The queue length in an <Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">G</Emphasis>/1 batch arrival retrial queue

An M/G/1 retrial queue with batch arrivals is studied. The queue length K _μ is decomposed into the sum of two independent random variables. One corresponds to the queue length K _∞ of a standard M/G/1 batch arrival queue, and another is compound-Poisson distributed. In the case of the distribution of the batch size being light-tailed, the tail asymptotics of K _μ are investigated through the relation between K _∞ and its service times.     

Characterization of finite simple group <Emphasis Type="Italic">D</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>(2)

Let G be a finite group, and let π _e (G) be the spectrum of G, that is, the set of all element orders of G. In 1987, Shi Wujie put forward the following conjecture. If G is a finite group and M is a non-abelian simple group, then G ≅ M if and only if |G| = |M| and π _e (G) = π _e (M). In this short paper, we prove that if G is a finite group, then G ≅ M if and only if |G| = |M| and π _e (G) = π _e (M), where M = D _n (2) and n is even.     

The Recursive Solution for Geom/G/1（E,SV） queue with Feedback and Single Server Vacation

Using recursive method,this paper studies the queue size properties at any epoch n + in Geom/G/1(E,SV) queueing model with feedback under LASDA (late arrival system with delayed access) setup.Some new results about the recursive expressions of queue size distribution at different epoch (n+,n,n-) are obtained.Furthermore the important relations between stationary queue size distribution at different epochs are discovered.The results are different from the relations given in M/G/1 queueing system.The model discussed in this paper can be widely applied in many kinds of communications and computer network.     

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.     

The Exceptional Compact Symmetric Spaces <Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript> and <Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript>/<Emphasis Type="Italic">SO</Emphasis>(4) as Tubes

In the present paper we discuss in detail the cohomogeneity one isometric actions of the Lie groups SU(3) × SU(3) and SU(3) on the exceptional compact symmetric spaces G₂ and G₂/SO(4), respectively. We show that the principal orbits coincide with the tubular hypersurfaces around the totally geodesic singular orbits, and the symmetric spaces G₂ and G₂/SO(4) can be thought of as compact tubes around SU(3) and P², respectively. Moreover, we determine the radii of these tubes and describe the shape operators of the principal orbits. Finally, we apply these results to compute the volumes of the two symmetric spaces.The author was partially supported by the Hungarian National Science and Research Foundation OTKA T032478.     

Two new heuristics for the <Emphasis Type="Italic">GI</Emphasis>/<Emphasis Type="Italic">G</Emphasis>/<Emphasis Type="Italic">n</Emphasis>/0 queueing loss system with examples based on the two-phase Coxian distribution

In this paper, we introduce a new heuristic approach for the numerical analysis of queueing systems. In particular, we study the general, multi-server queueing loss system, the GI/G/n/0 queue, with an emphasis on the calculation of steady-state loss probabilities. Two new heuristics are developed, called the GM Heuristic and the MG Heuristic, both of which make use of an exact analysis of the corresponding single-server GI/G/1/0 queue. The GM Heuristic also uses an exact analysis of the GI/M/n/0 queue, while the MG Heuristic uses an exact analysis of the M/G/n/0 queue. Experimental results are based on the use of two-phase Coxian distributions for both the inter-arrival time and the service time; these include an error analysis for each heuristic and the derivation of experimental probability bounds for the loss probability. For the class of problems studied, it is concluded that there are likely to be many situations where the accuracy of the GM Heuristic is adequate for practical purposes. Methods are also developed for combining the GM and MG Heuristics. In some cases, this leads to approximations that are significantly more accurate than those obtained by the individual heuristics.     

The <Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">M</Emphasis>/1 queue with synchronized abandonments

In this paper we present a detailed analysis of a single server Markovian queue with impatient customers. Instead of the standard assumption that customers perform independent abandonments, we consider situations where customers abandon the system simultaneously. Moreover, we distinguish two abandonment scenarios; in the first one all present customers become impatient and perform synchronized abandonments, while in the second scenario we exclude the customer in service from the abandonment procedure. Furthermore, we extend our analysis to the M/M/c queue under the second abandonment scenario.     

Maximal subgroups of <Emphasis Type="Italic">GL</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>(<Emphasis Type="Italic">D</Emphasis>) with finite conjugacy classes

Let D be an infinite division ring. A famous result due to Herstein says that every non-central element of D has infinitely many conjugates and so, if D ^* is an FC-group, then D is a field. Let M be a maximal subgroup of GL _n (D), where n ≥ 1. In this paper, we prove that if M is an FC-group, then it is the multiplicative group of some maximal subfield of M _n (D). Moreover, if M is algebraic over Z(D), then [D : Z(D)] < ∞.     

