Fluid model driven by an M/G/1 queue with multiple exponential vacations

This paper studies a fluid model driven by an M/G/1 queue with multiple exponential vacations. By introducing various vacation strategies to the fluid model, we can provide greater flexibility for the design and control of input rate and output rate. The Laplace transform of the steady-state distribution of the buffer content is expressed through the minimal positive solution to a crucial equation. Then the performance measure-mean buffer content, which is independent of the vacation parameter, is obtained. Finally, with some numerical examples, the parameter effect on the mean buffer content is presented.     

On the global stability of SIS, SIR and SIRS epidemic models with standard incidence

In this paper, we establish the global stability conditions of classic SIS, SIR and SIRS epidemic models with constant recruitment, disease-induced death and standard incidence rate. We will make ingenious linear combination of known functions, common quadratic and Volterra-type, and of a new class of functions, we call composite-Volterra function, for obtain a suitable Lyapunov functions. In particular, for SIRS model we prove the global stability of the endemic equilibrium under a condition of parameters.     

Combined m-consecutive and k-out-of-n sliding window systems

This paper proposes a new model that generalizes the linear multi-state sliding window system. In this model the system consists of n linearly ordered multi-state elements. Each element can have different states: from complete failure up to perfect functioning. A performance rate is associated with each state. The system fails if at least one of the following two conditions is met: (1) there exist at least m consecutive overlapping groups of r adjacent elements having the cumulative performance lower than V; (2) there exist at least k arbitrarily located groups of r adjacent elements having the cumulative performance lower than W. An algorithm for system reliability evaluation is suggested which is based on an extended universal moment generating function. Examples of evaluating system reliability and elements’ reliability importance indices are presented. Optimal sequencing of system elements is demonstrated.     

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.     

Diffusion-equation model of slightly loaded M/M/1 queue

Another derivation of the diffusion approximation of the M/M/1 queue is presented, which results in a new boundary condition. The model proposed approximates the time-dependent behavior of the M/M/1 system for all values of channel utilization.     

Reducing lost-sales rate in (T, R, L) inventory model with controllable lead time

In order to establish a good image and to enhance customer’s loyalty, many efforts such as upgrading the servicing facilities, maintaining a high quality of products and increasing expenditure on advertisement could be made by a selling shop. Naturally, an extra-added cost must be spent for these efforts and it is expected to have a result to reduce the shortage cost of lost-sales and the total expected annual cost. This paper explores a probabilistic inventory model with optimal lost-sales caused by investment due to two different types of cost functions. We consider that the lead time can be shortened at an extra crashing cost, which depends on the length of the lead time. Moreover, we assume that the lost-sales rate can also be reduced by capital investment. The purpose of this paper is to establish a (T, R, L) inventory model with controllable lead time and to analyze the effects of increasing two different types of investments to reduce the lost-sales rate, in which the review period, lead time and lost-sales rate are treated as decision variables. We first formulate the basic periodic review model mathematically with the capital investment to reduce lost-sales rate. Then two models are discussed, one with normally distributed protection interval demand and another with distribution-free case. For each model, two investment cost functional forms, logarithmic and power, are employed for lost-sales rate reduction. Two computational algorithms with the help of the software Matlab are furnished to determine the optimal solution. In addition, six numerical examples and sensitivity analysis are presented to illustrate the theoretical results and obtain some managerial insights. Finally, the effect of lost-sales rate reduction is investigated. By framing this new model, we observe that a significant amount of savings can be easily achieved to increase the competitive edge in business. The results in the numerical examples indicate that the savings of expected annual total cost are realized through lost-sales reduction.     

Some n by dn linear complementarity problems

The purpose of this paper is to study some recent applications of the n by dn LCP solvable by a parametric principal pivoting algorithm (PPP algorithm). Often, the LCPs arising from these applications give rise to large systems of linear equations which can be solved fairly efficiently by exploiting their special structures. First, it is shown that by analyzing the n by dn LCP we could study the problem of solving a system of equations and the (nonlinear) complementarity problem when the function involved is separable. Next, we examine conditions under which the PPP algorithm is applicable to a general LCP, and then present examples of LCPs arising from various applications satisfying the conditions; included among them is the n by dn LCP with a certain $\text{P}$-property. Finally we study a special class of n by dn LCPs which do not possess the $\text{P}$-property but to which the PPP algorithm is still applicable; a major application of this class of problems is a certain economic spatial equilibrium model with piecewise linear prices.     

