M/G/∞ with batch arrivals

Let p_∞(n) be the distribution of the number N(∞) in the system at ergodicity for systems with an infinite number of servers, batch arrivals with general batch size distribution and general holding times. This distribution is of importance to a variety of studies in congestion theory, inventory and storage systems. To obtain this distribution, a more general problem is addressed. In this problem, each epoch of a Poisson process gives rise to an independent stochastic function on the lattice of integers, which may be viewed as stochastic impulse response. A continuum analogue to the lattice process is also provided.     

Time-limited polling systems with batch arrivals and?phase-type service times

In this paper, we develop a general framework to analyze polling systems with either the autonomous-server or the time-limited service discipline. According to the autonomous-server discipline, the server continues servicing a queue for a certain period of time. According to the time-limited service discipline, the server continues servicing a queue for a certain period of time or until the queue becomes empty, whichever occurs first. We consider Poisson batch arrivals and phase-type service times. It is known that these disciplines do not satisfy the well-known branching property in polling systems. Therefore, hardly any exact results exist in the literature. Our strategy is to apply an iterative scheme that is based on relating in closed-form the joint queue-lengths at the beginning and the end of a server visit to a queue. These kernel relations are derived using the theory of absorbing Markov chains.     

On the GI/M/∞ queue with batch arrivals of constant size

In this note, the GI/M/ queue with batch arrivals of constant sizek is investigated. It is shown that the stationary probabilities that an arriving batch findsi customers in the system can be computed in terms of the corresponding binomial moments (Jordan's formula), which are determined by a recursive relation. This generalizes well-known results by Takács [12] for GI/M/. Furthermore, relations between batch arrival- and time-stationary probabilities are given.     

Real time disruption management for a two-stage batch production–inventory system with reliability considerations

In this research, a two-stage batch production–inventory system is introduced. In this system, the production may be disrupted, for a given period of time, either at one or both stages. In this paper, firstly, a mathematical model has been developed to suggest a recovery plan for a single occurrence of disruption at either stage. Secondly, multiple disruptions have been considered, for which a new disruption may or may not affect the recovery plan of earlier disruptions. We propose a new approach that deals with a series of disruptions over a period of time, which can be implemented for disruption recovery on a real time basis. In this approach, the model formulated for single disruption has been integrated to generate initial solutions for individual disruptions and the solutions have been revised for multiple dependent disruptions with changed parameters. With the proposed approach, an optimal recovery plan can be obtained in real time, whenever the production system experiences either a sudden disruption or a series of disruptions, at different points in time. Some numerical examples and a real-world case study are presented to explain the benefits of our proposed approach.     

Numerical identification of constitutive functions in scalar nonlinear convection–diffusion equations with application to batch sedimentation

A fast and simple method for the identification of nonlinear constitutive functions in scalar convection–diffusion equations is presented. No a priori information is needed on the form of the constitutive functions, which are obtained as continuous piecewise affine functions. Accurate and frequent measurements in space and time are required. Synthetic data of batch sedimentation of particles in a liquid and traffic flow are chosen as examples where a convective flux function and a function modelling compression are identified. Real data should first undergo a denoising procedure, which is also presented. It consists of a sequence of convex optimization problems, whose constraints originate from fundamental physical properties. The methodology is applied on data from a batch sedimentation experiment of activated sludge in wastewater treatment.     

On the gi/m/∞ service system with batch arrivals and different types of service distributions

We study the infinite-server system with batch arrivals ands different types of customers. With probabilityp _i an arriving customer is of typei (i=1,..., s) and requires an exponentially distributed service time with parameter _i (G ^GI /M ₁ ...M _s /). For theGI ^GI /M ₁...M _s / system it is shown that the binomial moments of thes-variate distribution of the number of type-i customers in the system at batch arrival epochs are determined by a recurrence relation and, in steady state, can be computed recursively. Furthermore, forG ^GI /M ₁...M _s /, relations between the distributions (and their binomial moments) of the system size vector at batch arrival and random epochs are given. Thus, earlier results by Takács [14], Gastwirth [9], Holman et al. [11], Brandt et al. [3] and Franken [6] are generalized.     

Obtaining confidence limits for safe creep life in the presence of multi batch hierarchical databases: An application to 18Cr–12Ni–Mo steel

The creep and creep rupture properties of 18Cr–12Ni–Mo steel tubes have been analysed using the Wilshire equations. The observed behaviour patterns are then briefly discussed in terms of the dislocation processes governing creep strain accumulation. A suitable statistical framework for analysing both the single and multi batch data available on this material is then specified. It is shown that ignoring the hierarchical nature present in many creep data bases, which has been the approach used until now when using the Wilshire equations, leads to a serious and significant underestimate of the predicted safe life for this material. The model allows accurate predictions, with associated levels of confidence, of long-term properties by extrapolation of short-term test results for this steel.     

<Emphasis Type="Italic">GI</Emphasis>/<Emphasis Type="Italic">G</Emphasis>/1/∞ batch arrival queueing system with a single exponential vacation

In the article the queueing system of GI/G/1 type with batch arrival of customers and a single exponentially distributed vacation period at the end of every busy period is considered. Basic characteristics of transient state of the system are investigated: the first busy period, the first vacation period and the number of customers served during the first busy period. New results for the Laplace transform of the joint distribution of these three variables are obtained in dependence on the initial conditions of the system. This material is based upon work supported by the Polish Ministry of Scientific Research and Information Technology under Grant No. 3 T11C 014 26.