In this paper we study a system consisting of c parallel servers with possibly different service ... more In this paper we study a system consisting of c parallel servers with possibly different service rates. Jobs arrive according to a Poisson stream and generate an exponentially distributed workload. An arriving job joins the shortest queue, where in case of multiple shortest queues, one of these queues is selected according to some arbitrary probability distribution. If the maximum difference between the lengths of the c queues exceeds some threshold value T, then one job switches from the longest to the shortest queue, where in case of multiple longest queues, the queue loosing a job is selected according to some arbitrary probability distribution. It is shown that the matrix-geometric approach is very well suited to find the equilibrium probabilities of the queue lengths. The interesting point is that a proper choice for the state space partitioning depends on the aspect one is interested in. Using one partitioning of the state space an explicit ergodicity condition can be derived ...
We consider a polling system with two queues, exhaustive service, no switchover times, and expone... more We consider a polling system with two queues, exhaustive service, no switchover times, and exponential service times with rate µ in each queue. The waiting cost depends on the position of the queue relative to the server: it costs a customercper time unit to wait in the busy queue (where the server is) anddper time unit in the idle queue (where there is no server). Customers arrive according to a Poisson process with rate λ. We study the control problem of how arrivals should be routed to the two queues in order to minimize the expected waiting costs and characterize individually and socially optimal routeing policies under three scenarios of available information at decision epochs: no, partial, and complete information. In the complete information case, we develop a new iterative algorithm to determine individually optimal policies (which are symmetric Nash equilibria), and show that such policies can be described by a switching curve. We use Markov decision processes to compute t...
Communications in Statistics. Stochastic Models, 1990
... 2. Equilibrium Equations For simplicity of notation the exponential servers have service time... more ... 2. Equilibrium Equations For simplicity of notation the exponential servers have service times with unit mean and the Poisson anival process has a rate 2p with 0 < p < 1. The parallel queue system can be represented by a continuous time Markov process, whose state space ...
Probability in the Engineering and Informational Sciences, 2009
We consider a two-queue model with state-dependent setups, in which a single server alternately s... more We consider a two-queue model with state-dependent setups, in which a single server alternately serves the two queues. The high-priority queue is served exhaustively, whereas the low-priority queue is served according to the k-limited strategy. A setup at a queue is incurred only if there are customers waiting at the polled queue. We obtain the transforms of the queue length and sojourn time distributions under the assumption of Poisson arrivals, generally distributed service times, and generally distributed setup times. The interest for this model is fueled by an application in the field of logistics. It is shown how the results of this analysis can be applied in the evaluation of a stochastic two-item single-capacity production system. From these results we can conclude that significant cost reductions are possible by bounding the production runs of the low-priority item, which indicates the potential of the k-limited service discipline as priority rule in production environments.
In this paper we study a system consisting of c parallel servers with possibly different service ... more In this paper we study a system consisting of c parallel servers with possibly different service rates. Jobs arrive according to a Poisson stream and generate an exponentially distributed workload. An arriving job joins the shortest queue, where in case of multiple shortest queues, one of these queues is selected according to some arbitrary probability distribution. If the maximum difference between the lengths of the c queues exceeds some threshold value T, then one job switches from the longest to the shortest queue, where in case of multiple longest queues, the queue loosing a job is selected according to some arbitrary probability distribution. It is shown that the matrix-geometric approach is very well suited to find the equilibrium probabilities of the queue lengths. The interesting point is that a proper choice for the state space partitioning depends on the aspect one is interested in. Using one partitioning of the state space an explicit ergodicity condition can be derived ...
We consider a polling system with two queues, exhaustive service, no switchover times, and expone... more We consider a polling system with two queues, exhaustive service, no switchover times, and exponential service times with rate µ in each queue. The waiting cost depends on the position of the queue relative to the server: it costs a customercper time unit to wait in the busy queue (where the server is) anddper time unit in the idle queue (where there is no server). Customers arrive according to a Poisson process with rate λ. We study the control problem of how arrivals should be routed to the two queues in order to minimize the expected waiting costs and characterize individually and socially optimal routeing policies under three scenarios of available information at decision epochs: no, partial, and complete information. In the complete information case, we develop a new iterative algorithm to determine individually optimal policies (which are symmetric Nash equilibria), and show that such policies can be described by a switching curve. We use Markov decision processes to compute t...
Communications in Statistics. Stochastic Models, 1990
... 2. Equilibrium Equations For simplicity of notation the exponential servers have service time... more ... 2. Equilibrium Equations For simplicity of notation the exponential servers have service times with unit mean and the Poisson anival process has a rate 2p with 0 < p < 1. The parallel queue system can be represented by a continuous time Markov process, whose state space ...
Probability in the Engineering and Informational Sciences, 2009
We consider a two-queue model with state-dependent setups, in which a single server alternately s... more We consider a two-queue model with state-dependent setups, in which a single server alternately serves the two queues. The high-priority queue is served exhaustively, whereas the low-priority queue is served according to the k-limited strategy. A setup at a queue is incurred only if there are customers waiting at the polled queue. We obtain the transforms of the queue length and sojourn time distributions under the assumption of Poisson arrivals, generally distributed service times, and generally distributed setup times. The interest for this model is fueled by an application in the field of logistics. It is shown how the results of this analysis can be applied in the evaluation of a stochastic two-item single-capacity production system. From these results we can conclude that significant cost reductions are possible by bounding the production runs of the low-priority item, which indicates the potential of the k-limited service discipline as priority rule in production environments.
Uploads
Papers by I.j.b.f. Adan