Algorithmic approximations for the busy period distribution of the M/M/c retrial queue

In this paper we deal with the main multiserver retrial queue of M/M/c type with exponential repeated attempts. This model is known to be analytically intractable due to the spatial heterogeneity of the underlying Markov chain, caused by the retrial feature. For this reason several models have been proposed for approximating its stationary distribution, that lead to satisfactory numerical implementations. This paper extends these studies by developing efficient algorithmic procedures for calculating the busy period distribution of the main approximation models of Wilkinson [Wilkinson, R.I., 1956. Theories for toll traffic engineering in the USA, The Bell System Technical Journal 35, 421–514], Falin [Falin, G.I., 1983. Calculations of probability characteristics of a multiline system with repeated calls, Moscow University Computational Mathematics and Cybernetics 1, 43–49] and Neuts and Rao [Neuts, M.F., Rao, B.M., 1990. Numerical investigation of a multiserver retrial model, Queueing Systems 7, 169–190]. Moreover, we develop stable recursive schemes for the computation of the busy period moments. The corresponding distributions for the total number of customers served during a busy period are also studied. Several numerical results illustrate the efficiency of the methods and reveal interesting facts concerning the behavior of the M/M/c retrial queue.     

Permanence of a delayed SIR epidemic model with density dependent birth rate

In this paper, we consider the permanence of a modified delayed SIR epidemic model with density dependent birth rate which is proposed in [M. Song, W. Ma, Asymptotic properties of a revised SIR epidemic model with density dependent birth rate and time delay, Dynamic of Continuous, Discrete and Impulsive Systems, 13 (2006) 199–208]. It is shown that global dynamic property of the modified delayed SIR epidemic model is very similar as that of the model in [W. Ma, Y. Takeuchi, T. Hara, E. Beretta, Permanence of an SIR epidemic model with distributed time delays, Tohoku Math. J. 54 (2002) 581–591; W. Ma, M. Song, Y. Takeuchi, Global stability of an SIR epidemic model with time delay, Appl. Math. Lett. 17 (2004) 1141–1145].     

On the solutions to x′(t) = A(t) x (t) over all A(t), where P ⩽ A(t) ⩽ Q

Let P_ij and q_ij be positive numbers for i ≠ j, i, j = 1, …, n, and consider the set of matrix differential equations x′(t) = A(t) x(t) over all A(t), where a_ij(t) is piecewise continuous, a_ij(t) = ?∑_{i ≠ j}a_ij(t), and p_ij ? a_ij(t) ? q_ij all t. A solution x is also to satisfy ∑_{i = 1}ⁿx_i(0) = 1. Let C_t denote the set of all solutions, evaluated at t to equations described above. It is shown that $\text{C}{}_{\mathrm{t}}$, the topological closure of C_t, is a compact convex set for each t. Further, the set valued function $\text{C}{}_{\mathrm{t}}$, of t is continuous and $\text{limit}{}_{\mathrm{t\; \to \; \infty }}\text{C}\text{?}{}_{\mathrm{t}}\text{= ∩}\text{C}\text{?}{}_{\mathrm{t}}$.     

Weak solution of (r, (x, u(x)) u′(x))′ = (Fu)′(x) with r(0, u(0))u′(0) = ku(0), r(L, u(L))u′ (L) = hu(L) and k, h are suitable elements of [0, ∞]

We consider weak solutions to the nonlinear boundary value problem (r, (x, u(x)) u′(x))′ = (Fu)′(x) with r(0, u(0)) u′(0) = ku(0), r(L, u(L)) u′(L) = hu(L) and k, h are suitable elements of [0, ∞]. In addition to studying some new boundary conditions, we also relax the constraints on r(x, u) and (Fu)(x). r(x, u) > 0 may have a countable set of jump discontinuities in u and r(x, u)^?1?L_q((0, L) × (0, p)). F is an operator from a suitable set of functions to a subset of L_p(0, L) which have nonnegative values. F includes, among others, examples of the form (Fu)(x) = (1 ? H(x ? x₀)) u(x₀), (Fu)(x) = ∫_x^Lf(y, u(y)) dy where f(y, u) may have a countable set of jump discontinuities in u or F may be chosen so that (Fu)′(x) = ? g(x, u(x)) u′(x) ? q(x) u(x) ? f(x, u(x)) where q is a distributional derivative of an L₂(0, L) function.     

