We focus on a particular connection between queueing and risk models in a multi-dimensional setti... more We focus on a particular connection between queueing and risk models in a multi-dimensional setting. We first consider the joint workload process in a queueing model with parallel queues and simultaneous arrivals at the queues. For the case that the service times are ordered (from largest in the first queue to smallest in the last queue) we obtain the Laplace-Stieltjes transform of the joint stationary workload distribution. Using a multivariate duality argument between queueing and risk models, this also gives the Laplace transform of the survival probability of all books in a multivariate risk model with simultaneous claim arrivals and the same ordering between claim sizes. Other features of the paper include a stochastic decomposition result for the workload vector, and an outline how the two-dimensional risk model with a general two-dimensional claim size distribution (hence without ordering of claim sizes) is related to a known Riemann boundary value problem. 2010 Mathematics Subject Classification. 60K25; 91B30. Key words and phrases. Queues with simultaneous arrivals, workload, stochastic decomposition, duality, multivariate risk model. The authors gratefully acknowledge stimulating discussions with Yifan Xu (Binghamton University). Serban Badila is being supported by Project 613.001.017 of the Netherlands Organisation for Scientific Research (NWO).
We investigate Markovian queues that are examined by a controller at random times determined by a... more We investigate Markovian queues that are examined by a controller at random times determined by a Poisson process. Upon examination, the controller sets the service speed to be equal to the minimum of the current number of customers in the queue and a certain maximum service speed; this service speed prevails until the next examination time. We study the resulting two-dimensional Markov process of queue length and server speed, in particular two regimes with time scale separation, specifically for infinitely frequent and infinitely long examination times. In the intermediate regime the analysis proves to be extremely challenging. To gain further insight into the model dynamics we then analyse two variants of the model in which the controller is just an observer and does not change the speed of the server.
In this paper we discuss some aspects of the asymptotic behavior of Discrete Event Dynamic System... more In this paper we discuss some aspects of the asymptotic behavior of Discrete Event Dynamic Systems (DEDS), in which the activity times are random variables. The main result is that a central limit theorem holds for DEDS and consequently that the cycletime of the system is asymptotically normally distributed.
In the paper a probabilistic coupling between the M/G/1 processor sharing queue and the M/M/1 fee... more In the paper a probabilistic coupling between the M/G/1 processor sharing queue and the M/M/1 feedback queue, with general feedback probabilities, is established. This coupling is then used to prove the almost sure convergence of sojourn times in the feedback model to sojourn times in the M/G/1 processor sharing queue. Using the theory of regenerative processes it follows that for stable queues the stationary distribution of the sojourn time in the feedback model converges in law to the corresponding distribution in the processor sharing model. The results do not depend on Poisson arrival times, but are also valid for general arrival processes.
Journal of Statistical Mechanics: Theory and Experiment, 2018
DOI to the publisher's website. • The final author version and the galley proof are versions of t... more DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal. If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the "Taverne" license above, please follow below link for the End User Agreement:
This paper studies a two-level traffic shaper. The cell arrival process is modelled by an on–off ... more This paper studies a two-level traffic shaper. The cell arrival process is modelled by an on–off fluid source. The output rate of the shaper is regulated by the content of a finite token bank. Our interest focuses on the stationary joint distribution of the content of the cell buffer and the token bank, from which various performance measures such as
We consider the joint dynamics of the instantaneous and exponentially averaged queue length in an... more We consider the joint dynamics of the instantaneous and exponentially averaged queue length in an M/M/1/K queue. A system of ordinary differential equations is derived for the joint stationary distribution of the instantaneous and the exponentially averaged queue length. The solution of the system of equations is obtained in a few special cases. Three different numerical approaches are presented to find the stationary distribution in the general case. Some results obtained with the numerical methods are presented and the efficiency of the numerical approaches is discussed. In addition, we describe how the model can be extended to a more complex situation which contains a rejection mechanism that randomly drops incoming customers with a dropping probability that depends on the current state of the averaged queue length.
We consider the joint dynamics of the instantaneous and exponentially averaged queue length in an... more We consider the joint dynamics of the instantaneous and exponentially averaged queue length in an M/M/1/K queue. A system of ordinary differential equations is derived for the joint stationary distribution of the instantaneous and the exponentially averaged queue length. The solution of the system of equations is obtained in a few special cases. Three different numerical approaches are presented to find the stationary distribution in the general case. Some results obtained with the numerical methods are presented and the efficiency of the numerical approaches is discussed. In addition, we describe how the model can be extended to a more complex situation which contains a rejection mechanism that randomly drops incoming customers with a dropping probability that depends on the current state of the averaged queue length.
queueing model with gated random order of service. In this service discipline customers are first... more queueing model with gated random order of service. In this service discipline customers are first gathered in an unordered waiting room before they are put in random order in an ordered service queue at the moment that this latter queue becomes empty. The second model is a tandem queue with coupled processors. The system consists of two stations with Poisson
. In this note we consider the fluid queue driven by an M=M=1 queue as analysed byVirtamo and Nor... more . In this note we consider the fluid queue driven by an M=M=1 queue as analysed byVirtamo and Norros [5]. We show that the stationary buffer content in this model can be easilyanalysed by looking at embedded time points. This approach gives the stationary buffer contentdistribution in terms of the modified Bessel function of the first kind of order one.
