Optimization and sensitivity analysis of controlling arrivals in the queueing system with single working vacation

This paper analyzes the F-policy M/M/1/K queueing system with working vacation and an exponential startup time. The F-policy deals with the issue of controlling arrivals to a queueing system, and the server requires a startup time before allowing customers to enter the system. For the queueing systems with working vacation, the server can still provide service to customers rather than completely stop the service during a vacation period. The matrix-analytic method is applied to develop the steady-state probabilities, and then obtain several system characteristics. We construct the expected cost function and formulate an optimization problem to find the minimum cost. The direct search method and Quasi-Newton method are implemented to determine the optimal system capacity K, the optimal threshold F and the optimal service rates (μ_B,μ_V) at the minimum cost. A sensitivity analysis is conducted to investigate the effect of changes in the system parameters on the expected cost function. Finally, numerical examples are provided for illustration purpose.     

Steady-state probability of the randomized server control system with second optional service, server breakdowns and startup

This paper deals with the 〈N,p〉-policy M/G/1 queue with server breakdowns and general startup times, where customers arrive to demand the first essential service and some of them further demand a second optional service. Service times of the first essential service channel are assumed to follow a general distribution and that of the second optional service channel are another general distribution. The server breaks down according to a Poisson process and his repair times obey a general distribution in the first essential service channel and second optional service channel, respectively. The server operation starts only when N (N≥1) customers have accumulated, he requires a startup time before each busy period. When the system becomes empty, turn the server off with probability p (p∈[0,1]) and leave it on with probability (1?p). The method of maximum entropy principle is used to develop the approximate steady-state probability distribution of the queue length in the M/G(G, G)/1 queueing system. A study of the derived approximate results, compared to the established exact results for three different 〈N,p〉-policy queues, suggests that the maximum entropy principle provides a useful method for solving complex queueing systems.     

Boundary trace embedding theorems for variable exponent Sobolev spaces

We study boundary trace embedding theorems for variable exponent Sobolev space W1,p(⋅)(Ω). Let Ω be an open (bounded or unbounded) domain in RN satisfying strong local Lipschitz condition. Under the hypotheses that p∈L^∞(Ω), 1?infp(x)?supp(x)<N, |∇p|∈Lγ(⋅)(Ω), where γ∈L^∞(Ω) and infγ(x)>N, we prove that there is a continuous boundary trace embedding W1,p(⋅)(Ω)→Lq(⋅)(∂Ω) provided q(⋅), a measurable function on ∂Ω, satisfies condition for x∈∂Ω.     

Property (w) and perturbations II

This note is a continuation of a previous article [P. Aiena, M.T. Biondi, Property (w) and perturbations, J. Math. Anal. Appl. 336 (2007) 683-692] concerning the stability of property (w), a variant of Weyl's theorem, for a bounded operator T acting on a Banach space, under finite-dimensional perturbations K commuting with T. A counterexample shows that property (w) in general is not preserved under finite-dimensional perturbations commuting with T, also under the assumption that T is a-isoloid.     

Dynamic inventory and pricing policy in a capacitated stochastic inventory system with fixed ordering cost

We study a dynamic inventory and pricing optimization problem in a periodic review inventory system with setup cost and finite ordering capacity in each period. We show that the optimal inventory control is characterized by an (s,s^′,p) policy in four regions of the starting inventory level.     

Local “superlinearity” and “sublinearity” for the p-Laplacian

We study the existence, nonexistence and multiplicity of positive solutions for a family of problems −Δpu=fλ(x,u), , where Ω is a bounded domain in RN, N>p, and λ>0 is a parameter. The family we consider includes the well-known nonlinearities of Ambrosetti-Brezis-Cerami type in a more general form, namely λa(x)uq+b(x)ur, where 0?q<p−1<r?p^∗−1. Here the coefficient a(x) is assumed to be nonnegative but b(x) is allowed to change sign, even in the critical case. Preliminary results of independent interest include the extension to the p-Laplacian context of the Brezis-Nirenberg result on local minimization in and , a C1,α estimate for equations of the form −Δpu=h(x,u) with h of critical growth, a strong comparison result for the p-Laplacian, and a variational approach to the method of upper-lower solutions for the p-Laplacian.     

