arrive according to a Poisson process and the i.i.d. demands follow some (rather) arbitrary distribution. The inventory is replenished at a constant rate only when production is on. It will be proved that a long-run optimal stationary policy exists and is described by two parameters: s and S. As soon as the inventory gets below s, production is switched on, while as soon as the inventory hits S, production is switched off. We also provide a dynamic-programming based numerical method to efficiently characterize the optimal policy. In passing we mention that in (s, S)-policy literature, see e.g. and references therein, an important objective of knowing that the optimal policy is of the (s, S)-type is to use this to find the optimal values for s and S. For our approach, however, it is not necessary to know the structure of the optimal policy; it is a corollary that the optimal stationary policy has an (s, S)-structure and can be found simply by iteration (bisection).