Polling systems and multitype branching processes

The joint queue length process in polling systems with and without switchover times is studied. If the service discipline in each queue satisfies a certain property it is shown that the joint queue length process at polling instants of a fixed queue is a multitype branching process (MTBP) with immigration. In the case of polling models with switchover times, it turns out that we are dealing with an MTBP with immigration in each state, whereas in the case of polling models without switchover times we are dealing with an MTBP with immigration in state zero. The theory of MTBPs leads to expressions for the generating function of the joint queue length process at polling instants. Sufficient conditions for ergodicity and moment calculations are also given

Approximation of discrete-time polling systems via structured Markov chains

We devise an approximation of the marginal queue length distribution in discrete-time polling systems with batch arrivals and fixed packet sizes. The polling server uses the Bernoulli service discipline and Markovian routing. The 1-limited and exhaustive service disciplines are special cases of the Bernoulli service discipline, and traditional cyclic routing is a special case of Markovian routing. The key step of our approximation is the translation of the polling system to a structured Markov chain, while truncating all but one queue. Numerical experiments show that the approximation is very accurate in general. Our study is motivated by networks on chips with multiple masters (e.g., processors) sharing a single slave (e.g., memory)

A saturated tree network of polling stations with flow control

We consider a saturated tree network with flow control. The network consists of two layers of polling stations, and all polling stations use the random polling service discipline. We obtain the equilibrium distribution of the network using a Markov chain approach. This equilibrium distribution can be used to efficiently compute the division of throughput over packets from different sources. Our study shows that this throughput division is determined by an interaction between the flow control limits, buffer sizes, and the service discipline parameters. A numerical study provides more insight in this interaction. The study is motivated by networks on chips where multiple masters share a single slave, operating under flow control

Duality and equivalencies in closed tandem queueing networks

Equivalence relations between closed tandem queueing networks are established. Four types of models are under consideration: single-server infinite-capacity buffer queues, infinite-server queues with resequencing, single-server unit-capacity buffer queues with blocking before service, and single-server unit-capacity buffer queues with blocking after service. Using a customer/server duality we show that in a network consisting of single-server infinite-capacity-buffer queues, customer-dependent service times and server-dependent service times yield equivalent performance characteristics. We further show that for closed tandem queueing networks, a system consisting of single-server infinite-capacity-buffer queues (resp. infinite-server queues with resequencing) and a system consisting of single-serverunit-capacity-buffer queues with blocking before service (resp. blocking after service) have equivalent performance behaviors. As applications of these equivalence properties, we obtain new results on the analysis of symmetric closed tandem networks, where all the service times are independent and identically distributed random variables. In particular, we obtain a closed-form expression for the throughput of networks with unit-capacity-buffer queues and blocking before service when the service times are exponentially distributed. We also prove the monotonicity of throughput (of queues) with respect to the number of queues and number of customers in these models. This last property in turn implies the existence of nonzero asymptotic throughput when the number of queues and number of customers go to infinity. Keywords: Equivalence, customer/server duality, closed tandem queueing network, infinite server with resequencing, queues with blocking, customer-dependent service, symmetric queueing network

Condition based maintenance policies under imperfect maintenance at scheduled and unscheduled opportunities

