The Factorization of Queueing Equations and Their Interpretation

There are many queueing systems, including the M ^x /M ^y /c queue, the GI ^x /M/c queue and the M/D/c queue, in which the distribution of the queue length at certain epochs is determined by a Markov chain with the following structure. Except for a number of boundary states, all columns of the transition matrix are identical except for a shift which assures that there is always the same element occupying the main diagonal. This paper describes how one can find the equilibrium distribution for such Markov chains. Typically, this problem is solved by factorizing of a certain expression characterizing the repeated columns. In this paper, we show that this factorization has a probabilistic significance and we use this result to develop new approaches for finding the equilibrium distribution in question.     

Spatial mapping of price competition using logit-type market share models and store-level scanner-data

This paper proposes a methodology to obtain reliable spatial maps of price competition using store-level scanner data. Specifically, a procedure to obtain a symmetric matrix of similarities between brands considering their substitutability depending on price variations is proposed. The matrix is derived from a market response model where price cross-effects are split into two components. The first component accounts for the fact that price variation in one brand can have different effects to price variation in other brands (ie j → j′≠j′ → j). The second component accounts for the fact that the price of each brand can have different effects across competing brands (ie j → j′≠j → j^″). The matrix is obtained by imposing symmetry on this second component of price cross-effects. The parameterization of this symmetric matrix of similarities as the distances between the spatial representations of brands allows us to obtain the positioning maps. The proposed approach is illustrated through an empirical application.     

Clifford Theory for Glider Representations

Classical Clifford theory studies the decomposition of simple G-modules into simple H-modules for some normal subgroup H ? G. In this paper we deal with chains of normal subgroups 1?G ₁?· · ·?G _d = G, which allow to consider fragments and in particular glider representations. These are given by a descending chain of vector spaces over some field K and relate different representations of the groups appearing in the chain. Picking some normal subgroup H ? G one obtains a normal subchain and one can construct an induced fragment structure. Moreover, a notion of irreducibility of fragments is introduced, which completes the list of ingredients to perform a Clifford theory.     

A multiple replenishment contract with ARIMA demand processes

This paper is concerned with a multiple replenishment contract with a purchase price discount in a supply chain. The chain is composed of one supplier, one buyer and consumers for a product. The replenishment contract is based upon the well-known (s, Q) policy, but allows us to contract replenishments at a future time with a price discount. Owing to the larger forecast error of future demand, the buyer should keep a higher level of safety stock to provide the same level of service as the usual (s, Q) policy. However, the buyer can reduce his purchase cost by ordering a larger quantity at a discounted price. Hence, there exists a trade-off between the price discount and the inventory holding cost. For the ARIMA demand processes, we present a model for the contract and an algorithm to find the number of the future replenishments. Computational experiments show that the algorithm finds the global optimum solution very quickly.     

Markov Chains on Graded Posets

We consider two types of discrete-time Markov chains where the state space is a graded poset and the transitions are taken along the covering relations in the poset. The first type of Markov chain goes only in one direction, either up or down in the poset (an up chain or down chain). The second type toggles between two adjacent rank levels (an up-and-down chain). We introduce two compatibility concepts between the up-directed transition probabilities (an up rule) and the down-directed (a down rule), and we relate these to compatibility between up-and-down chains. This framework is used to prove a conjecture about a limit shape for a process on Young’s lattice. Finally, we settle the questions whether the reverse of an up chain is a down chain for some down rule and whether there exists an up or down chain at all if the rank function is not bounded.     

Quasiplanar Diagrams and Slim Semimodular Lattices

For elements x and y in the (Hasse) diagram D of a finite bounded poset P, x is on the left of y, written as x λ y, if x and y are incomparable and x is on the left of all maximal chains through y. Being on the right, written as x ? y, is defined analogously. The diagram D is quasiplanar if λ and ? are transitive and for any pair (x,y) of incomparable elements, if x is on the left of some maximal chain through y, then x λ y. A planar diagram is quasiplanar, and P has a quasiplanar diagram iff its order dimension is at most 2. We are interested in diagrams only up to similarity. A finite lattice is slim if it is join-generated by the union of two chains. The main result gives a bijection between the set of (the similarity classes of) finite quasiplanar diagrams and that of (the similarity classes of) planar diagrams of finite slim semimodular lattices. This bijection allows one to describe finite posets of order dimension at most 2 by finite slim semimodular lattices, and conversely. As a corollary, we obtain that there are exactly (n?2)! quasiplanar diagrams of size n.     

