1,474 research outputs found
Super-Exponential Solution in Markovian Supermarket Models: Framework and Challenge
Marcel F. Neuts opened a key door in numerical computation of stochastic
models by means of phase-type (PH) distributions and Markovian arrival
processes (MAPs). To celebrate his 75th birthday, this paper reports a more
general framework of Markovian supermarket models, including a system of
differential equations for the fraction measure and a system of nonlinear
equations for the fixed point. To understand this framework heuristically, this
paper gives a detailed analysis for three important supermarket examples: M/G/1
type, GI/M/1 type and multiple choices, explains how to derive the system of
differential equations by means of density-dependent jump Markov processes, and
shows that the fixed point may be simply super-exponential through solving the
system of nonlinear equations. Note that supermarket models are a class of
complicated queueing systems and their analysis can not apply popular queueing
theory, it is necessary in the study of supermarket models to summarize such a
more general framework which enables us to focus on important research issues.
On this line, this paper develops matrix-analytical methods of Markovian
supermarket models. We hope this will be able to open a new avenue in
performance evaluation of supermarket models by means of matrix-analytical
methods.Comment: Randomized load balancing, supermarket model, matrix-analytic method,
super-exponential solution, density-dependent jump Markov process, Batch
Markovian Arrival Process (BMAP), phase-type (PH) distribution, fixed poin
Estimating process batch flow times in a two-stage stochastic flowshop with overlapping operations.
Processes; Time;
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Performance modelling of wormhole-routed hypercubes with bursty traffice and finite buffers
An open queueing network model (QNM) is proposed for wormhole-routed hypercubes with finite
buffers and deterministic routing subject to a compound Poisson arrival process (CPP) with geometrically
distributed batches or, equivalently, a generalised exponential (GE) interarrival time distribution. The GE/G/1/K
queue and appropriate GE-type flow formulae are adopted, as cost-effective building blocks, in a queue-by-queue
decomposition of the entire network. Consequently, analytic expressions for the channel holding time, buffering
delay, contention blocking and mean message latency are determined. The validity of the analytic approximations
is demonstrated against results obtained through simulation experiments. Moreover, it is shown that the wormholerouted
hypercubes suffer progressive performance degradation with increasing traffic variability (burstiness)
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
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