The G/M/1 queue revisited

The G/M/1 queue is one of the classical models of queueing theory. The goal of this paper is two-fold: (a) To introduce new derivations of some well-known results, and (b) to present some new results for the G/M/1 queue and its variants. In particular, we pay attention to the G/M/1 queue with a set-up time at the start of each busy period, and the G/M/1 queue with exceptional first service. Dedicated to Arie Hordijk on his 65th birthday, in friendship and admiration.     

The (S−1,S) inventory model and its counterparts in queueing theory

We explore the relationship between the ($S?1,S$) inventory model and three well-known queueing models: the Erlang loss system, the machine-repair model and a two-node Jackson network. Exploiting this relationship allows us to obtain key performance measures of the ($S?1,S$) model, like the so-called virtual outdating time, the number of items on the shelf in steady state, the long-run rate of unsatisfied demands and the distribution of the empty shelf period.     

The Single Server Queue with Mixing Dependencies

Methodology and Computing in Applied Probability - We study a single server queue, where a certain type of dependence is introduced between the service times, or between the inter-arrival times, or...     

Decomposition results for stochastic storage processes and queues with alternating Lévy inputs

In this paper we generalize known workload decomposition results for Lévy queues with secondary jump inputs and queues with server vacations or service interruptions. Special cases are polling systems with either compound Poisson or more general Lévy inputs. Our main tools are new martingale results, which have been derived in a companion paper.     

Disentangling Reaction Processes of Zeolites within Single‐Oriented Channels

Establishing structure–reactivity relationships for specific channel orientations of zeolites is vital to developing new, superior materials for various applications, including oil and gas conversion processes. Herein, a well‐defined model system was developed to build structure–reactivity relationships for specific zeolite‐channel orientations during various catalytic reaction processes, for example, the methanol‐ and ethanol‐to‐hydrocarbons (MTH and ETH) process as well as oligomerization reactions. The entrapped and effluent hydrocarbons from single‐oriented zeolite ZSM‐5 channels during the MTH process were monitored by using operando UV/Vis diffuse reflectance spectroscopy (DRS) and on‐line mass spectrometry (MS), respectively. The results reveal that the straight channels favor the formation of internal coke, promoting the aromatic cycle. Furthermore, the sinusoidal channels produce aromatics, (e.g., toluene) that further grow into larger polyaromatics (e.g., graphitic coke) leading to deactivation of the zeolites. This underscores the importance of careful engineering of materials to suppress coke formation and tune product distribution by rational control of the location of zeolite acid sites and crystallographic orientations.     

On Lévy-driven vacation models with correlated busy periods and service interruptions

This paper considers queues with server vacations, but departs from the traditional setting in two ways: (i) the queueing model is driven by Lévy processes rather than just compound Poisson processes; (ii) the vacation lengths depend on the length of the server’s preceding busy period. Regarding the former point: the Lévy process active during the busy period is assumed to have no negative jumps, whereas the Lévy process active during the vacation is a subordinator. Regarding the latter point: where in a previous study (Boxma et al. in Probab. Eng. Inf. Sci. 22:537–555, 2008) the durations of the vacations were positively correlated with the length of the preceding busy period, we now introduce a dependence structure that may give rise to both positive and negative correlations. We analyze the steady-state workload of the resulting queueing (or: storage) system, by first considering the queue at embedded epochs (viz. the beginnings of busy periods). We show that this embedded process does not always have a proper stationary distribution, due to the fact that there may occur an infinite number of busy-vacation cycles in a finite time interval; we specify conditions under which the embedded process is recurrent. Fortunately, irrespective of whether the embedded process has a stationary distribution, the steady-state workload of the continuous-time storage process can be determined. In addition, a number of ramifications are presented. The theory is illustrated by several examples.     

Joint distribution of sojourn time and queue length in the M/G/1 queue with (in) finite capacity

For the M/G/1 queue we study the joint distribution of the number of customers x present immediately before an arrival epoch and of the residual service time ζ of the customer in service at this epoch. The correlation coefficient ? (x, ζ) is shown to be positive (negative) when the service time distribution is DFR (IFR). The result for the joint distribution of x and ζ leads to the joint distribution of x, of the sojourn time s of the arriving customer and of the number of customers z left behind by this customer at his departure. ?(x, s), ?(z, s) and ?(x, z) are shown to be positive; ?(x, s) and ?(z, s) are compared in some detail.Subsequently the M/G/1 queue with finite capacity is considered; the joint distributions of x and ζ and of x and s are derived. These results may be used to study the cycle time distribution in a two-stage cyclic queue.     

Milling of Organic Solids in a Jet Mill. Part 2: Checking the Validity of the Predicted Rate of Breakage Function

The particle size distribution of fine chemicals in the solid state, like active pharmaceutical ingredients, is often a critical parameter. To achieve the desired particle size distribution, milling of such materials is usually the method of choice. Since these chemicals are often scarcely available, experimental optimization of milling is not possible. Therefore, a model to predict the milling conditions has been developed. The model estimates the rate of breakage function, and needs mechanical properties like hardness and yield strength as input to calculate the rate of breakage function. This paper attempts to check the validity of the model by a series of experiments. A comparison of the experimental results with the outcomes of the model using five different model compounds has been performed. It appears that the rate of breakage function can be estimated by: The model is able to rank the compounds by degree of fracture as an effect of milling. It was also possible to perform a quantitative prediction of the impact of milling pressure on the milling behavior. Finally, it appeared that the prediction of the large particles in the distribution was significantly better than small ones. Because the oversized material is usually the most critical parameter, the conclusion is that the model has acceptable practical applicability.