Analysis of the M x /G/1 Queue with a Random Setup Time

Abstract

We consider an M ^x /G/1 queueing system with a random setup time, where the service of the first unit at the commencement of each busy period is preceded by a random setup time, on completion of which service starts. For this model, the queue size distributions at a random point of time as well as at a departure epoch and some important performance measures are known [see Choudhury, G. An M ^x /G/1 queueing system with setup period and a vacation period. Queueing Sys. 2000, 36, 23–38]. In this paper, we derive the busy period distribution and the distribution of unfinished work at a random point of time. Further, we obtain the queue size distribution at a departure epoch as a simple alternative approach to Choudhury4 Choudhury, G. 2000. An M^x/G/1 queueing system with setup period and a vacation period. Queueing Syst., 36: 23–38. [CROSSREF][Crossref], [Web of Science ®] , [Google Scholar]. Finally, we present a transform free method to obtain the mean waiting time of this model.     

Queue length asymptotics for the multiple-server queue with heavy-tailed Weibull service times

Queueing Systems - We study the occurrence of large queue lengths in the GI / GI / d queue with heavy-tailed Weibull-type service times. Our analysis hinges on a...     

A non-well-founded primitive recursive tree provably well-founded for co-r.e. sets

 We construct by diagonalization a non-well-founded primitive recursive tree, which is well-founded for co-r.e. sets, provable in Σ₁ ⁰. It follows that the supremum of order-types of primitive recursive well-orderings, whose well-foundedness on co-r.e. sets is provable in Σ₁ ⁰, equals the limit of all recursive ordinals ω₁ ^ck . RID="ID=" <E5>Mathematics Subject Classification (2000):&ensp;03B30</E5>, 03F15 RID="ID=" Supported by the Deutschen Akademie der Naturforscher Leopoldina grant #BMBF-LPD 9801-7 with funds from the Bundesministerium f&uuml;r Bildung, Wissenschaft, Forschung und Technologie. RID="ID=" I would like to thank A. SETZER for his hospitality during my stay in Uppsala in December 1998 &ndash; these investigations are inspired by a discussion with him; S. BUSS for his hospitality during my stay at UCSD and for valuable remarks on a previous version of this paper; and M. M&Ouml;LLERFELD for remarks on a previous title. Received: 27 July 2000 / Published online: 25 February 2002 RID=" ID=" <E5>Mathematics Subject Classification (2000):&ensp;03B30</E5>, 03F15 RID=" ID=" Supported by the Deutschen Akademie der Naturforscher Leopoldina grant #BMBF-LPD 9801-7 with funds from the Bundesministerium f&uuml;r Bildung, Wissenschaft, Forschung und Technologie. RID=" ID=" I would like to thank A. SETZER for his hospitality during my stay in Uppsala in December 1998 &ndash; these investigations are inspired by a discussion with him; S. BUSS for his hospitality during my stay at UCSD and for valuable remarks on a previous version of this paper; and M. M&Ouml;LLERFELD for remarks on a previous title.     

An approach to two-sided M&amp;A fits based on a cross-efficiency evaluation with contrasting attitudes

Two-sided mergers and acquisitions (M&A) fits have been regarded as a critical step, which should always be taken by a bidder company when trying to identify suitable target companies prior to an M&A. This paper proposes an approach to two-sided M&A fits based on a cross-efficiency model with contrasting attitudes. In this approach, firstly, feasible M&A fits are screened using a preference function from an M&A fit matrix, according to the preferences of both the bidder companies and target companies in terms of efficiency and return to scale. Secondly, two-sided M&A fits are selected from a feasible M&A fit matrix, according to the value of cross-efficiency with contrasting attitudes. This allows for the existence of contrasting attitudes of peers toward an M&A fit, as opposed to the aggressive, benevolent or neutral cross-efficiency evaluation which consists of just one attitude (either aggressive, benevolent or indifferent). Finally, an illustrative example is given to explain the feasibility and validity of the two-sided M&A fit strategy.     