In this paper we determine the equilibrium distribution for a general class of Markov processes o... more In this paper we determine the equilibrium distribution for a general class of Markov processes on a semi-infinite strip. We expose a method to express the equilibrium probabilities of the Markov process as a finite sum of terms, which are geometric in the unbounded variable. The geometric factors are the roots inside the unit circle of a determinantal equation. By
We focus on a particular connection between queueing and risk models in a multi-dimensional setti... more We focus on a particular connection between queueing and risk models in a multi-dimensional setting. We first consider the joint workload process in a queueing model with parallel queues and simultaneous arrivals at the queues. For the case that the service times are ordered (from largest in the first queue to smallest in the last queue) we obtain the Laplace-Stieltjes transform of the joint stationary workload distribution. Using a multivariate duality argument between queueing and risk models, this also gives the Laplace transform of the survival probability of all books in a multivariate risk model with simultaneous claim arrivals and the same ordering between claim sizes. Other features of the paper include a stochastic decomposition result for the workload vector, and an outline how the two-dimensional risk model with a general two-dimensional claim size distribution (hence without ordering of claim sizes) is related to a known Riemann boundary value problem. 2010 Mathematics Subject Classification. 60K25; 91B30. Key words and phrases. Queues with simultaneous arrivals, workload, stochastic decomposition, duality, multivariate risk model. The authors gratefully acknowledge stimulating discussions with Yifan Xu (Binghamton University). Serban Badila is being supported by Project 613.001.017 of the Netherlands Organisation for Scientific Research (NWO).
We investigate Markovian queues that are examined by a controller at random times determined by a... more We investigate Markovian queues that are examined by a controller at random times determined by a Poisson process. Upon examination, the controller sets the service speed to be equal to the minimum of the current number of customers in the queue and a certain maximum service speed; this service speed prevails until the next examination time. We study the resulting two-dimensional Markov process of queue length and server speed, in particular two regimes with time scale separation, specifically for infinitely frequent and infinitely long examination times. In the intermediate regime the analysis proves to be extremely challenging. To gain further insight into the model dynamics we then analyse two variants of the model in which the controller is just an observer and does not change the speed of the server.
In this paper we discuss some aspects of the asymptotic behavior of Discrete Event Dynamic System... more In this paper we discuss some aspects of the asymptotic behavior of Discrete Event Dynamic Systems (DEDS), in which the activity times are random variables. The main result is that a central limit theorem holds for DEDS and consequently that the cycletime of the system is asymptotically normally distributed.
In the paper a probabilistic coupling between the M/G/1 processor sharing queue and the M/M/1 fee... more In the paper a probabilistic coupling between the M/G/1 processor sharing queue and the M/M/1 feedback queue, with general feedback probabilities, is established. This coupling is then used to prove the almost sure convergence of sojourn times in the feedback model to sojourn times in the M/G/1 processor sharing queue. Using the theory of regenerative processes it follows that for stable queues the stationary distribution of the sojourn time in the feedback model converges in law to the corresponding distribution in the processor sharing model. The results do not depend on Poisson arrival times, but are also valid for general arrival processes.
Journal of Statistical Mechanics: Theory and Experiment, 2018
DOI to the publisher's website. • The final author version and the galley proof are versions of t... more DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal. If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the "Taverne" license above, please follow below link for the End User Agreement:
This paper studies a two-level traffic shaper. The cell arrival process is modelled by an on–off ... more This paper studies a two-level traffic shaper. The cell arrival process is modelled by an on–off fluid source. The output rate of the shaper is regulated by the content of a finite token bank. Our interest focuses on the stationary joint distribution of the content of the cell buffer and the token bank, from which various performance measures such as
We consider the joint dynamics of the instantaneous and exponentially averaged queue length in an... more We consider the joint dynamics of the instantaneous and exponentially averaged queue length in an M/M/1/K queue. A system of ordinary differential equations is derived for the joint stationary distribution of the instantaneous and the exponentially averaged queue length. The solution of the system of equations is obtained in a few special cases. Three different numerical approaches are presented to find the stationary distribution in the general case. Some results obtained with the numerical methods are presented and the efficiency of the numerical approaches is discussed. In addition, we describe how the model can be extended to a more complex situation which contains a rejection mechanism that randomly drops incoming customers with a dropping probability that depends on the current state of the averaged queue length.
We consider the joint dynamics of the instantaneous and exponentially averaged queue length in an... more We consider the joint dynamics of the instantaneous and exponentially averaged queue length in an M/M/1/K queue. A system of ordinary differential equations is derived for the joint stationary distribution of the instantaneous and the exponentially averaged queue length. The solution of the system of equations is obtained in a few special cases. Three different numerical approaches are presented to find the stationary distribution in the general case. Some results obtained with the numerical methods are presented and the efficiency of the numerical approaches is discussed. In addition, we describe how the model can be extended to a more complex situation which contains a rejection mechanism that randomly drops incoming customers with a dropping probability that depends on the current state of the averaged queue length.
queueing model with gated random order of service. In this service discipline customers are first... more queueing model with gated random order of service. In this service discipline customers are first gathered in an unordered waiting room before they are put in random order in an ordered service queue at the moment that this latter queue becomes empty. The second model is a tandem queue with coupled processors. The system consists of two stations with Poisson
. In this note we consider the fluid queue driven by an M=M=1 queue as analysed byVirtamo and Nor... more . In this note we consider the fluid queue driven by an M=M=1 queue as analysed byVirtamo and Norros [5]. We show that the stationary buffer content in this model can be easilyanalysed by looking at embedded time points. This approach gives the stationary buffer contentdistribution in terms of the modified Bessel function of the first kind of order one.
In this paper we determine the equilibrium distribution for a general class of Markov processes o... more In this paper we determine the equilibrium distribution for a general class of Markov processes on a semi-infinite strip. We expose a method to express the equilibrium probabilities of the Markov process as a finite sum of terms, which are geometric in the unbounded variable. The geometric factors are the roots inside the unit circle of a determinantal equation. By
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