3,936 research outputs found

    Fixed points for multi-class queues

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    Burke's theorem can be seen as a fixed-point result for an exponential single-server queue; when the arrival process is Poisson, the departure process has the same distribution as the arrival process. We consider extensions of this result to multi-type queues, in which different types of customer have different levels of priority. We work with a model of a queueing server which includes discrete-time and continuous-time M/M/1 queues as well as queues with exponential or geometric service batches occurring in discrete time or at points of a Poisson process. The fixed-point results are proved using interchangeability properties for queues in tandem, which have previously been established for one-type M/M/1 systems. Some of the fixed-point results have previously been derived as a consequence of the construction of stationary distributions for multi-type interacting particle systems, and we explain the links between the two frameworks. The fixed points have interesting "clustering" properties for lower-priority customers. An extreme case is an example of a Brownian queue, in which lower-priority work only occurs at a set of times of measure 0 (and corresponds to a local time process for the queue-length process of higher priority work).Comment: 25 page

    Waiting times in queueing networks with a single shared server

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    We study a queueing network with a single shared server that serves the queues in a cyclic order. External customers arrive at the queues according to independent Poisson processes. After completing service, a customer either leaves the system or is routed to another queue. This model is very generic and finds many applications in computer systems, communication networks, manufacturing systems, and robotics. Special cases of the introduced network include well-known polling models, tandem queues, systems with a waiting room, multi-stage models with parallel queues, and many others. A complicating factor of this model is that the internally rerouted customers do not arrive at the various queues according to a Poisson process, causing standard techniques to find waiting-time distributions to fail. In this paper we develop a new method to obtain exact expressions for the Laplace-Stieltjes transforms of the steady-state waiting-time distributions. This method can be applied to a wide variety of models which lacked an analysis of the waiting-time distribution until now

    Metastability of Queuing Networks with Mobile Servers

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    We study symmetric queuing networks with moving servers and FIFO service discipline. The mean-field limit dynamics demonstrates unexpected behavior which we attribute to the meta-stability phenomenon. Large enough finite symmetric networks on regular graphs are proved to be transient for arbitrarily small inflow rates. However, the limiting non-linear Markov process possesses at least two stationary solutions. The proof of transience is based on martingale techniques

    Analysis of a polling system modeling QoS differentiation in WLANs

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    This paper investigates a polling system with a random polling scheme, a 1-limited service discipline and deterministic service requirement modeling WLANs with QoS differentation capability. The system contains high and low priority queues that are distinguished via the probability of being served next. We propose a new iteration algorithm to approximate the waiting time of customers in the high and low priority queues. As shown by simulation results, our approximation is accurate for light to moderately loaded networks

    FCFS Parallel Service Systems and Matching Models

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    We consider three parallel service models in which customers of several types are served by several types of servers subject to a bipartite compatibility graph, and the service policy is first come first served. Two of the models have a fixed set of servers. The first is a queueing model in which arriving customers are assigned to the longest idling compatible server if available, or else queue up in a single queue, and servers that become available pick the longest waiting compatible customer, as studied by Adan and Weiss, 2014. The second is a redundancy service model where arriving customers split into copies that queue up at all the compatible servers, and are served in each queue on FCFS basis, and leave the system when the first copy completes service, as studied by Gardner et al., 2016. The third model is a matching queueing model with a random stream of arriving servers. Arriving customers queue in a single queue and arriving servers match with the first compatible customer and leave immediately with the customer, or they leave without a customer. The last model is relevant to organ transplants, to housing assignments, to adoptions and many other situations. We study the relations between these models, and show that they are closely related to the FCFS infinite bipartite matching model, in which two infinite sequences of customers and servers of several types are matched FCFS according to a bipartite compatibility graph, as studied by Adan et al., 2017. We also introduce a directed bipartite matching model in which we embed the queueing systems. This leads to a generalization of Burke's theorem to parallel service systems
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