Alan Krinik

Transient probabilities of a single server priority queueing system

Transient probability functions of a single server preemptive (repeat-identical) two priority queueing system are determined using the randomization solution form and lattice path combinatorics.

Gambler's Ruin with Catastrophes and Windfalls

We compute ruin probabilities, in both infinite-time and finite-time, for a Gambler’s Ruin problem with both catastrophes and windfalls in addition to the customary win/loss probabilities. For constant transition probabilities, the infinite-time ruin probabilities are derived using difference equations. Finite-time ruin probabilities of a system having constant win/loss probabilities and variable catastrophe/windfall probabilities are determined using lattice path combinatorics. Formulae for expected time till ruin and the expected duration of gambling are also developed. The ruin probabilities (in infinite-time) for a system having variable win/loss/catastrophe probabilities but no windfall probability are found. Finally, the infinite-time ruin probabilities of a system with variable win/loss/catastrophe/windfall probabilities are determined.

Taylor series solution of the M/M/1 queueing system

Abstract A Taylor series method for determining the transient probabilities of the classical single server queueing system is presented. The method is direct, practical and avoids Bessel function theory, Laplace transform theory and complex analysis. The resulting Taylor series are proved to converge for all time under arbitrary initial conditions. A method to obtain explicit Taylor series representations is demonstrated.

Stochastic Processes and Functional Analysis: A Volume of Recent Advances in Honor of M. M. Rao

This extraordinary compilation is an expansion of the recent American Mathematical Society Special Session celebrating M. M. Raos distinguished career and includes most of the presented papers as well as ancillary contributions from session invitees. This book shows the effectiveness of abstract analysis for solving fundamental problems of stochastic theory, specifically the use of functional analytic methods for elucidating stochastic processes, as made manifest in M. M. Raos prolific research achievements. Featuring a biography of M. M. Rao, a complete bibliography of his published works, and meditations from former students, the book includes contributions from over 30 notable researchers.

Markov Processes with Constant Transition Rates of Size One & Two

The two classical single server queueing systems, M/M/1 and M/M/1/N are generalized to allow constant transition rates of size two in addition to the standard constant transitions rates of size one. In terms of the queueing models, these new systems each allow customers to arrive or be served instantly in pairs as well as individually. The steady state distributions are explicitly determined and a condition for the existence of a steady state distribution is established in the infinite-state space case. Assuming that a steady state condition prevails, the canonical performance measures are determined. Expressions for the average number of customers in either system or queue are derived. Formulae for the average waiting time that a customer spends in each system or queue are also developed.