Motivated by the cost savings that can be obtained by sharing resources in a network context, we consider a stylized, yet representative model, for the coordination of maintenance and service logistics for a geographic network of assets. Capital assets, such as wind turbines in a wind park, require maintenance throughout their long lifetimes. Two types of preventive maintenance are considered: planned maintenance at periodic, scheduled opportunities, and opportunistic maintenance at unscheduled opportunities. The latter type of maintenance arises due to the network context: when an asset in the network fails, this constitutes an opportunity for preventive maintenance for the other assets in the network. \u3cbr/\u3eSo as to increase the realism of the model at hand and its applicability to various sectors, we consider the option of not-deferring and of deferring planned maintenance after the occurrence of opportunistic maintenance. We also assume that preventive maintenance may not always restore the condition of the system to `as good as new'. By formulating this problem as a semi-Markov decision process, we characterize the optimal policy as a control limit policy (depending on the remaining time until the next planned maintenance) that indicates on the one hand when it is optimal to perform preventive maintenance and on the other hand when maintenance resources should be shared if an opportunity in the network arises. In order to facilitate managerial insights on the effect of each parameter on the cost, we provide a closed-form expression for the long-run rate of cost for any given control limit policy (depending on the remaining time until the next planned maintenance) and compare the costs (under the optimal policy) to these of sub-optimal policies that neglect the opportunity for resource sharing. We illustrate our findings using data from the wind energy industry

Condition-based maintenance at both scheduled and unscheduled opportunities

Motivated by original equipment manufacturer (OEM) service and maintenance practices we consider a single component subject to replacements at failure instances and two types of preventive maintenance opportunities: scheduled, which occur due to periodic system reviews of the equipment, and unscheduled, which occur due to failures of other components in the system. Modelling the state of the component appropriately and incorporating a realistic cost structure for corrective maintenance as well as condition-based maintenance (CBM), we derive the optimal CBM policy. In particular, we show that the optimal long-run average cost policy for the model at hand is a control-limit policy, where the control limit depends on the time until the next scheduled opportunity. Furthermore, we explicitly calculate the long-run average cost for any given control-limit time dependent policy and compare various policies numerically

Condition-based maintenance policies under imperfect maintenance at scheduled and unscheduled opportunities

\u3cp\u3eMotivated by the cost savings that can be obtained by sharing resources in a network context, we consider a stylized, yet representative, model for the coordination of maintenance and service logistics for a geographic network of assets. Capital assets, such as wind turbines in a wind park, require maintenance throughout their long lifetimes. Two types of preventive maintenance are considered: planned maintenance at periodic, scheduled opportunities, and opportunistic maintenance at unscheduled opportunities. The latter type of maintenance arises due to the network context: When an asset in the network fails, this constitutes an opportunity for preventive maintenance for the other assets in the network. So as to increase the realism of the model at hand and its applicability to various sectors, we consider the option of not-deferring and of deferring planned maintenance after the occurrence of opportunistic maintenance. We also assume that preventive maintenance may not always restore the condition of the system to ‘as good as new.’ By formulating this problem as a semi-Markov decision process, we characterize the optimal policy as a control limit policy (depending on the remaining time until the next planned maintenance) that indicates on the one hand when it is optimal to perform preventive maintenance and on the other hand when maintenance resources should be shared if an opportunity in the network arises. In order to facilitate managerial insights on the effect of each parameter on the cost, we provide a closed-form expression for the long-run rate of cost for any given control limit policy (depending on the remaining time until the next planned maintenance) and compare the costs (under the optimal policy) to those of suboptimal policies that neglect the opportunity for resource sharing. We illustrate our findings using data from the wind energy industry.\u3c/p\u3

The single server queue with synchronized services

We consider a single server queueing system with generally distributed synchronized services. More specifically, customers arrive according to a Poisson process and there is a single server that provides service, if there is at least one customer present in the system. Upon the initialization of a service, all present customers start to receive service simultaneously. We consider the gated version of the model, that is, customers who arrive during a service time do not receive service immediately but wait for the beginning of the next service time. At service completion epochs, all served customers decide simultaneously and independently whether they will leave the system or stay for another service. The probability that a served customer gets another service is the same for all customers. We study the model and derive its main performance measures that include the equilibrium distribution of the number of customers at service completion epochs and in continuous time, the busy period and the sojourn time distributions. Moreover, we prove some limiting results regarding the behavior of the system in the extreme cases of the synchronization level. Several variants and extensions of the model are also discussed

The M/G/1 processor sharing queue as the almost sure limit of feedback queues

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