Adjoining batch Markov arrival processes of a Markov chain

A batch Markov arrival process (BMAP) X* = (N, J) is a 2-dimensional Markov process with two components, one is the counting process N and the other one is the phase process J. It is proved that the phase process is a time-homogeneous Markov chain with a finite state-space, or for short, Markov chain. In this paper, a new and inverse problem is proposed firstly: given a Markov chain J, can we deploy a process N such that the 2-dimensional process X* = (N, J) is a BMAP? The process X* = (N, J) is said to be an adjoining BMAP for the Markov chain J. For a given Markov chain the adjoining processes exist and they are not unique. Two kinds of adjoining BMAPs have been constructed. One is the BMAPs with fixed constant batches, the other one is the BMAPs with independent and identically distributed (i.i.d) random batches. The method we used in this paper is not the usual matrix-analytic method of studying BMAP, it is a path-analytic method. We constructed directly sample paths of adjoining BMAPs. The expressions of characteristic (D _k , k = 0, 1, 2 · · ·) and transition probabilities of the adjoining BMAP are obtained by the density matrix Q of the given Markov chain J. Moreover, we obtained two frontal Theorems. We present these expressions in the first time.     

Finite 2-groups whose length of chain of nonnormal subgroups is at most 2

Assume that G is a finite non-Dedekind p-group. D. S. Passman introduced the following concept: we say that H₁ < H₂ < ? < H_k is a chain of nonnormal subgroups of G if each H_i ? G and if |H_i : H_i?1| = p for i = 2, 3,…, k. k is called the length of the chain. chn(G) denotes the maximum of the lengths of the chains of nonnormal subgroups of G. In this paper, finite 2-groups G with chn(G) ? 2 are completely classified up to isomorphism.     

Probabilistic stable rules and Nash equilibrium in two-sided matching problems

We study many-to-many matching with substitutable and cardinally monotonic preferences. We analyze stochastic dominance (sd) Nash equilibria of the game induced by any probabilistic stable matching rule. We show that a unique match is obtained as the outcome of each sd-Nash equilibrium. Furthermore, individual-rationality with respect to the true preferences is a necessary and sufficient condition for an equilibrium outcome. In the many-to-one framework, the outcome of each equilibrium in which firms behave truthfully is stable for the true preferences. In the many-to-many framework, we identify an equilibrium in which firms behave truthfully and yet the equilibrium outcome is not stable for the true preferences. However, each stable match for the true preferences can be achieved as the outcome of such equilibrium.     

Chain Blockers and Convoluted Catalan Numbers

We give new interpretations of Catalan and convoluted Catalan numbers in terms of trees and chain blockers. For a poset P we say that a subset A ? P is a chain blocker if it is an inclusionwise minimal subset of P that contains at least one element from every maximal chain. In particular, we study the set of chain blockers for the class of posets P = C _a × C _b where C _i is the chain 1 < ? < i. We show that subclasses of these chain blockers are counted by Catalan and convoluted Catalan numbers.     

Some minimax problems in lexicographic order

In this paper, minimax theorems and saddle points for a class of vector-valued mappings f(x, y) = u(x)+β(x)v(y) are first investigated in the sense of lexicographic order, where u, v are two general vector-valued mappings and β is a non-negative real-valued function. Then, by applying the existence theorem of lexicographic saddle point, we investigate a lexicographic equilibrium problem and establish an equivalent relationship between the lexicographic saddle point theorem and existence theorem of a lexicographic equilibrium problem for vector-valued mappings.     

<Emphasis Type="Italic">M</Emphasis><Superscript><Emphasis Type="Italic">X</Emphasis></Superscript>/<Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">c</Emphasis> Queue with catastrophes and state-dependent control at idle time

We consider an M ^X /M/c queue with catastrophes and state-dependent control at idle time. Properties of the queues which terminate when the servers become idle are first studied. Recurrence, equilibrium distribution, and equilibrium queue-size structure are studied for the case of resurrection and no catastrophes. All of these properties and the first effective catastrophe occurrence time are then investigated for the case of resurrection and catastrophes. In particular, we obtain the Laplace transform of the transition probability for the absorbing M ^X /M/c queue.     

Stability of Manpower Systems

Members of an organization belong to one of the grades (1, 2,... k), and movements between grades are governed by a substochastic matrix P. The model can be deterministic, stochastic, or partially stochastic; and we may or may not insist that the total size be restored to a fixed quantity each year. Various notions of the stability of a structure x = (x₁, x₂, … x _k ), where x _i represents the number, or proportion, in grade i are considered. Mainly, but not exclusively, the case when P is upper triangular is considered.     

