Ryszard Szekli

Stochastic ordering and dependence in applied probability

1 Univariate Ordering.- 1.1 Construction of iid random variables.- 1.2 Strong ordering.- 1.3 Convex ordering.- 1.4 Conditional orderings.- 1.5 Relative inverse function orderings.- 1.6 Dispersive ordering.- 1.7 Compounding.- 1.8 Integral orderings for queues.- 1.9 Relative inverse orderings for queues.- 1.10 Loss systems.- 2 Multivariate Ordering.- 2.1 Strassens theorem.- 2.2 Coupling constructions.- 2.3 Conditioning.- 2.4 Markov processes.- 2.5 Point processes on R, martingales.- 2.6 Markovian queues and Jackson networks.- 2.7 Poissonian flows and product formula.- 2.8 Stochastic ordering of Markov processes.- 2.9 Stochastic ordering of point processes.- 2.10 Renewal processes.- 2.11 Comparison of replacement policies.- 2.12 Stochastically monotone networks.- 2.13 Queues with MR arrivals.- 3 Dependence.- 3.1 Association.- 3.2 MTP2.- 3.3 A general theory of positive dependence.- 3.4 Multivariate orderings and dependence.- 3.5 Negative association.- 3.6 Independence via uncorrelatedness.- 3.7 Association for Markov processes.- 3.8 Dependencies in Markovian networks.- 3.9 Dependencies in Markov renewal queues.- 3.10 Associated point processes.- A.- A.1 Probability spaces.- A.2 Distribution functions.- A.3 Examples of distribution functions.- A.4 Other characteristics of probability measures.- A.5 Random variables equal in distribution.- A.6 Bibliography.

M/M/1 Queueing systems with inventory

We derive stationary distributions of joint queue length and inventory processes in explicit product form for various M/M/1-systems with inventory under continuous review and different inventory management policies, and with lost sales. Demand is Poisson, service times and lead times are exponentially distributed. These distributions are used to calculate performance measures of the respective systems. In case of infinite waiting room the key result is that the limiting distributions of the queue length processes are the same as in the classical M/M/1/∞-system.

MR/GI/1 QUEUES WITH POSITIVELY CORRELATED ARRIVAL STREAM

The effects of dependencies (such as association) in the arrival process to a single server queue on mean queue lengths and mean waiting times are studied. Markov renewal arrival processes with a particular transition matrix for the underlying Markov chain are used which allow us to change dependency properties without at the same time changing distributional conditions. It turns out that correlations do not seem to be pure effects, and three main factors are studied: (a) differences in the mean interarrival times in the underlying Markov renewal process, (b) intensity in the Markov renewal jump process, (c) variability in the point processes underlying the Markov renewal process. It is shown that the mean queue length can be made arbitrarily large in the class of queues with the same interarrival distributions and the same service time distributions (with fixed smaller than one traffic intensity), by making (a) large enough and (b) small enough. The existence of the moments of interest is confirmed and some stochastic comparison results for actual waiting times

Stochastic ordering and thinning of point processes

We study the stochastic ordering of random measures and point processes generated by a partial order [mu]

ON THE CONCAVITY OF THE WAITING-TIME DISTRIBUTION IN SOME GI/G/1 QUEUES

In this paper the concavity property for the distribution of a geometric random sum (geometric compound) X, + .. + X is established under the assumption that X, are i.i.d. and have a DFR distribution. From this and the fact that the actual waiting time in GI/G/1 queues can be written as a geometric random sum, the concavity of the waiting-time distribution in GI/G/1 queues with a DFR service-time distribution is derived. The DFR property of the idle-period distribution in specialized GI/G/1 queues is also mentioned.

DEPENDENCIES IN MARKOVIAN NETWORKS

Monotonicity and correlation results for queueing network processes, generalized birth-death procgsses and generalized migration processes are obtained with respect to various orderings of the state space. We prove positive (e.g. association) and negative (e.g. negative association) correlations in space and positive correlations in time for different situations, in steady state as well as in the transient phase of the system. This yields exact bounds for joint probabilities in terms of their independent versions.

Compensator conditions for stochastic ordering of point processes

Sufficient conditions are given under which two simple point processes on the positive half-line can be stochastically compared as random elements of D (0,∞) or R ∞ + Using a martingale approach to point processes, the conditions are proposed via a compensator function family. Appropriate versions of the processes being compared are constructed on the same probability space. The results are illustrated by replacement policies and semi-Markov point processes.

Comparison of replacement policies via point processes

First, some basic concepts from the theory of point processes are recalled and expanded. Then some notions of stochastic comparisons, which compare whole processes, are introduced. The use of these notions is illustrated by stochastically comparing renewal and related processes. Finally, applications of the different notions of stochastic ordering of point processes to many replacement policies are given.

A queueing theoretical proof of increasing property of Polya frequency functions

Let X1,...,Xn be independent random variables with PF2 densities and [phi] an increasing function. Then E([phi](X1,...,Xn) [Sigma]i=1n X1 = s) is increasing in s, almost surely (Efron, 1965). We put this theorem into the context of queueing theory and provide an elementary proof for non-negative random variables.

Strong stationary duality for Möbius monotone Markov chains

For Markov chains with a finite, partially ordered state space, we show strong stationary duality under the condition of Möbius monotonicity of the chain. We give examples of dual chains in this context which have no downwards transitions. We illustrate general theory by an analysis of nonsymmetric random walks on the cube with an interpretation for unreliable networks of queues.