Experimental numerical studies on a supercomputer of natural convection in an enclosure with localized heating

We present particle simulations of natural convection of a symmetrical, nonlinear, three-dimensional cavity flow problem. Qualitative studies are made in an enclosure with localized heating. The assumption is that particles interact locally by means of a compensating Lennard-Jones type force F, whose magnitude is given by −G/r^p + H/r^q.

In this formula, the parameters G, H, p, q depend upon the nature of the interacting particles and r is the distance between two particles. We also consider the system to be under the influence of gravity. Assuming that there are n particles, the equations relating position, velocity and acceleration at time t_k = kΔt, K = 0, 1, 2, …, are solved simultaneously using the “leap-frog” formulas. The basic formulas relating force and acceleration are Newton's dynamical equations F_i,k = m_ia_i,k, I = 1, 2, 3, …, n, where m_i is the mass of the ith particle.

Extensive and varied computations on a CRAY X - MP/24 are described and discussed, and comparisons are made with the results of others.     

Properties of Noncentral Dirichlet Distributions

Let X₁,…, X_r+1 be independent random variables, X_iGa (a_i, θ, δ_i), i = 1,…, r + 1. Define and V_i = X_i/X_r+1, i = 1,…, r. Then, (U₁,…, U_r) and (V₁,…, V_r) follow noncentral Dirichlet Type 1 and Type 2 distributions, respectively. In this article several properties of these distributions and their connections with the uniform, the noncentral multivariate-F and the noncentral multivariate-t distributions are discussed.     

Asymptotic analysis and simple approximation of the loss probability of the GI/M/c/K queue

We consider the GI^X/M/c/K queues with partial rejection or total rejection, and find an asymptotic behavior of loss probability of the GI^X/M/c/K queue as K tends to infinity. The asymptotic loss probability is expressed only in terms of the roots of the characteristic equation and the boundary probabilities of the corresponding GI^X/M/c queue. Numerical examples show that the asymptotic loss probability is a quite accurate approximation for the loss probability of the GI^X/M/c/K queue even when the system capacity K is moderate.     

Order-configuration functions: Mathematical characterizations and applications to digital signal and image processing

We call a function f in n variables an order-configuration function if for any x₁,…, x_n such that x_i1 … x_in we have f(x₁,…, x_n) = x_t, where t is determined by the n-tuple (i₁,…, i_n) corresponding to that ordering. Equivalently, it is a function built as a minimum of maxima, or a maximum of minima. Well-known examples are the minimum, the maximum, the median, and more generally rank functions, or the composition of rank functions. Such types of functions are often used in nonlinear processing of digital signals or images (for example in the median or separable median filter, min-max filters, rank filters, etc.). In this paper we study the mathematical properties of order-configuration functions and of a wider class of functions that we call order-subconfiguration functions. We give several characterization theorems for them. We show through various examples how our concepts can be used in the design of digital signal filters or image transformations based on order-configuration functions.     

A systolic algorithm for solving dense linear systems

For an arbitrary n × n matrix A and an n × 1 column vector b, we present a systolic algorithm to solve the dense linear equations Ax = b. An important consideration is that the pivot row can be changed during the execution of our systolic algorithm. The computational model consists of n linear systolic arrays. For 1 ≤ i ≤ n, the i^th linear array is responsible to eliminate the i^th unknown variable x_i of x. This algorithm requires 4n time steps to solve the linear system. The elapsed time unit within a time step is independent of the problem size n. Since the structure of a PE is simple and the same type PE executes the identical instructions, it is very suitable for VLSI implementation. The design process and correctness proof are considered in detail. Moreover, this algorithm can detect whether A is singular or not.     

Infinite words and biprefix codes

Let A be an alphabet and ƒ be a right infinite word on A. If ƒ is not ultimately periodic then there exists an infinite set {v_ii0} of (finite) words on A such that ƒ=v₀v₁…v_i…, {v_ii1} is a biprefix code and v_i≠v_j for positive integers i≠j.     