Block LU factorizations of M–matrices

Summary. It is well known that any nonsingular M–matrix admits an LU factorization into M–matrices (with L and U lower and upper triangular respectively) and any singular M–matrix is permutation similar to an M–matrix which admits an LU factorization into M–matrices. Varga and Cai establish necessary and sufficient conditions for a singular M–matrix (without permutation) to allow an LU factorization with L nonsingular. We generalize these results in two directions. First, we find necessary and sufficient conditions for the existence of an LU factorization of a singular M-matrix where L and U are both permitted to be singular. Second, we establish the minimal block structure that a block LU factorization of a singular M–matrix can have when L and U are M–matrices. Received November 21, 1994 / Revised version received August 4, 1997     

On the multi-phase M/G/1 queueing system with random feedback

In this paper we consider an M/G/1 queue with k phases of heterogeneous services and random feedback, where the arrival is Poisson and service times has general distribution. After the completion of the i-th phase, with probability θ _i the (i + 1)-th phase starts, with probability p _i the customer feedback to the tail of the queue and with probability 1 − θ _i − p _i  = q _i departs the system if service be successful, for i = 1, 2 , . . . , k. Finally in kth phase with probability p _k feedback to the tail of the queue and with probability 1 − p _k departs the system. We derive the steady-state equations, and PGF’s of the system is obtained. By using them the mean queue size at departure epoch is obtained.     

Asymptotic expansions for the sojourn time distribution in the M/G/1-PS queue

We consider the M/G/1 queue with a processor sharing server. We study the conditional sojourn time distribution, conditioned on the customer’s service requirement, as well as the unconditional distribution, in various asymptotic limits. These include large time and/or large service request, and heavy traffic, where the arrival rate is only slightly less than the service rate. Our results demonstrate the possible tail behaviors of the unconditional distribution, which was previously known in the cases G = M and G = D (where it is purely exponential). We assume that the service density decays at least exponentially fast. We use various methods for the asymptotic expansion of integrals, such as the Laplace and saddle point methods.     

Cauchy problem for the Zakharov System Arising from Ion-Acoustic Modes with low regularity data

We prove local well-posedness results for the Zakharov System Arising from Ion-Acoustic Modes in two spacial dimension with large initial data in low regularity Sobolev space \(&nbsp;&nbsp; (\dot{H}^1 \bigcup H^{\frac{1}{2}})\times L^2 \times H^{-1}&nbsp; \).&nbsp;&nbsp;Using &rdquo;derivative sharing&rdquo;, the local well-posedness results in \( (\dot{H}^1 \bigcup H^{\frac{1}{2}-\delta})\times H^{\delta} \times H^{-1+\delta}\)&nbsp;&nbsp;are also obtained, for any &nbsp;0&nbsp;\(\leq \delta \leq&nbsp;1/2 \).     

Onsager’s Conjecture on the Energy Conservation for Solutions of Euler Equations in Bounded Domains

Journal of Nonlinear Science - The Onsager’s conjecture has two parts: conservation of energy, if the exponent is larger than 1&nbsp;/&nbsp;3, and the possibility of dissipative Euler...     

On the integrability of the differential systems in dimension two and of the polynomial differential systems in arbitrary dimension

This is a survey on recent results providing sufficient conditions for the existence of a first integral, first for vector fields defined on real surfaces, and second for polynomial vector fields in \(R^n\) or \(C^n\) with \(n\geq 2\). We also&nbsp;provide an open question and some applications based on the existence of such first integrals. &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;     

On the Lang–Trotter conjecture for two elliptic curves

The Ramanujan Journal - We extend the author’s earlier computation and give coefficient formulas for the (quasimodular) Poincaré square series of weight 3&nbsp;/&nbsp;2 and...     

On the Diophantine equation $$x^2+3^a41^b=y^n $$ x 2 + 3 a 41 b = y n

Periodica Mathematica Hungarica - In this paper we find all positive integer solutions (x,&nbsp;y,&nbsp;n,&nbsp;a,&nbsp;b) of the equation in the title for non negative integers a...     

Double Hopf bifurcation and chaos in Liu system with delayed feedback

In this paper, we consider the stability of equilibria,&nbsp;Hopf and&nbsp;double Hopf bifurcation in Liu system with delay&nbsp;feedback. Firstly,&nbsp;we identify the critical values for stability switches and&nbsp;Hopf&nbsp;bifurcationusing the method of bifurcation&nbsp;analysis. When we choose&nbsp;appropriate feedback strength and delay, two symmetrical nontrivial&nbsp;equilibria of Liusystem can be controlled to be stable at the same&nbsp;time, and the stable bifurcating periodic solutions occur in the&nbsp;neighborhood of the two equilibria at the same time. Secondly, by&nbsp;applying the normal form method and center manifold theory,the&nbsp;normal form near the double Hopf bifurcation, as well as&nbsp;classifications of local dynamics are analyzed. Furthermore, we give&nbsp;the bifurcation diagram to illustrate numerically that a family of&nbsp;stable periodic solutions bifurcated from Hopf bifurcation occur in&nbsp;a large region of delay and the Liu system with delay can appear the&nbsp;phenomenon of ``chaos switchover''.     