Asymptotically cylindrical 7-manifolds of holonomy <Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript> with applications to compact irreducible <Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript>-manifolds

We construct examples of exponentially asymptotically cylindrical (EAC) Riemannian 7-manifolds with holonomy group equal to G ₂. To our knowledge, these are the first such examples. We also obtain EAC coassociative calibrated submanifolds. Finally, we apply our results to show that one of the compact G ₂-manifolds constructed by Joyce by desingularisation of a flat orbifold T ⁷/Γ can be deformed to give one of the compact G ₂-manifolds obtainable as a generalized connected sum of two EAC SU(3)-manifolds via the method of Kovalev (J Reine Angew Math 565:125–160, 2003).     

Scheduling policies in the <Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">G</Emphasis>/1 make-to-stock queue

In this paper, we analyse a production/inventory system modelled as an M/G/1 make-to-stock queue producing different products requiring different and general production times. We study different scheduling policies including the static first-come-first-served, preemptive and non-preemptive priority disciplines. For each static policy, we exploit the distributional Little's law to obtain the steady-state distribution of the number of customers in the system and then find the optimal inventory control policy and the cost. We additionally provide the conditions under which it is optimal to produce a product according to a make-to-order policy. We further extend the application area of a well-known dynamic scheduling heuristic, Myopic(T), for systems with non-exponential service times by permitting preemption. We compare the performance of the preemptive-Myopic(T) heuristic alongside that of the static preemptive-bμ rule against the optimal solution. The numerical study we have conducted demonstrates that the preemptive-Myopic(T) policy is superior between the two and yields costs very close to the optimal.     

Reality properties of conjugacy classes in<Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript>

LetG be an algebraic group over a fieldk. We callg εG(k) real ifg is conjugate tog ⁻¹ inG(k). In this paper we study reality for groups of typeG ₂ over fields of characteristic different from 2. LetG be such a group overk. We discuss reality for both semisimple and unipotent elements. We show that a semisimple element inG(k) is real if and only if it is a product of two involutions inG(k). Every unipotent element inG(k) is a product of two involutions inG(k). We discuss reality forG ₂ over special fields and construct examples to show that reality fails for semisimple elements inG ₂ over ℚ and ℚ_p. We show that semisimple elements are real forG ₂ overk withcd(k) ≤ 1. We conclude with examples of nonreal elements inG ₂ overk finite, with characteristick not 2 or 3, which are not semisimple or unipotent.     

Quasirecognition by prime graph of the simple group <Superscript>2</Superscript><Emphasis Type="Italic">F</Emphasis><Subscript>4</Subscript>(<Emphasis Type="Italic">q</Emphasis>)

As the main result, we show that if G is a finite group such that Γ(G) = Γ(² F ₄(q)), where q = 2^2m+1 for some m ≧ 1, then G has a unique nonabelian composition factor isomorphic to ² F ₄(q). We also show that if G is a finite group satisfying |G| =|² F ₄(q)| and Γ(G) = Γ(² F ₄(q)), then G ≅ ² F ₄(q). As a consequence of our result we give a new proof for a conjecture of W. Shi and J. Bi for ² F ₄(q). The third author was supported in part by a grant from IPM (No. 87200022).     

Packet loss characteristics for <Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">G</Emphasis>/1/<Emphasis Type="Italic">N</Emphasis> queueing systems

In this contribution we investigate higher-order loss characteristics for M/G/1/N queueing systems. We focus on the lengths of the loss and non-loss periods as well as on the number of arrivals during these periods. For the analysis, we extend the Markovian state of the queueing system with the time and number of admitted arrivals since the instant where the last loss occurred. By combining transform and matrix techniques, expressions for the various moments of these loss characteristics are found. The approach also yields expressions for the loss probability and the conditional loss probability. Some numerical examples then illustrate our results.     

