1,410 research outputs found
Arrival first queueing networks with applications in kanban production systems
In this paper we introduce a new class of queueing networks called {\it arrival first networks}. We characterise its transition rates and derive the relationship between arrival rules, linear partial balance equations, and product form stationary distributions. This model is motivated by production systems operating under a kanban protocol. In contrast with the conventional {\em departure first networks}, where a transition is initiated by service completion of items at the originating nodes that are subsequently routed to the destination nodes (push system), in an arrival first network a transition is initiated by the destination nodes of the items and subsequently those items are processed at and removed from the originating nodes (pull system). These are similar to the push and pull systems in manufacturing systems
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
Loss systems in a random environment
We consider a single server system with infinite waiting room in a random
environment. The service system and the environment interact in both
directions. Whenever the environment enters a prespecified subset of its state
space the service process is completely blocked: Service is interrupted and
newly arriving customers are lost. We prove an if-and-only-if-condition for a
product form steady state distribution of the joint queueing-environment
process. A consequence is a strong insensitivity property for such systems.
We discuss several applications, e.g. from inventory theory and reliability
theory, and show that our result extends and generalizes several theorems found
in the literature, e.g. of queueing-inventory processes.
We investigate further classical loss systems, where due to finite waiting
room loss of customers occurs. In connection with loss of customers due to
blocking by the environment and service interruptions new phenomena arise.
We further investigate the embedded Markov chains at departure epochs and
show that the behaviour of the embedded Markov chain is often considerably
different from that of the continuous time Markov process. This is different
from the behaviour of the standard M/G/1, where the steady state of the
embedded Markov chain and the continuous time process coincide.
For exponential queueing systems we show that there is a product form
equilibrium of the embedded Markov chain under rather general conditions. For
systems with non-exponential service times more restrictive constraints are
needed, which we prove by a counter example where the environment represents an
inventory attached to an M/D/1 queue. Such integrated queueing-inventory
systems are dealt with in the literature previously, and are revisited here in
detail
Load Balancing in Large-Scale Systems with Multiple Dispatchers
Load balancing algorithms play a crucial role in delivering robust
application performance in data centers and cloud networks. Recently, strong
interest has emerged in Join-the-Idle-Queue (JIQ) algorithms, which rely on
tokens issued by idle servers in dispatching tasks and outperform power-of-
policies. Specifically, JIQ strategies involve minimal information exchange,
and yet achieve zero blocking and wait in the many-server limit. The latter
property prevails in a multiple-dispatcher scenario when the loads are strictly
equal among dispatchers. For various reasons it is not uncommon however for
skewed load patterns to occur. We leverage product-form representations and
fluid limits to establish that the blocking and wait then no longer vanish,
even for arbitrarily low overall load. Remarkably, it is the least-loaded
dispatcher that throttles tokens and leaves idle servers stranded, thus acting
as bottleneck.
Motivated by the above issues, we introduce two enhancements of the ordinary
JIQ scheme where tokens are either distributed non-uniformly or occasionally
exchanged among the various dispatchers. We prove that these extensions can
achieve zero blocking and wait in the many-server limit, for any subcritical
overall load and arbitrarily skewed load profiles. Extensive simulation
experiments demonstrate that the asymptotic results are highly accurate, even
for moderately sized systems
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
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