Kernels and some operations in edge-coloured digraphs

Let D be an edge-coloured digraph, V(D) will denote the set of vertices of D; a set N⊆V(D) is said to be a kernel by monochromatic paths of D if it satisfies the following two conditions: For every pair of different vertices u,v∈N there is no monochromatic directed path between them and; for every vertex x∈V(D)−N there is a vertex y∈N such that there is an xy-monochromatic directed path.In this paper we consider some operations on edge-coloured digraphs, and some sufficient conditions for the existence or uniqueness of kernels by monochromatic paths of edge-coloured digraphs formed by these operations from another edge-coloured digraphs.     

Super-vertex-antimagic total labelings of disconnected graphs

Let G=(V,E) be a finite, simple and non-empty (p,q)-graph of order p and size q. An (a,d)-vertex-antimagic total labeling is a bijection f from V(G)∪E(G) onto the set of consecutive integers 1,2,…,p+q, such that the vertex-weights form an arithmetic progression with the initial term a and the common difference d, where the vertex-weight of x is the sum of values f(xy) assigned to all edges xy incident to vertex x together with the value assigned to x itself, i.e. f(x). Such a labeling is called super if the smallest possible labels appear on the vertices.In this paper, we will study the properties of such labelings and examine their existence for disconnected graphs.     

Soliton solutions of a BBM(m, n) equation with generalized evolution

We consider a BBM(m, n) equation which is a generalization of the celebrated Benjamin-Bona-Mahony equation with generalized evolution term. By using two solitary wave ansatze in terms of sech^p(x) and tanh^p(x) functions, we find exact analytical bright and dark soliton solutions for the considered model. The physical parameters in the soliton solutions are obtained as function of the dependent model coefficients. The conditions of existence of solitons are presented. Note that, it is always useful and desirable to construct exact analytical solutions especially soliton-type envelope for the understanding of most nonlinear physical phenomena.     

Mapping properties for oscillatory integrals in d-dimensions

For aj,bj?1, j=1,2,…,d, we prove that the operator maps into itself for , where , and k(x,y)=φ(x,y)eig(x,y), φ(x,y) satisfies (1.2) (e.g. φ(x,y)=|x−y|^iτ,τ real) and the phase g(x,y)=xa⋅yb. We study operators with more general phases and for these operators we require that aj,bj>1, j=1,2,…,d, or al=bl?1 for some l∈{1,2,…,d}.     

Bounds on isoperimetric values of trees

Let G=(V,E) be a finite, simple and undirected graph. For S⊆V, let δ(S,G)={(u,v)∈E:u∈S and v∈V−S} be the edge boundary of S. Given an integer i, 1≤i≤|V|, let the edge isoperimetric value of G at i be defined as b_e(i,G)=min_S⊆V;|S|=i|δ(S,G)|. The edge isoperimetric peak of G is defined as b_e(G)=max_1≤j≤|V|b_e(j,G). Let b_v(G) denote the vertex isoperimetric peak defined in a corresponding way. The problem of determining a lower bound for the vertex isoperimetric peak in complete t-ary trees was recently considered in [Y. Otachi, K. Yamazaki, A lower bound for the vertex boundary-width of complete k-ary trees, Discrete Mathematics, in press (doi:10.1016/j.disc.2007.05.014)]. In this paper we provide bounds which improve those in the above cited paper. Our results can be generalized to arbitrary (rooted) trees.The depth d of a tree is the number of nodes on the longest path starting from the root and ending at a leaf. In this paper we show that for a complete binary tree of depth d (denoted as ), and where c₁, c₂ are constants. For a complete t-ary tree of depth d (denoted as ) and d≥clogt where c is a constant, we show that and where c₁, c₂ are constants. At the heart of our proof we have the following theorem which works for an arbitrary rooted tree and not just for a complete t-ary tree. Let T=(V,E,r) be a finite, connected and rooted tree — the root being the vertex r. Define a weight function w:V→N where the weight w(u) of a vertex u is the number of its successors (including itself) and let the weight index η(T) be defined as the number of distinct weights in the tree, i.e η(T)=|{w(u):u∈V}|. For a positive integer k, let ?(k)=|{i∈N:1≤i≤|V|,b_e(i,G)≤k}|. We show that .     