Behavior of the beta-splines with values of the parameters beta2 negative

In this paper we study the behavior of the beta-spline functions in the case the parameter β₂(i) is negative. We prove that a negative value exists so that if , the beta-spline functionsN_i(u) are positive. Moreover, if the control vertices are such that x₀ x_m−1, we have proved that the design curve keeps the properties already proved in the case β₂(i) 0.     

Bivariate hermite subdivision

A subdivision scheme for constructing smooth surfaces interpolating scattered data in R³ is proposed. It is also possible to impose derivative constraints in these points. In the case of functional data, i.e., data are given in a properly triangulated set of points {(x_i, y_i)}_i=1^N from which none of the pairs (x_i,y_i) and (x_j,y_j) with i≠j coincide, it is proved that the resulting surface (function) is C¹. The method is based on the construction of a sequence of continuous splines of degree 3. Another subdivision method, based on constructing a sequence of splines of degree 5 which are once differentiable, yields a function which is C² if the data are not ‘too irregular’. Finally the approximation properties of the methods are investigated.     

Uniform and optimal schemes for stiff initial-value problems

We formulate a class of difference schemes for stiff initial-value problems, with a small parameter ε multiplying the first derivative. We derive necessary conditions for uniform convergence with respect to the small parameter ε, that is the solution of the difference scheme u_i^h satisfies |u_i^h−u(x_i)| Ch, where C is independent of h and ε. We also derive sufficient conditions for uniform convergence and show that a subclass of schemes is also optimal in the sense that |u_i^h−u(x_i)| C min (h, ε). Finally, we show that this class contains higher-order schemes.     

Entropy maximisation and open queueing networks with priorities and blocking

A review is carried out on the characterisation and algorithmic implementation of an extended product-form approximation, based on the principle of maximum entropy (ME), for a wide class of arbitrary finite capacity open queueing network models (QNMs) with service and space priorities. A single server finite capacity GE/GE/1/N queue with R (R>1) distinct priority classes, compound Poisson arrival processes (CPPs) with geometrically distributed batches and generalised exponential (GE) service times is analysed via entropy maximisation, subject to suitable GE-type queueing theoretic constraints, under preemptive resume (PR) and head-of-line (HOL) scheduling rules combined with complete buffer sharing (CBS) and partial buffer sharing (PBS) management schemes stipulating a sequence of buffer thresholds {N=(N₁,…,N_R),0<N_i≤N_i−1,i=2,…,R}. The GE/GE/1/N queue is utilised, in conjunction with GE-type first two moment flow approximation formulae, as a cost-effective building block towards the establishment of a generic ME queue-by-queue decomposition algorithm for arbitrary open QNMs with space and service priorities under repetitive service blocking with random destination (RS-RD). Typical numerical results are included to illustrate the credibility of the ME algorithm against simulation for various network topologies and define experimentally pessimistic GE-type performance bounds. Remarks on the extensions of the ME algorithm to other types of blocking mechanisms, such as repetitive service blocking with fixed destination (RS-FD) and blocking-after-service (BAS), are included.     

Confluent general-factorial difference and derivative operators and polynomial-exponential interpolation

A previous application of the Newton divided difference series of the displacement function E^z = (1 + Δ)^z = e ^Dz, where the operators Δ and D are the variables, to purely exponential interpolation employing general-factorial differences and derivatives, {Pi;^m_i=0 (Δ - S_i)}f(0) and {Pi;^m_i=0 (D - t_i)}f(0), in which the s_i's and t_i's are distinct[1], is here extended to mixed polynomial-exponential interpolation where the s_i's and t_i's are no longer distinct.     

On canonical number systems

Let be such that d1,p_d=1,p₀2 and . We are proving in this note a new criterion for the pair to be a canonical number system. This enables us to prove that if p₂,…,p_d−1,∑_i=1^dp_i0 and p₀>2∑_i=1^d|p_i|, then is a canonical number system.     