Irreducibility of the Laplacian eigenspaces of some homogeneous spaces

Mathematische Zeitschrift - For a compact homogeneous space G&nbsp;/&nbsp;K, we study the problem of existence of G-invariant Riemannian metrics such that each eigenspace of the Laplacian...     

Widening requirements capture with soft methods: an investigation of hybrid M&amp;S studies in health care

A simulation study consists of several stages: problem formulation, model implementation, verification and validation, experimentation and output data analysis. The application of multiple techniques in the model implementation stage is referred to as hybrid simulation, which we distinguish in this paper from a hybrid M&S study, the latter referring to studies that apply methods and techniques from disciplines like Operations Research (OR), Systems Engineering and Computer Science to one or more stages of a simulation study. Our paper focuses on the contribution of soft OR methods in the problem formulation stage of a simulation study (and by extension a hybrid M&S study). Soft Systems Methodology (SSM) has, arguably, been the most widely used qualitative approach for eliciting system requirements. In this paper, we present Qualitative System Dynamics (QSD), a soft systems method, as having potential use in the problem formulation stage of a healthcare M&S study. The contribution of this paper is thus twofold: (1) a review of the literature in SSM for healthcare operations management and (2) an examination of QSD as an additional soft OR method, complementing (rather than supplanting) existing approaches, which can further aid the understanding of the system in the problem formulation/conceptual modelling stage of a hybrid M&S study.     

A Class of Invertible Subelliptic Operators in S(m,g)-Classes

Results in Mathematics - Given a self-adjoint second order differential operator L with positive characteristic and subellipticity of order 1&nbsp;≤&nbsp;τ &lt;&nbsp;2....     

Cesàro matrix and (1, 1; r)-convex sequences

Positivity - This paper deals with (1,&nbsp;1;&nbsp;r)-convexity of sequences. First, we prove several results on the sets of (1,&nbsp;1;&nbsp;r)-convex sequences for various values...     

Hermite’s theorem via Galois cohomology

Archiv der Mathematik - An 1861 theorem of Hermite asserts that for every field extension E&nbsp;/&nbsp;F of degree 5 there exists an element of E whose minimal polynomial over F is of the...     

Trends in the evolution of models & modeling perspectives on mathematical learning and problem solving

In this paper we briefly outline the models and modelling (M&M) perspective of mathematical thinking and learning relevant for the 21st century. Models and modeling (M&M) research often investigates the nature of understandings and abilities that are needed in order for students to be able to use what they have (presumably) learned in the classroom in “real life” situations beyond school Nonetheless, M&M perspectives evolved out of research on concept development more than research on problem solving; and, rather than being preoccupied with the kind of word problems emphasized in textbooks and standardized tests, we focus on (simulations of) problem solving “in the wild.” Also, we give special attention to the fact that, in a technology-basedage of information, significant changes are occurring in the kinds of “mathematical thinking” that is coming to be needed in the everyday lives of ordinary people in the 21^st century—as well as in the lives of productive people in future-oriented fields that are heavy users of mathematics, science, and technology.     

Existence of Generalized Traveling Waves in Time Recurrent and Space Periodic Monostable Equations

This paper is concerned with the extension of the concepts and theories of traveling wave solutions of time and space periodic monostable equations to time recurrent and space periodic ones.&nbsp; It first introduces the concept of generalized traveling wave solutions of time recurrent and space&nbsp;periodic monostable equations, which extends the concept of periodic traveling wave solutions of time and space periodic monostable equations to time recurrent and space periodic ones.&nbsp;It then proves that in the direction of any unit vector \(\xi\), there is \(c^*(\xi)\) such that for any \(c&gt;c^*(\xi)\), a generalized traveling wave solution in the direction of \(\xi\) with averaged propagation speed \(c\) exists. It also proves that if the time recurrent and space periodic&nbsp;monostable equation is indeed time periodic, then \(c^*(\xi)\) is the minimal wave speed in the direction of&nbsp;\(\xi\)&nbsp;and the generalized traveling wave solution in the direction of&nbsp;\(\xi\)&nbsp;with averaged speed \(c&gt;c^*(\xi)\) is a periodic traveling wave solution with speed \(c\), which recovers the existing results on the existence of periodic traveling wave solutions in the direction of&nbsp;\(\xi\)&nbsp;with speed greater than the minimal speed in that direction.