Sri Gopal Mohanty

Sooner and later waiting time problems for success and failure runs in higher order Markov dependent trials

The probability generating functions of the waiting times for the first success run of length k and for the sooner run and the later run between a success run of length k and a failure run of length r in the second order Markov dependent trials are derived using the probability generating function method and the combinatorial method. Further, the systems of equations of 2.mconditional probability generating functions of the waiting times in the m-th order Markov dependent trials are given. Since the systems of equations are linear with respect to the conditional probability generating functions, they can be solved exactly, and hence the probability generating functions of the waiting time distributions are obtained. If m is large, some computer algebra systems are available to solve the linear systems of equations.

Start-Up Demonstration Tests Under Markov Dependence Model with Corrective Actions

A general probability model for a start-up demonstration test is studied. The joint probability generating function of some random variables appearing in the Markov dependence model of the start-up demonstration test with corrective actions is derived by the method of probability generating function. By using the probability generating function, several characteristics relating to the distribution are obtained.

Success runs of lengthk in Markov dependent trials

The geometric type and inverse Polýa-Eggenberger type distributions of waiting time for success runs of lengthk in two-state Markov dependent trials are derived by using the probability generating function method and the combinatorial method. The second is related to the minimal sufficient partition of the sample space. The first two moments of the geometric type distribution are obtained. Generalizations to ballot type probabilities of which negative binomial probabilities are special cases are considered. Since the probabilities do not form a proper distribution, a modification is introduced and new distributions of orderk for Markov dependent trials are developed.

On a multiserver Markovian queueing system with balking and reneging

Abstract In this paper a multi-server queueing model with balking and reneging is considered. Steady-state distribution of the number of customers in the system is obtained. An expression for the average loss of customers during a fixed duration of time is also advanced.

The transient solution of M/M/1 queues under (M,N)-policy. A combinatorial approach

Abstract It has been demonstrated by several authors, that combinatorial methods can be successfully applied to derive certain probability distributions in queuing theory. Recently the authors have obtained the transient solution of (0,N)-policy M/M/1 queues with an arbitrary number of initial customers, by considering their discrete-time analogue and by using combinatorial arguments. In this note, we derive the transient solution of M/M/1 queues under (M,N)-policy by an alternative combinatorial method in which lattice paths with diagonal steps are counted. As a special case the result for ordinary M/M/1 queues is checked.

The combinatorics of birth–death processes and applications to queues

In this paper we present a combinatorial technique which allows the derivation of the transition functions of general birth-death processes. This method provides a flexible tool for the transient analysis of Markovian queueing systems with state dependent transition rates, like M/M/c models or systems with balking and reneging.

On the transient behavior of a finite birth-death process with an application

Abstract The transient distribution of the number in the system and the distribution of the length of a busy period for a finite birth and death process is derived by solving the system of linear equations of Laplace-Transforms and finding the inversions through the properties of tridiagonal symmetric matrices. It is proved that the distributions of a busy period is hyperexponential. The steady-state solution of the number of customers in the system is verified without any difficulty. The numerical solution of the number of customers in the system and the busy period is possible by the use of a high speed computer for which a multi-server queueing system with balking and reneging serves as an illustration. Some numerical comparisons are made with the randomization method.

Enumeration of higher-dimensional paths under restrictions

Abstract The enumeration of lattice paths lying between two boundaries in two dimensional space has been done and the explicit expression is a determinant. By considering a natural extension of the boundaries in higher dimensional space, a generalized recurrence relation is established, the solution of which gives the number of paths in higher dimension not crossing the boundaries, in a determinant form.

Transient analysis of queues with heterogeneous arrivals

In this paper we consider a discrete time queueing model where the time axis is divided into time slots of unit length. The model satisfies the following assumptions: (i) an event is either an arrival of typei of batch sizebi, i=1,...,r with probabilityαi or is a depature of a single customer with probabilityγ or zero depending on whether the queue is busy or empty; (ii) no more than one event can occur in a slot, therefore the probability that neither an arrival nor a departure occurs in a slot is 1−γ−⌆iαi or 1−⌆iαi according as the queue is busy or empty; (iii) events in different slots are independent. Using a lattice path representation in higher dimensional space we will derive the time dependent joint distribution of the number of arrivals of various types and the number of completed services. The distribution for the corresponding continuous time model is found by using weak convergence.

On an inclusion-exclusion formula based on the reflection principle

Abstract The n -candidate ballot problem corresponding to the standard Young tableau has been solved recently by Zeilberger (Discrete Math. 44 (1983) 325–326) by using the reflection principle. In this paper, a refinement of Zeilbergers approach is provided in which the reflection principle is formulated through the symmetric group and an inclusion-exclusion formula for the counting problem is developed. This approach reveals the nature in which successive applications of the reflection principle work.