Kanwar Sen

GERT Analysis of

An m-consecutive-k-out-of-n:F system, introduced by W.S. Griffith, consists of an ordered linear sequence of n i.i.d. components that fails iff there are at least m non-overlapping runs of k consecutive failed components. However, a situation may occur in which an ordered linear sequence of n i.i.d. components fails iff there are at least m non-overlapping runs of at least k consecutive failed components. We call such a system an m-consecutive-at least-k-out-of-n:F system. This paper presents a graphical evaluation and review technique (GERT) analysis of both types of systems providing closed form explicit formulae for reliability evaluation in a unified manner. GERT, besides providing a visual picture of the system, helps to analyse the system in a less inductive manner. Numerical examples for each system are studied in detail by computing the reliability for various combinations of sets of values of the parameters involved. It is observed that m-consecutive-at least-k-out-of-n:F systems are more reliable than m-consecutive-k-out-of-n:F systems as the number of possible state combinations leading to systems failure are larger in the latter. Mathematica is used for systematic computations. Numerical investigations illustrate the efficiency of GERT in reliability analysis of such systems. In comparison with the existing formulae of m-consecutive-k-out-of-n:F systems for i.i.d. components, the formula obtained by GERT analysis, to be referred to as GERT-F, is much more efficient owing to its significantly low computational time, and easy implementation

m

This paper proposes a new model, a combined m-consecutive-k-out-of-n : F & consecutive-kc-out-of-n : F system, consisting of n components ordered in a line such that the system fails iff there exist at least kc consecutive failed components, or at least m non-overlapping runs of k consecutive failed components, where kc < mk. The components are assumed to be i.i.d. An algorithm for system reliability evaluation, along with an illustrative numerical example, is provided based on the analysis of the system using graphical evaluation and review technique (GERT). The algorithm is also evaluated in terms of the order of computational time. This model has applications to various complex systems such as infrared (IR) detecting and signal processing, and bank automatic payment systems.

-Consecutive-

This paper addresses the heterogeneous redundancy allocation problem in multi-state series-parallel reliability structures with the objective to minimize the total cost of system design satisfying the given reliability constraint and the consumer load demand. The demand distribution is presented as a piecewise cumulative load curve and each subsystem is allowed to consist of parallel redundant components of not more than three types. The system uses binary capacitated components chosen from a list of available products to provide redundancy so as to increase system performance and reliability. The components are characterized by their feeding capacity, reliability and cost. A system that consists of elements with different reliability and productivity parameters has the capacity strongly dependent upon the selection of constituent components. A binomial probability based method to compute exact system reliability index is suggested. To analyze the problem and suggest an optimal/near-optimal system structure, an ant colony optimization algorithm has been presented. The solution approach consists of a series of simple steps as used in early ant colony optimization algorithms dealing with other optimization problems and offers straightforward analysis. Four multi-state system design problems have been solved for illustration. Two problems are taken from the literature and solved to compare the algorithm with the other existing methods. The other two problems are based upon randomly generated data. The results show that the method can be appealing to many researchers with regard to the time efficiency and yet without compromising over the solution quality.

k

Little attention has been given to the use of combinatorial methods which could prove to be much easier yielding results in situations which otherwise become intractable. We have discussed in this paper how combinatoric analysis of lattice paths representing the queueing process lead to the transient solutions without lending into complicated analysis of the techniques used earlier. Specially we will be discussing the transient solution of M/M/1 queueing model with (0,k) control policy which perhaps has not been discussed in the literature. Known results have also been verified.

-Out-of-

Some of the known results in transient behaviour of the Markovian queueing models such as the distribution of X(r)-number of units in the system at time t, joint distribution of length of the busy period and the number of units served, and the maximum length during a busy period have been derived using the combinatoric analysis of lattice paths representing the queueing process thus unifying the earlier results on transient solution of Markovian queueing models. AMS Subject Classification: Primary 60k25; secondary 6OCO5. Considerable attention has been paid to obtain the transient solution for the system size and the distribution for the length of a busy period of an M/M/l queue- ing system. A number of methods have been put forward and most of them require setting up of differential difference equations. Champernowne (1956) has used the combinatorial method for the first time for the same problem. Takacs (1962, 1967) adopted combinatorial approach which involved the use of celebrated ballot theorem to study sequences of inter-arrival and service times. Mohanty (1979) in- vestigated some results treating queueing system as a random walk on the lattice in the plane and then used generating functions. Neuts (1964) developed a new ap- proach avoiding the use of generating function and Rouches theorem. Recently Mohanty and Panny (1990) obtained transient results by employing the technique of first discretizing the continuous time model and then representing the same by a random walk path. In this paper, we have rederived transient solutions by first discretizing the model and then using the combinatorics through the so called lattice path method. The classical transient results for M/M/l queueing system provided little insight into the behaviour of the queueing system through a fixed operation

n

This paper aims at deriving explicit transient queue length distribution for GI/M/1 system and busy period analysis of bulk queue GIb/M/1 through lattice paths (LPs) combinatorics. The general interarrival time distribution is approximated by two-phase Cox distribution, C2, that has Markovian property, enabling us to represent the processes by two-dimensional LPs. As distributions C2 cover a wide class of distributions that have rational Laplace–Stieltjes transforms (LSTs) with square coefficient of variation lying in [12,∞), the results obtained are applicable to a large class of real life situations. Some numerical results for the C2b/M/1 model are also given.

Systems

A test is proposed for testing bivariate exponentiality against the Bivariate Increasing Failure Rate (BIFR) class of alternatives. The test statistic is a function of U-statistics and hence asymptotically normally distributed and consistent

Combined

We consider a {0,1}-valuedm-th order stationary Markov chain. We study the occurrences of runs where two 1’s are separated byat most/exactly/at least k 0’s under the overlapping enumeration scheme wherek≥0 and occurrences of scans (at leastk1 successes in a window of length at mostk, 1≤k1≤k) under both non-overlapping and overlapping enumeration schemes. We derive the generating function of first two types of runs. Under the conditions, (1) strong tendency towards success and (2) strong tendency towards reversing the state, we establish the convergence of waiting times of ther-th occurrence of runs and scans to Poisson type distributions. We establish the central limit theorem and law of the iterated logarithm for the number of runs and scans up to timen.

m

Abstract In this paper, on approximating the general service time distribution by Coxian two-phase distribution, C 2 , explicit busy period density function for non-Markovian finite queues M/G/1/ N has been obtained by using lattice paths combinatorics (LPC). The results obtained are specifically important as it seems that no explicit transient solutions are available for finite queues. Two-phase Cox distribution, C 2 , has Markovian property and is thus amenable to lattice path (LP) analysis. The transient results obtained are applicable to almost any real life finite queuing system M/G/1/ N .

-Consecutive-

Abstract In this paper, based on Polya–Eggenberger model we have developed a generalized Polya–Eggenberger model of order k via lattice path approach. It generates a generalized Polya–Eggenberger distribution Kanwar Sen and Mishra (Sankhya Ser. A Part-2 58 (1996) 243). It generates a number of other new discrete distributions of order k , namely, uniform distribution of order k , beta-binomial distribution of order k , factorial distribution of order k , beta-Pascal distribution of order k , and Haight distribution of order k , as particular cases, besides the already known ones. Distributions of order k can be applied in obtaining reliability of some complex systems, viz, consecutive- k -out-of- n : F system.