Sojourn Times in the <Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">PH</Emphasis>/1 Processor Sharing Queue

We give in this paper an algorithm to compute the sojourn time distribution in the processor sharing, single server queue with Poisson arrivals and phase type distributed service times. In a first step, we establish the differential system governing the conditional sojourn times probability distributions in this queue, given the number of customers in the different phases of the PH distribution at the arrival instant of a customer. This differential system is then solved by using a uniformization procedure and an exponential of matrix. The proposed algorithm precisely consists of computing this exponential with a controlled accuracy. This algorithm is then used in practical cases to investigate the impact of the variability of service times on sojourn times and the validity of the so-called reduced service rate (RSR) approximation, when service times in the different phases are highly dissymmetrical. For two-stage PH distributions, we give conjectures on the limiting behavior in terms of an M/M/1 PS queue and provide numerical illustrative examples.This revised version was published online in June 2005 with corrected coverdate     

Complex<Emphasis Type="BoldItalic">n</Emphasis>-dimensional manifolds with a real<Emphasis Type="BoldItalic">n</Emphasis><Superscript>2</Superscript>-dimensional automorphism group

The main theorem of this article is a characterization of non compact simply connected complete Kobayashi hyperbolic complex manifold of dimension n≽ 2 with real n ²-dimensional holomorphic automorphism group. Together with the earlier work [11, 12] and [13] of Isaev and Krantz, this yields a complete classification of the simply-connected, complete Kobayashi hyperbolic manifolds with dim_ℝ Aut (M) ≽ (dim_ℂ M)².     

Extremal <Emphasis Type="Italic">G</Emphasis>-invariant eigenvalues of the Laplacian of <Emphasis Type="Italic">G</Emphasis>-invariant metrics

The study of extremal properties of the spectrum often involves restricting the metrics under consideration. Motivated by the work of Abreu and Freitas in the case of the sphere S ² endowed with S ¹-invariant metrics, we consider the subsequence of the spectrum of a Riemannian manifold M which corresponds to metrics and functions invariant under the action of a compact Lie group G. If G has dimension at least 1, we show that the functional λ _k ^G admits no extremal metric under volume-preserving G-invariant deformations. If, moreover, M has dimension at least three, then the functional is unbounded when restricted to any conformal class of G-invariant metrics of fixed volume. As a special case of this, we can consider the standard O(n)-action on S ⁿ ; however, if we also require the metric to be induced by an embedding of S ⁿ in , we get an optimal upper bound on .      

Graded central polynomials for the algebra <Emphasis Type="Italic">M</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>(<Emphasis Type="Italic">K</Emphasis>)

Let M _n (K) be the algebra of all n × n matrices over an infinite field K. This algebra has a natural ℤ _n -grading and a natural ℤ-grading. Finite bases for its ℤ _n -graded identities and for its ℤ-graded identities are known. In this paper we describe finite generating sets for the ℤ _n -graded and for the ℤ-graded central polynomials for M _n (K) Partially supported by CNPq 620025/2006-9     

Influence of <Emphasis Type="Italic">M</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-supplemented subgroups on the structure of <Emphasis Type="Italic">p</Emphasis>-modular subgroups

A subgroup K of G is M _p -supplemented in G if there exists a subgroup B of G such that G = KB and TB < G for every maximal subgroup T of K with |K: T| = p ^α. We study the structure of the chief factor of G by using M _p -supplemented subgroups and generalize the results of Monakhov and Shnyparkov by involving the relevant results about the p-modular subgroup O ^p (G) of G.     

A Bijection for the Alperin Weight Conjecture in <Emphasis Type="Italic">S</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

The Alperin weight conjecture states that if G is a finite group and p is a prime, then the number of irreducible Brauer characters of a group G should be equal to the number of conjugacy classes of p-weights of G. This conjecture is known to be true for the symmetric group S _n , however there is no explicit bijection given between the two sets. In this paper we develop an explicit bijection between the p-weights of S _n and a certain set of partitions that is known to have the same cardinality as the irreducible Brauer characters of S _n . We also develop some properties of this bijection, especially in relation to a certain class of partitions whose corresponding Specht modules over fields of characteristic p are known to be irreducible.     

On the Theory of <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>(<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">q</Emphasis></Subscript>)-Banach Lattices

Given 1≤ p,q < ∞, let BL_pL_q be the class of all Banach lattices X such that X is isometrically lattice isomorphic to a band in some L_p(L_q)-Banach lattice. We show that the range of a positive contractive projection on any BL_pL_q-Banach lattice is itself in BL_pL_q. It is a consequence of this theorem and previous results that BL_pL_q is first-order axiomatizable in the language of Banach lattices. By studying the pavings of arbitrary BL_pL_q-Banach lattices by finite dimensional sublattices that are themselves in this class, we give an explicit set of axioms for BL_pL_q. We also consider the class of all sublattices of L_p(L_q)-Banach lattices; for this class (when p/q is not an integer) we give a set of axioms that are similar to Krivine’s well-known axioms for the subspaces of L_p-Banach spaces (when p/2 is not an integer). We also extend this result to the limiting case q = ∞.