A fast algorithm for parametric curve plotting

In parametric curve plotting by means of line segments, a curve r = r(t), t[t₀, t_u] is given, and a set of ordered points r(t_i, iN, t_ie[t₀, t_u] is specified as the vertices of an inscribed polygon of the curve. There are several, analytical, numerical or intuitive ways to derive these vertices obtaining a smooth polygonal approximation. The methods, which can be found in the literature, either belong to some special curves or involve a considerable waste of computing time.

In this paper, we consider an algorithm that appears as a subroutine in the whole program. The subroutine allows the main program to space points as a function of a distance interval for any parametric curve. The design of the routine for performing this spacing is outlined and two examples are shown.     

Heaps and heapsort on secondary storage

A heap structure designed for secondary storage is suggested that tries to make the best use of the available buffer space in primary memory. The heap is a complete multi-way tree, with multi-page blocks of records as nodes, satisfying a generalized heap property. A special feature of the tree is that the nodes may be partially filled, as in B-trees. The structure is complemented with priority-queue operations insert and delete-max. When handling a sequence of S operations, the number of page transfers performed is shown to be O(∑_{i = 1}^S(1/P) log_(M/P)(N_i/P)), where P denotes the number of records fitting into a page, M the capacity of the buffer space in records, and N_i, the number of records in the heap prior to the ith operation (assuming P 1 and S> M c · P, where c is a small positive constant). The number of comparisons required when handling the sequence is O(∑_{i = 1}^S log₂ N_i). Using the suggested data structure we obtain an optimal external heapsort that performs O((N/P) log_(M/P)(N/P)) page transfers and O(N log₂ N) comparisons in the worst case when sorting N records.     

Computational complexity of distributed detection problems with information constraints

For a system consisting of a set of sensors S = {S₁, S₂, …, S_m} and a set of objects O = {O₁, O₂, …, O_n}, there are information constraints given by a relation R S × O such that (S_i, O_j) R if and only if S_i is capable of detecting O_j. Each (S_i, O_j) R is assigned a confidence factor (a positive real number) which is either explicitly given or can be efficiently computed. Given that a subset of sensors have detected obstacles, the detection problem is to identify a subset H O with the maximum confidence value. The computational complexity of the detection problem, which depends on the nature of the confidence factor and the information constraints, is the main focus of this paper. This problem exhibits a myriad of complexity levels ranging from a worst-case exponential (in n) lower bound in a general case to an O(m + n) time solvability. We show that the following simple versions of a detection problem are computationally intractable: (a) deterministic formulation, where confidence factors are either 0 or 1; (b) uniform formulation where (S_i, O_j) R, for all S_i S, O_j O; (c) decomposable systems under multiplication operation. We then show that the following versions are solvable in polynomial (in n) time: (a) single object detection; (b) probabilistically independent detection; (c) decomposable systems under additive and nonfractional multiplicative measures; and (d) matroid systems.     

Moments of conjugacy classes of binary words

For each nonempty binary word w=c₁c₂c_q, where c_i{0,1}, the nonnegative integer ∑_i=1^q (q+1−i)c_i is called the moment of w and is denoted by M(w). Let [w] denote the conjugacy class of w. Define M([w])={M(u): u[w]}, N(w)={M(u)−M(w): u[w]} and δ(w)=max{M(u)−M(v): u,v[w]}. Using these objects, we obtain equivalent conditions for a binary word to be an -word (respectively, a power of an -word). For instance, we prove that the following statements are equivalent for any binary word w with |w|2: (a) w is an -word, (b) δ(w)=|w|−1, (c) w is a cyclic balanced primitive word, (d) M([w]) is a set of |w| consecutive positive integers, (e) N(w) is a set of |w| consecutive integers and 0N(w), (f) w is primitive and [w]St.     

