Exact Product Form Solutions Research Articles

Queueing analysis for operations modeling in port logistics

Purpose The use of queueing network models was stimulated by the appearance (1975) of the exact product form solution of a class of open, closed and mixed queueing networks obeying the local balance principle and solved, a few years later, by the popular mean value analysis algorithm (1980). Since then, research efforts have been produced to approximate solutions for non-exponential services and non-pure random mechanisms in customer processing and routing. The purpose of this paper is to examine the suitability of modeling choices and solution approaches consolidated in other domains with respect to two key logistic processes in container terminals. Design/methodology/approach In particular, the analytical solution of queueing networks is assessed for the vessel arrival-departure process and the container internal transfer process with respect to a real terminal of pure transshipment. Findings Numerical experiments show the extent to which a decomposition-based approximation, under fixed or state-dependent arrival rates, may be suitable for the approximate analysis of the queueing network models. Research limitations/implications The limitation of adopting exponential service time distributions and Poisson flows is highlighted. Practical implications Comparisons with a simulation-based solution deliver numerical evidence on the companion use of simulation in the daily practice of managing operations in a finite-time horizon under complex policies. Originality/value Discussion of some open modeling issues and encouraging results provide some guidelines on future research efforts and/or suitable adaption to container terminal logistics of the large body of techniques and algorithms available nowadays for supporting long-run decisions.

A survey of product form queueing networks with blocking and their equivalences

Queueing network models have been extensively used to represent and analyze resource sharing systems, such as production, communication and information systems. Queueing networks with blocking are used to represent systems with finite capacity resources and with resource constraints. Different blocking mechanisms have been defined and analyzed in the literature to represent distinct behaviors of real systems with limited resources. Exact product form solutions of queueing networks with blocking have been derived, under special constraints, for different blocking mechanisms. In this paper we present a survey of product form solutions of queueing networks with blocking and equivalence properties among different blocking network models. By using such equivalences we can extend product form solutions to queueing network models with different blocking mechanisms. The equivalence properties include relationships between open and closed product form queueing networks with different blocking mechanisms.

Exact analysis of queueing networks with multiple job classes and blocking-after-service

Two models of closed queueing networks with blocking-after-service and multiple job classes are analyzed. The first model is a network withN stations and each station has either type II or type III. The second model is a star-like queueing network, also called a central server model, in which the stations may have either type I or type IV, with the condition that the neighbors of these stations must be of type II or type III such that blocking will be caused only by this set of station types. Exact product form solutions are obtained for the equilibrium state probabilities in both models. Formulae for performance measures such as throughput and the mean number of jobs are also derived.

Computational algorithms for networks of queues with rejection blocking

Open, closed and mixed queueing networks with reversible routing, multiple job classes and rejection blocking are investigated. In rejection blocking networks blocking event occurs when upon completion of its service of a particular station's server, a job attempts to proceed to its next station. If, at that moment, its destination station is full, the job is rejected. The job goes back to the server of the source station and immediately receives a new service. This is repeated until the next station releases a job and a place becomes available. In the model jobs may change their class membership and general service time distributions depending on the job class are allowed. Two station types are considered: Either the scheduling discipline is symmetric, in which case the service time distributions are allowed to be general and dependent on the job class or the service time distributions at a station are all identical exponential distributions, in which case more general scheduling disciplines are allowed. An exact product form solution for equilibrium state probabilities is presented. Using the exact product form solution of the equilibrium state distribution, algorithms for computation of performance measures, such as mean number of jobs and throughputs, are derived. The complexity of the algorithms is discussed.

Dual and selfdual networks of queues with rejection blocking

Open and closed queueing networks with rejection blocking are investigated. Using the concept of holes, a duality is derived for pairs of networks. This duality equates equilibrium state probabilities and throughputs of the given network and its dual. As an application an exact product form solution for certain closed queueing networks with rejection blocking are derived. In this case, duality provides a simple method to compute performance measures. In some cases, a network and its dual have the same structure. Duality provides relations among performance measures of different stations in this case.

Exact Product Form Solution for Queueing Networks with Blocking

This work investigates closed queueing networks with blocking composed of two stations with multiple servers. Blocking occurs when a job wanting to enter a full station is forced to remain in its source station, thus blocking the source station until room is available at the destination station. This type of blocking is known as classical blocking. We show that, for a two-station closed queueing network with blocking, there exists an equivalent nonblocking network with the same state space structure. Utilizing this concept, we demonstrate that two-station closed queueing networks with blocking have product form solutions.