The G/M/r machine interference model

This paper solves the machine interference problem in which N automatic machines are maintained by a team of r operatives. Repair times are assumed to follow a negative exponential distribution, and running times for each of the machines are assumed to be independently and identically distributed. It is shown that the solution to this G/M/r model is identical in most important respects to that for the M/M/r model.     

Enumeration of (p,q)-parking functions

Parking functions are central in many aspects of combinatorics. We define in this communication a generalization of parking functions which we call (p₁,…,p_k)-parking functions. We give a characterization of them in terms of parking functions and we show that they can be interpreted as recurrent configurations in the sandpile model for some graphs. We also establish a correspondence with a Lukasiewicz language, which enables to enumerate (p₁,…,p_k)-parking functions as well as increasing ones.     

Approximation of M/M/s/K retrial queue with nonpersistent customers

We consider the M/M/s/K retrial queues in which a customer who is blocked to enter the service facility may leave the system with a probability that depends on the number of attempts of the customer to enter the service facility. Approximation formulae for the distributions of the number of customers in service facility, waiting time in the system and the number of retrials made by a customer during its waiting time are derived. Approximation results are compared with the simulation.     

Factorization of Q(h(T)(x)) over a finite field where Q(x) is irreducible and h(T)(x) is linear—I

Let GF(q) be the finite field of order q, let Q(x) be an irreducible polynomial in GF(q)(x), and let h(T)(x) be a linear polynomial in GF(q)[x], where T:x→x^q. We use properties of the linear operator h(T) to give conditions for Q(h(T)(x)) to have a root of arbitrary degree k over GF(q), and we describe how to count the irreducible factors of Q(h(T)(x)) of degree k over GF(q). In addition we compare our results with those Ore which count the number of irreducible factors belonging to a linear polynomial having index k.     

Upper bounds for the risk in the α — t utility function

Fishburn's α — t model is an important example of a utility function involving a target for a random variable. Simple upper bounds for the risk function in this model are proposed for cases in which the probability distribution is either unknown, or produces complicated or intractable statements of the model.     

An M/G/1 queue with two phases of service subject to the server breakdown and delayed repair

This paper deals with the steady-state behaviour of an M/G/1 queue with an additional second phase of optional service subject to breakdowns occurring randomly at any instant while serving the customers and delayed repair. This model generalizes both the classical M/G/1 queue subject to random breakdown and delayed repair as well as M/G/1 queue with second optional service and server breakdowns. For this model, we first derive the joint distributions of state of the server and queue size, which is one of chief objectives of the paper. Secondly, we derive the probability generating function of the stationary queue size distribution at a departure epoch as a classical generalization of Pollaczek–Khinchin formula. Next, we derive Laplace Stieltjes transform of busy period distribution and waiting time distribution. Finally, we obtain some important performance measures and reliability indices of this model.     

Uniqueness of norm on L(G) and C(G) when G is a compact group

Let G be a compact group. If the trivial representation of G is not weakly contained in the left regular representation of G on L₀²(G) and X is either L^p(G) for 1<p?∞ or C(G), then we show that every complete norm |·| on X that makes translations from (X,|·|) into itself continuous is equivalent to ||·||_p or ||·||_∞ respectively. If 1<p?∞ and every left invariant linear functional on L^p(G) is a constant multiple of the Haar integral, then we show that every complete norm |·| on L^p(G) that makes translations from (L^p(G),|·|) into itself continuous and that makes the map t?L_t from G into bounded is equivalent to ||·||_p.