Stochastic bounds for a polling system

In this note we consider two queueing systems: a symmetric polling system with gated service at allN queues and with switchover times, and a single-server single-queue model with one arrival stream of ordinary customers andN additional permanently present customers. It is assumed that the combined arrival process at the queues of the polling system coincides with the arrival process of the ordinary customers in the single-queue model, and that the service time and switchover time distributions of the polling model coincide with the service time distributions of the ordinary and permanent customers, respectively, in the single-queue model. A complete equivalence between both models is accomplished by the following queue insertion of arriving customers. In the single-queue model, an arriving ordinary customer occupies with probabilityp _i a position at the end of the queue section behind theith permanent customer,i = l, ...,N. In the cyclic polling model, an arriving customer with probabilityp _i joins the end of theith queue to be visited by the server, measured from its present position.For the single-queue model we prove that, if two queue insertion distributions {p _{i, i} = l, ...,N} and {q _{i, i} = l, ...,N} are stochastically ordered, then also the workload and queue length distributions in the corresponding two single-queue versions are stochastically ordered. This immediately leads to equivalent stochastic orderings in polling models.Finally, the single-queue model with Poisson arrivals andp ₁ = 1 is studied in detail.Part of the research of the first author has been supported by the Esprit BRA project QMIPS.     

The <Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">G</Emphasis>/1+<Emphasis Type="Italic">G</Emphasis> queue revisited

We consider an M/G/1 queue with the following form of customer impatience: an arriving customer balks or reneges when its virtual waiting time, i.e., the amount of work seen upon arrival, is larger than a certain random patience time. We consider the number of customers in the system, the maximum workload during a busy period, and the length of a busy period. We also briefly treat the analogous model in which any customer enters the system and leaves at the end of his patience time or at the end of his virtual sojourn time, whichever occurs first.     

Tensor products of Archimedean partially ordered vector spaces

We study the tensor product of two directed Archimedean partially ordered vector spaces X and Y by means of Riesz completions. With the aid of the Fremlin tensor product of the Riesz completions of X and Y we show that the projective cone in X ? Y is contained in an Archimedean cone. The smallest Archimedean cone containing the projective cone satisfies an appropriate universal mapping property.     

Power series approximations for two-class generalized processor sharing systems

We develop power series approximations for a discrete-time queueing system with two parallel queues and one processor. If both queues are nonempty, a customer of queue 1 is served with probability β, and a customer of queue 2 is served with probability 1−β. If one of the queues is empty, a customer of the other queue is served with probability 1. We first describe the generating function U(z ₁,z ₂) of the stationary queue lengths in terms of a functional equation, and show how to solve this using the theory of boundary value problems. Then, we propose to use the same functional equation to obtain a power series for U(z ₁,z ₂) in β. The first coefficient of this power series corresponds to the priority case β=0, which allows for an explicit solution. All higher coefficients are expressed in terms of the priority case. Accurate approximations for the mean stationary queue lengths are obtained from combining truncated power series and Padé approximation.     

On the longest service time in a busy period of the M⧸G⧸1 queue

This paper considers the supremum m of the service times of the customers served in a busy period of the M?G?1 queueing system. An implicit expression for the distribution m(w) of m is derived. This expression leads to some bounds for m(w), while it can also be used to obtain numerical results. The tail behaviour of m(w) is investigated, too. The results are particularly useful in the analysis of a class of tandem queueing systems.     

Does back-bonding involve bonding orbitals in boryl complexes? A theoretical DFT study

Theoretical calculations at the DFT (B3LYP) level have been undertaken on tris- and bis(boryl) complexes. Two model d(6) complexes [Rh(PH(3))(3)(BX(2))(3) and Rh(PH(3))(4)(BX(2))(2)(+), X = OH and H] have been studied. In the model tris(boryl) complex (X = OH) we find a fac structure as a minimum, in accordance with the experimental data. The mer geometries are found to be higher in energy. Analysis of the energetic ordering in mer isomers shows that back-bonding in these complexes involves a bonding Rh-B orbital (and not a d-block orbital as usual). This surprising behavior is rationalized through a qualitative MO analysis and quantitative NBO analysis. Results on the bis(boryl) complex confirm the preceding analysis. Full optimization of unsubstituted (X = H) complexes leads to structures in which the BH(2) moieties are coupled. In the optimal geometry of the bis(boryl) complex, the B(2)H(4) ligand resembles the transition state of the C(2v)-->D(2d) interconversion of the isolated B(2)H(4) species. In the tris(boryl) complex, we find a B(3)H(6) ligand in which the B(3) atoms define an isosceles triangle with one hydrogen bridging the shorter B-B bond.     

A Tandem Queue with a Gate Mechanism

Inspired by a problem regarding cable access networks, we consider a two station tandem queue with Poisson arrivals. At station 1 we operate a gate mechanism, leading to batch arrivals at station 2. Upon arrival at station 1, customers join a queue in front of a gate. Whenever all customers present at the service area of station 1 have received service, the gate before as well as a gate behind the service facility open. Customers leave the service area and enter station 2 (as a batch), while all customers waiting at the gate in front of station 1 are admitted into the service area. For station 1 we analyse the batch size and the time between two successive gate openings, as well as waiting and sojourn times of individual customers for different service disciplines. For station 2, we investigate waiting times of batch customers, where we allow that service times may depend on the size of the batch and also on the interarrival time. In the analysis we use Wiener–Hopf factorization techniques for Markov modulated random walks.     

Shot-noise queueing models

Queueing Systems - We provide a survey of so-called shot-noise queues: queueing models with the special feature that the server speed is proportional to the amount of work it faces. Several results...     

Redundancy scheduling with scaled Bernoulli service requirements

Queueing Systems - Redundancy scheduling has emerged as a powerful strategy for improving response times in parallel-server systems. The key feature in redundancy scheduling is replication of a job...