<Emphasis Type="Italic">km</Emphasis>th price sealed-bid auctions with general independent values and equilibrium linear mark-ups

A generalised bidding model is developed to calculate a bidder’s expected profit and auctioners expected revenue/payment for both a General Independent Value and Independent Private Value (IPV) kmth price sealed-bid auction (where the mth bidder wins at the kth bid payment) using a linear (affine) mark-up function. The Common Value (CV) assumption, and highbid and lowbid symmetric and asymmetric First Price Auctions and Second Price Auctions are included as special cases. The optimal n bidder symmetric analytical results are then provided for the uniform IPV and CV models in equilibrium. Final comments concern implications, the assumptions involved and prospects for further research.     

Waiting times and stopping probabilities for patterns in Markov chains

Suppose that C is a finite collection of patterns. Observe a Markov chain until one of the patterns in C occurs as a run. This time is denoted by τ. In this paper, we aim to give an easy way to calculate the mean waiting time E(τ) and the stopping probabilities P(τ = τA)with A ∈ C, where τA is the waiting time until the pattern A appears as a run.     

On <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>-estimates of derivatives of univalent rational functions

We study the growth of the quantity ∫_T|R′(z)|dm(z) for rational functions R of degree n which are bounded and univalent in the unit disk and prove that this quantity can grow like n ^γ , γ > 0, as n → ∞. Some applications of this result to problems of regularity of boundaries of Nevanlinna domains are considered. We also discuss a related result of Dolzhenko, which applies to general (non-univalent) rational functions.     

Projective bichains

An algebra with two binary operations · and +  that are commutative, associative, and idempotent is called a bisemilattice. A bisemilattice that satisfies Birkhoff’s equation x · (x + y) =  x + (x · y) is a Birkhoff system. Each bisemilattice determines, and is determined by, two semilattices, one for the operation +  and one for the operation ·. A bisemilattice for which each of these semilattices is a chain is called a bichain. In this note, we characterize the finite bichains that are weakly projective in the variety of Birkhoff systems as those that do not contain a certain three-element bichain. As subdirectly irreducible weak projectives are splitting, this provides some insight into the fine structure of the lattice of subvarieties of Birkhoff systems.     

A method for analysing operational complexity in supply chains

This paper proposes a method for analysing the operational complexity in supply chains by using an entropic measure based on information theory. The proposed approach estimates the operational complexity at each stage of the supply chain and analyses the changes between stages. In this paper a stage is identified by the exchange of data and/or material. Through analysis the method identifies the stages where the operational complexity is both generated and propagated (exported, imported, generated or absorbed). Central to the method is the identification of a reference point within the supply chain. This is where the operational complexity is at a local minimum along the data transfer stages. Such a point can be thought of as a ‘sink’ for turbulence generated in the supply chain. Where it exists, it has the merit of stabilising the supply chain by attenuating uncertainty. However, the location of the reference point is also a matter of choice. If the preferred location is other than the current one, this is a trigger for management action. The analysis can help decide appropriate remedial action. More generally, the approach can assist logistics management by highlighting problem areas. An industrial application is presented to demonstrate the applicability of the method.     

Perturbation of chains of de Branges spaces

We investigate the structure of the set of de Branges spaces of entire functions which are contained in a space L²(μ). Thereby, we follow a perturbation approach. The main result is a growth dependent stability theorem. Namely, assume that measures μ₁ and μ₂ are close to each other in a sense quantified relative to a proximate order. Consider the sections of corresponding chains of de Branges spaces C₁ and C₂ which consist of those spaces whose elements have finite (possibly zero) type with respect to the given proximate order. Then either these sections coincide or one is smaller than the other but its complement consists of only a (finite or infinite) sequence of spaces.

Among other situations, we apply—and refine—this general theorem in two important particular situations

1. (1)
   the measures μ₁ and μ₂ differ in essence only on a compact set; then stability of whole chains rather than sections can be shown
    
2. (2)
   the linear space of all polynomials is dense in L²(μ₂); then conditions for density of polynomials in the space L²(μ₂) are obtained.
    

In the proof of the main result, we employ a method used by P. Yuditskii in the context of density of polynomials. Another vital tool is the notion of the index of a chain, which is a generalisation of the index of determinacy of a measure having all power moments. We undertake a systematic study of this index, which is also of interest on its own right.

     

On the Heterogeneous M/G/n Blocking System in a Random Environment

This paper discusses the equilibrium behaviour of the generalized M/G/n blocking system with heterogeneous servers. The system is evolving in a random environment controlled by an irreducible, aperiodic, m-state Markov chain Z(t). This paper deals with the main characteristics, such as utilization, the average length of busy period of the system, the mean number of occupied servers and the probability of blocking. Finally, numerical examples illustrate the problem in question.