# Stochastic Comparisons for Non-Markov Processes

@article{Whitt1986StochasticCF, title={Stochastic Comparisons for Non-Markov Processes}, author={Ward Whitt}, journal={Math. Oper. Res.}, year={1986}, volume={11}, pages={608-618} }

A technique is developed for comparing a non-Markov process to a Markov process on a general state space with many possible stochastic orderings. Two such comparisons with a common Markov process yield a comparison between two non-Markov processes. The technique, which is based on stochastic monotonicity of the Markov process, yields stochastic comparisons of the limiting distributions and the marginal distributions at single time points, but not the joint distributions. These stochastic…

## 60 Citations

### A coupling technique for stochastic comparison of functions of Markov Processes

- MathematicsAdv. Decis. Sci.
- 2000

The aim of this work is to obtain explicit conditions (i.e., conditions on the transition rates) for the stochastic comparison of Markov Processes by using a direct technique developed in particular in Liggett (1985) for interacting particle systems, the comparison by coupling.

### Stochastic comparison of Markov queueing networks using coupling.

- Mathematics
- 2010

In this thesis, a thorough description of stochastic comparison using coupling for the probability kernels of Markov processes is presented and an explicit coupling which preserves a subrelation of the coordinate-wise order relation is constructed, allowing to conclude that the steady-state distributions of the breakdown models arecoord-wise comparable.

### Monotonicity in Generalized Semi-Markov Processes

- MathematicsMath. Oper. Res.
- 1992

Stochastic monotonicity of the event epoch sequences of generalized semi-Markov processes is established and several existing results previously established using special properties of individual systems are unified.

### Computational methods for stochastic relations and Markovian couplings

- MathematicsVALUETOOLS
- 2009

The main contributions of the paper are an algorithmic characterization of stochastic relations between finite spaces, and a truncation approach for comparing infinite-state Markov processes.

### Insensitive bounds for the stationary distribution of non-reversible Markov chains

- Mathematics, Computer ScienceJournal of Applied Probability
- 1988

A general method is developed to compute easy bounds of the weighted stationary probabilities for networks of queues which do not satisfy the standard product form, and the bounds are insensitive with respect to service-time distributions.

### Stochastic Comparative Statics in Markov Decision Processes

- Computer ScienceMath. Oper. Res.
- 2021

This work derives both comparative statics results and stochastic comparativestatics results showing how the current and future optimal decisions change in response to changes in the single-period payoff function, the discount factor, the initial state of the system, and the transition probability function.

### COMPARISON OF LEVEL CROSSING TIMES FOR MARKOV AND SEMI-MARKOV PROCESSSES FÁTIMA

- Mathematics
- 2005

We address, using a sample-path approach, the level crossing ordering of stochastic processes with respect to integral stochastic orders closed for convolution. We extend results of Di Crescenzo and…

### Strong Stochastic Bounds for the Stationary Distribution of a Class of Multicomponent Performability Models

- MathematicsOper. Res.
- 1998

It is shown that the performability of multicomponent systems that do not satisfy these rules can be bounded by tractable modifications.

### Stochastic relations of random variables and processes

- Mathematics
- 2008

This paper generalizes the notion of stochastic order to a relation between probability measures over arbitrary measurable spaces. This generalization is motivated by the observation that for the…

## References

SHOWING 1-10 OF 18 REFERENCES

### Comparing Semi-Markov Processes

- MathematicsMath. Oper. Res.
- 1980

The construction is accomplished for semi-Markov processes for which all subprobability transition rates are absolutely continuous with failure rates uniformly bounded over finite intervals by representing the two semi- Markov processes as compositions of discrete-time stochastic processes with a sequence of Poisson processes.

### Comparing counting processes and queues

- MathematicsAdvances in Applied Probability
- 1981

Several partial orderings of counting processes are introduced and applied to compare stochastic processes in queueing models. The conditions for the queueing comparisons involve the counting…

### Stochastically monotone Markov Chains

- Mathematics
- 1968

SummaryA real-valued discrete time Markov Chain {Xn} is defined to be stochastically monotone when its one-step transition probability function pr {Xn+1≦y¦ Xn=x} is non-increasing in x for every…

### Comparison methods for queues and other stochastic models

- Mathematics
- 1986

Comparison properties of random variables and stochastic processes are given and are illustrated by application to various queueing models and questions in experimental design, renewal and reliability theory, PERT networks and branching processes.

### New stochastic orderings for Markov processes on partially ordered spaces

- MathematicsThe 23rd IEEE Conference on Decision and Control
- 1984

A unified theory of stochastic ordering for Markov processes on partially ordered state spaces is developed and found to be quite useful when analyzing multi-dimensional Stochastic models such as queueing networks.

### Poisson Arrivals See Time Averages

- MathematicsOper. Res.
- 1982

This paper presents a proof of this result under one basic assumption: the process being observed cannot anticipate the future jumps of the Poisson process.

### An operator-analytic approach to the Jackson network

- Computer Science, MathematicsJournal of Applied Probability
- 1984

Operator methods are used in this paper to systematically analyze the behavior of the Jackson network. Here, we consider rarely treated issues such as the transient behavior, and arbitrary…

### A family of bounds for the transient behavior of a Jackson network

- Computer ScienceJournal of Applied Probability
- 1986

Using operator methods, a family of stochastic bounds for the Jackson network is derived and new types of partial orders for Stochastic processes that are not equivalent to sample-path orderings are suggested.

### Open networks of queues: their algebraic structure and estimating their transient behavior

- MathematicsAdvances in Applied Probability
- 1984

We develop the mathematical machinery in this paper to construct a very general class of Markovian network queueing models. Each node has a heterogeneous class of customers arriving at their own…