1,470 research outputs found
A Numerical Approach to Stability of Multiclass Queueing Networks
The Multi-class Queueing Network (McQN) arises as a natural multi-class
extension of the traditional (single-class) Jackson network. In a single-class
network subcriticality (i.e. subunitary nominal workload at every station)
entails stability, but this is no longer sufficient when jobs/customers of
different classes (i.e. with different service requirements and/or routing
scheme) visit the same server; therefore, analytical conditions for stability
of McQNs are lacking, in general. In this note we design a numerical
(simulation-based) method for determining the stability region of a McQN, in
terms of arrival rate(s). Our method exploits certain (stochastic) monotonicity
properties enjoyed by the associated Markovian queue-configuration process.
Stochastic monotonicity is a quite common feature of queueing models and can be
easily established in the single-class framework (Jackson networks); recently,
also for a wide class of McQNs, including first-come-first-serve (FCFS)
networks, monotonicity properties have been established. Here, we provide a
minimal set of conditions under which the method performs correctly.
Eventually, we illustrate the use of our numerical method by presenting a set
of numerical experiments, covering both single and multi-class networks
Optimization of polling systems with Bernoulli schedules
Optimization;Polling Systems;Queueing Theory;operations research
Multiclass multiserver queueing system in the Halfin-Whitt heavy traffic regime. Asymptotics of the stationary distribution
We consider a heterogeneous queueing system consisting of one large pool of
identical servers, where is the scaling parameter. The
arriving customers belong to one of several classes which determines the
service times in the distributional sense. The system is heavily loaded in the
Halfin-Whitt sense, namely the nominal utilization is where
is the spare capacity parameter. Our goal is to obtain bounds on the
steady state performance metrics such as the number of customers waiting in the
queue . While there is a rich literature on deriving process level
(transient) scaling limits for such systems, the results for steady state are
primarily limited to the single class case.
This paper is the first one to address the case of heterogeneity in the
steady state regime. Moreover, our results hold for any service policy which
does not admit server idling when there are customers waiting in the queue. We
assume that the interarrival and service times have exponential distribution,
and that customers of each class may abandon while waiting in the queue at a
certain rate (which may be zero). We obtain upper bounds of the form
on both and the number of idle servers. The bounds
are uniform w.r.t. parameter and the service policy. In particular, we show
that . Therefore, the
sequence is tight and has a uniform exponential tail
bound. We further consider the system with strictly positive abandonment rates,
and show that in this case every weak limit of
has a sub-Gaussian tail. Namely .Comment: 21 page
Monotonicity and error bounds for networks of Erlang loss queues
Networks of Erlang loss queues naturally arise when modelling finite communication systems without delays, among which, most notably are (i) classical circuit switch telephone networks (loss networks) and (ii) present-day wireless mobile networks. Performance measures of interest such as loss probabilities or throughputs can be obtained from the steady state distribution. However, while this steady state distribution has a closed product form expression in the first case (loss networks), it does not have one in the second case due to blocked (and lost) handovers. Product form approximations are therefore suggested. These approximations are obtained by a combined modification of both the state space (by a hypercubic expansion) and the transition rates (by extra redial rates). It will be shown that these product form approximations lead to (1) upper bounds for loss probabilities and \ud
(2) analytic error bounds for the accuracy of the approximation for various performance measures.\ud
The proofs of these results rely upon both monotonicity results and an analytic error bound method as based on Markov reward theory. This combination and its technicalities are of interest by themselves. The technical conditions are worked out and verified for two specific applications:\ud
(1)• pure loss networks as under (2)• GSM networks with fixed channel allocation as under.\ud
The results are of practical interest for computational simplifications and, particularly, to guarantee that blocking probabilities do not exceed a given threshold such as for network dimensioning
Monotonicity and error bounds for networks of Erlang loss queues
Networks of Erlang loss queues naturally arise when modelling finite communication systems without delays, among which, most notably\ud
(i) classical circuit switch telephone networks (loss networks) and\ud
(ii) present-day wireless mobile networks.\ud
\ud
Performance measures of interest such as loss probabilities or throughputs can be obtained from the steady state distribution. However, while this steady state distribution has a closed product form expression in the first case (loss networks), it has not in the second case due to blocked (and lost) handovers. Product form approximations are therefore suggested. These approximations are obtained by a combined modification of both the state space (by a hyper cubic expansion) and the transition rates (by extra redial rates). It will be shown that these product form approximations lead to\ud
\ud
- secure upper bounds for loss probabilities and\ud
- analytic error bounds for the accuracy of the approximation for various performance measures.\ud
\ud
The proofs of these results rely upon both monotonicity results and an analytic error bound method as based on Markov reward theory. This combination and its technicalities are of interest by themselves. The technical conditions are worked out and verified for two specific applications:\ud
\ud
- pure loss networks as under (i)\ud
- GSM-networks with fixed channel allocation as under (ii).\ud
\ud
The results are of practical interest for computational simplifications and, particularly, to guarantee blocking probabilities not to exceed a given threshold such as for network dimensioning.\u
Simple bounds for queueing systems with breakdowns
Computationally attractive and intuitively obvious simple bounds are proposed for finite service systems which are subject to random breakdowns. The services are assumed to be exponential. The up and down periods are allowed to be generally distributed. The bounds are based on product-form modifications and depend only on means. A formal proof is presented. This proof is of interest in itself. Numerical support indicates a potential usefulness for quick engineering and performance evaluation purposes
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