Computing steady state probabilities in λ(n)/G/1/K queue

In this paper a recursive method is developed to obtain the steady state probability distribution of the number in system at arbitrary and departure time epochs of a single server state-dependent arrival rate queue λ(n)/G/1/K in which the arrival process is Markovian with arrival rates λ(n) which depend on the number of customers n in the system and general service time distribution. It is assumed that there exists an integer K such that λ(n) > 0 for all 0 n < K and λ(n) = 0 for all n K. Numerical results have been presented for many queueing models by suitably defining the function λ(n). These include machine interference model, queues with balking, queues with finite waiting space and machine interference model with finite waiting space. These models have wide application in computer/communication networks.     

M/M/1-PS queue and size-aware task assignment

We consider a distributed server system in which heterogeneous servers operate under the processor sharing (PS) discipline. Exponentially distributed jobs arrive to a dispatcher, which assigns each task to one of the servers. In the so-called size-aware system, the dispatcher is assumed to know the remaining service requirements of some or all of the existing jobs in each server. The aim is to minimize the mean sojourn time, i.e., the mean response time. To this end, we first analyze an M/M/1-PS queue in the framework of Markov decision processes, and derive the so-called size-aware relative value of state, which sums up the deviation from the average rate at which sojourn times are accumulated in the infinite time horizon. This task turns out to be non-trivial. The exact analysis yields an infinite system of first order differential equations, for which an explicit solution is derived. The relative values are then utilized to develop efficient dispatching policies by means of the first policy iteration (FPI). Numerically, we show that for the exponentially distributed job sizes the myopic approach, ignoring the future arrivals, yields an efficient and robust policy when compared to other heuristics. However, in the case of highly asymmetric service rates, an FPI based policy outperforms it. Additionally, the size-aware relative value of an M/G/1-PS queue is shown to be sensitive with respect to the form of job size distribution, and indeed, the numerical experiments with constant job sizes confirm that the optimal decision depends on the job size distribution.     

Mutually nonblocking supervisory control of discrete event systems

A single maximally permissive and nonblocking supervisor to simultaneously fulfill several marked specifications pertaining to a single plant is investigated. Given a plant G and two marked specification languages K₁ and K₂, a supervisor S is said to be (K₁,K₂)-mutually nonblocking if (for i,j=1,2)

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. This means that when the closed-loop system marks a trace of K_i, then it is always able to continue to a trace of K_j, also marked in the closed-loop system. Thus, the controlled system can execute traces within one specification while always being able to continue a trace of the other and hence not blocking the other specification. A complete, globally nonblocking and (K₁,K₂)-mutually nonblocking supervisor such that L_m(G || S)K₁K₂ exists if and only if there exists a controllable mutually nonblocking sublanguage of the union of the specifications. There does exist a supremal such language. Furthermore, in the case that each specification is nonconflicting with respect to the prefix-closure of the other, this supremal language can be calculated by expressing it as the union of the supremal prefix-bounded sublanguages of the respective specifications. Finally, we show that the multiply nonblocking supervision of Thistle, Malhame, Hoang and Lafortune ((1997). Internal Report, Dept. de genie electrique et de genie informatique, Ecole Polytechnique de Montreal, Canada) is equivalent to globally and mutually nonblocking supervision.     

On the asymptotic execution time of multi-tasked processes on tandem processors

Consider an ordered set of processes, each consisting of tasks, to be processed by a system of K processors in tandem. Each process, immediately after havings its i-th (1≤i<K) task served by the i-th processor, is queued up on a first-come-first-served basis for processing of its (i+1)-st task by the (i+1)-st processor, until all its tasks have been served. Define the execution time to be the time to serve all the processes in the set.

Given that the processes form a stationary and ergodic sequence, as far as the processing times of the tasks are concernec, the asymptotic execution time is first explicitly computed, as the number of processes tends to infinity. This is done by using subadditive ergodic theory and certain results from the stability theory of tandem queueing networks. Then, design issues of such systems are discussed.