Labeling graphs with two distance constraints

Given a graph G and integers p,q,d₁ and d₂, with p>q, d₂>d₁?1, an L(d₁,d₂;p,q)-labeling of G is a function f:V(G)→{0,1,2,…,n} such that |f(u)−f(v)|?p if d_G(u,v)?d₁ and |f(u)−f(v)|?q if d_G(u,v)?d₂. A k-L(d₁,d₂;p,q)-labeling is an L(d₁,d₂;p,q)-labeling f such that max_v∈V(G)f(v)?k. The L(d₁,d₂;p,q)-labeling number ofG, denoted by , is the smallest number k such that G has a k-L(d₁,d₂;p,q)-labeling. In this paper, we give upper bounds and lower bounds of the L(d₁,d₂;p,q)-labeling number for general graphs and some special graphs. We also discuss the L(d₁,d₂;p,q)-labeling number of G, when G is a path, a power of a path, or Cartesian product of two paths.     

Entropy solutions for a doubly nonlinear elliptic problem with variable exponent

We study the boundary value problem in Ω, u=0 on ∂Ω, where Ω is a smooth bounded domain in RN (N?3) and is a p(x)-Laplace type operator with p(.):Ω→[1,+∞) a measurable function and b a continuous and nondecreasing function from R→R. We prove the existence and uniqueness of an entropy solution for L¹-data f.     

Variations of Y-dominating functions on graphs

Let Y be a subset of real numbers. A Y-dominating function of a graph G=(V,E) is a function f:V→Y such that for all vertices v∈V, where N_G[v]={v}∪{u|(u,v)∈E}. Let for any subset S of V and let f(V) be the weight of f. The Y-domination problem is to find a Y-dominating function of minimum weight for a graph G=(V,E). In this paper, we study the variations of Y-domination such as {k}-domination, k-tuple domination, signed domination, and minus domination for some classes of graphs. We give formulas to compute the {k}-domination, k-tuple domination, signed domination, and minus domination numbers of paths, cycles, n-fans, n-wheels, n-pans, and n-suns. Besides, we present a unified approach to these four problems on strongly chordal graphs. Notice that trees, block graphs, interval graphs, and directed path graphs are subclasses of strongly chordal graphs. This paper also gives complexity results for the problems on doubly chordal graphs, dually chordal graphs, bipartite planar graphs, chordal bipartite graphs, and planar graphs.     

Identifying codes and locating-dominating sets on paths and cycles

Let G=(V,E) be a graph and let r≥1 be an integer. For a set D⊆V, define N_r[x]={y∈V:d(x,y)≤r} and D_r(x)=N_r[x]∩D, where d(x,y) denotes the number of edges in any shortest path between x and y. D is known as an r-identifying code (r-locating-dominating set, respectively), if for all vertices x∈V (x∈V?D, respectively), D_r(x) are all nonempty and different. Roberts and Roberts [D.L. Roberts, F.S. Roberts, Locating sensors in paths and cycles: the case of 2-identifying codes, European Journal of Combinatorics 29 (2008) 72-82] provided complete results for the paths and cycles when r=2. In this paper, we provide results for a remaining open case in cycles and complete results in paths for r-identifying codes; we also give complete results for 2-locating-dominating sets in cycles, which completes the results of Bertrand et al. [N. Bertrand, I. Charon, O. Hudry, A. Lobstein, Identifying and locating-dominating codes on chains and cycles, European Journal of Combinatorics 25 (2004) 969-987].     

On operator valued sequences of multipliers and R-boundedness

In recent papers (cf. [J.L. Arregui, O. Blasco, (p,q)-Summing sequences, J. Math. Anal. Appl. 274 (2002) 812-827; J.L. Arregui, O. Blasco, (p,q)-Summing sequences of operators, Quaest. Math. 26 (2003) 441-452; S. Aywa, J.H. Fourie, On summing multipliers and applications, J. Math. Anal. Appl. 253 (2001) 166-186; J.H. Fourie, I. Röntgen, Banach space sequences and projective tensor products, J. Math. Anal. Appl. 277 (2) (2003) 629-644]) the concept of (p,q)-summing multiplier was considered in both general and special context. It has been shown that some geometric properties of Banach spaces and some classical theorems can be described using spaces of (p,q)-summing multipliers. The present paper is a continuation of this study, whereby multiplier spaces for some classical Banach spaces are considered. The scope of this research is also broadened, by studying other classes of summing multipliers. Let E(X) and F(Y) be two Banach spaces whose elements are sequences of vectors in X and Y, respectively, and which contain the spaces c₀₀(X) and c₀₀(Y) of all X-valued and Y-valued sequences which are eventually zero, respectively. Generally spoken, a sequence of bounded linear operators (un)⊂L(X,Y) is called a multiplier sequence from E(X) to F(Y) if the linear operator from c₀₀(X) into c₀₀(Y) which maps (xi)∈c₀₀(X) onto (unxn)∈c₀₀(Y) is bounded with respect to the norms on E(X) and F(Y), respectively. Several cases where E(X) and F(Y) are different (classical) spaces of sequences, including, for instance, the spaces Rad(X) of almost unconditionally summable sequences in X, are considered. Several examples, properties and relations among spaces of summing multipliers are discussed. Important concepts like R-bounded, semi-R-bounded and weak-R-bounded from recent papers are also considered in this context.     

Multiple positive solutions for a critical quasilinear elliptic system

The existence of cat_Ω(Ω) positive solutions for the p-Laplacian system with convex and Sobolev critical nonlinearities is obtained by some standard variational methods, whose key is to construct homotopies between Ω and levels of the functional J_λ,μ, and some analytical techniques.     

Endomorphisms of the shift dynamical system, discrete derivatives, and applications

All continuous endomorphisms f_∞ of the shift dynamical system S on the 2-adic integers Z₂ are induced by some , where n is a positive integer, B_n is the set of n-blocks over {0, 1}, and f_∞(x)=y₀y₁y₂… where for all i∈N, y_i=f(x_ix_i+1…x_i+n−1). Define D:Z₂→Z₂ to be the endomorphism of S induced by the map {(00,0),(01,1),(10,1),(11,0)} and V:Z₂→Z₂ by V(x)=−1−x. We prove that D, V°D, S, and V°S are conjugate to S and are the only continuous endomorphisms of S whose parity vector function is solenoidal. We investigate the properties of D as a dynamical system, and use D to construct a conjugacy from the 3x+1 function T:Z₂→Z₂ to a parity-neutral dynamical system. We also construct a conjugacy R from D to T. We apply these results to establish that, in order to prove the 3x+1 conjecture, it suffices to show that for any m∈Z⁺, there exists some n∈N such that R⁻¹(m) has binary representation of the form or .     

Rayleigh dissipation from the general recurrence of metrics in path spaces

Given a pair (metric g, symmetric 2-covariant tensor field H though as a Rayleigh dissipation) on a path space (manifold M, semispray S), the family of nonlinear connections N such that H equals the dynamical derivative of g with respect to (S,N) is determined by using the Obata tensors. In this way, we generalize the case of metric nonlinear connections as well as that of recurrent metrics. As applications, we treat firstly the case of Finslerian (α,β)-metrics finding all nonlinear connections for which the associated Finsler-Sasaki metric is exactly the dynamical derivative of the Riemannian-Sasaki metric. Secondly, we apply our results for the case of Beil metrics used in Relativity and field theories.     

Density of finite rank operators in the Banach space of p-compact operators

A Banach space X is said to have the kp-approximation property (kp-AP) if for every Banach space Y, the space F(Y,X) of finite rank operators is dense in the space Kp(Y,X) of p-compact operators endowed with its natural ideal norm kp. In this paper we study this notion that has been previously treated by Sinha and Karn (2002) in [15]. As application, the kp-AP of dual Banach spaces is characterized via density of finite rank operators in the space of quasi p-nuclear operators for the p-summing norm. This allows to obtain a relation between the kp-AP and Saphar's approximation property. As another application, the kp-AP is characterized in terms of a trace condition. Finally, we relate the kp-AP to the (p,p)-approximation property introduced in Sinha and Karn (2002) [15] for subspaces of Lp(μ)-spaces.