A stationary representation

2000 ◽  
pp. 53-58
Author(s):  
Dmitri R. Yafaev
Keyword(s):  
Stationary Representation

Exponential approximation of waiting time and queue size for queues in heavy traffic

1990 ◽  
Vol 22 (1) ◽  
pp. 230-240 ◽  
Author(s):  
Władysław Szczotka
Keyword(s):  
Waiting Time ◽  
Wide Class ◽  
Time Distribution ◽  
Heavy Traffic ◽  
Single Server ◽  
Queue Size ◽  
Exponential Approximation ◽  
Waiting Time Distribution ◽  
Stationary Waiting Time ◽  
Stationary Representation

An exponential approximation for the stationary waiting time distribution and the stationary queue size distribution for single-server queues in heavy traffic is given for a wide class of queues. This class contains for example not only queues for which the generic sequence, i.e. the sequence of service times and interarrival times, is stationary but also such queues for which the generic sequence is asymptotically stationary in some sense. The conditions ensuring the exponential approximation of the characteristics considered in heavy traffic are expressed in terms of the invariance principle for the stationary representation of the generic sequence and its first two moments.

Exponential approximation of waiting time and queue size for queues in heavy traffic

1990 ◽  
Vol 22 (01) ◽  
pp. 230-240 ◽  
Author(s):  
Władysław Szczotka
Keyword(s):  
Waiting Time ◽  
Wide Class ◽  
Time Distribution ◽  
Heavy Traffic ◽  
Single Server ◽  
Queue Size ◽  
Exponential Approximation ◽  
Waiting Time Distribution ◽  
Stationary Waiting Time ◽  
Stationary Representation

An exponential approximation for the stationary waiting time distribution and the stationary queue size distribution for single-server queues in heavy traffic is given for a wide class of queues. This class contains for example not only queues for which the generic sequence, i.e. the sequence of service times and interarrival times, is stationary but also such queues for which the generic sequence is asymptotically stationary in some sense. The conditions ensuring the exponential approximation of the characteristics considered in heavy traffic are expressed in terms of the invariance principle for the stationary representation of the generic sequence and its first two moments.

Entropy of killed-resurrected stationary Markov chains

2021 ◽  
Vol 58 (1) ◽  
pp. 177-196
Author(s):  
Servet Martínez
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Probability Measure ◽  
Markov Chains ◽  
Stochastic Matrix ◽  
Hitting Times ◽  
Stationary Distributions ◽  
Natural Factors ◽  
Absorbing States ◽  
Stationary Representation

AbstractWe consider a strictly substochastic matrix or a stochastic matrix with absorbing states. By using quasi-stationary distributions we show that there is an associated canonical Markov chain that is built from the resurrected chain, the absorbing states, and the hitting times, together with a random walk on the absorbing states, which is necessary for achieving time stationarity. Based upon the 2-stringing representation of the resurrected chain, we supply a stationary representation of the killed and the absorbed chains. The entropies of these representations have a clear meaning when one identifies the probability measure of natural factors. The balance between the entropies of these representations and the entropy of the canonical chain serves to check the correctness of the whole construction.

Is Germany's GDP Trend-Stationary? A Measurement-With-Theory Approach / Ist das deutsche BIP trendstationär: Ein Measurement-With-Theory Ansatz

2005 ◽  
Vol 225 (1) ◽  
Author(s):  
Bernd Lücke
Keyword(s):  
World War Ii ◽  
Unit Root ◽  
Linear Trend ◽  
Unit Root Tests ◽  
Theory Approach ◽  
World War ◽  
Standard Unit ◽  
Trend Model ◽  
Catch Up ◽  
Stationary Representation

SummaryThe time series properties of German GDP have been re-examined in recent research. Extending the sample to include GDP data from 1950 onwards, some researchers argued in favor of a trend-stationary rather than difference stationary representation of real log GDP. I show that this conclusion is based on an atheoretic trend model underlying the unit root tests. A simple linear trend model fails to take the post World-War II catch-up process properly into account. I use the Solow growth model to discriminate between transitional catch-up dynamics and longrun equilibrium growth. With the proper transformation of GDP data, I am able to use standard unit root tests and find that both ADF and KPSS tests suggest a difference stationary model. This evidence is supported by non-standard unit root tests which allow for polynomial trend representations.

Stationary representation of queues. II

1986 ◽  
Vol 18 (3) ◽  
pp. 849-859 ◽  
Author(s):  
Władysław Szczotka
Keyword(s):  
Asymptotic Behaviour ◽  
Waiting Time ◽  
Heavy Traffic ◽  
The Other ◽  
Traffic Situation ◽  
Queue Size ◽  
Stationary Representation

The paper is a continuation of [7]. One of the main results is as follows: if the sequence (w, v, u) is asymptotically stationary in some sense then (l, w, v, u) is asymptotically stationary in the same sense. The other main result deals with an asymptotic behaviour of the vector of the queue size and the waiting time in the heavy-traffic situation. This result resembles a formula of the Little type.

Stationary representation of queues. I

1986 ◽  
Vol 18 (3) ◽  
pp. 815-848 ◽  
Author(s):  
Władysław Szczotka
Keyword(s):  
Weak Convergence ◽  
Asymptotic Behaviour ◽  
Waiting Time ◽  
Initial Conditions ◽  
Probability Distributions ◽  
Stationary Sequence ◽  
Limiting Distribution ◽  
Stationary Representation

The paper deals with the asymptotic behaviour of queues for which the generic sequence is not necessarily stationary but is asymptotically stationary in some sense. The latter property is defined by an appropriate type of convergence of probability distributions of the sequences to the distribution of a stationary sequence We consider six types of convergence of to The main result is as follows: if the sequence of the distributions converges in one of six ways then the sequence of distributions of the sequences converges in the same way, independently of initial conditions. Furthermore the limiting distribution is the same as the limiting distribution obtained by the weak convergence of the distributions Here wk and w∗k denote the waiting time of the kth unit in the queue generated by (v, u) and (v0, u0) respectively.

Stationary representation of queues. II

1986 ◽  
Vol 18 (03) ◽  
pp. 849-859 ◽  
Author(s):  
Władysław Szczotka
Keyword(s):  
Asymptotic Behaviour ◽  
Waiting Time ◽  
Heavy Traffic ◽  
The Other ◽  
Traffic Situation ◽  
Queue Size ◽  
Stationary Representation

The paper is a continuation of [7]. One of the main results is as follows: if the sequence ( w, v, u ) is asymptotically stationary in some sense then ( l , w, v, u ) is asymptotically stationary in the same sense. The other main result deals with an asymptotic behaviour of the vector of the queue size and the waiting time in the heavy-traffic situation. This result resembles a formula of the Little type.

Stationary representation of queues. I

1986 ◽  
Vol 18 (03) ◽  
pp. 815-848 ◽  
Author(s):  
Władysław Szczotka
Keyword(s):  
Weak Convergence ◽  
Asymptotic Behaviour ◽  
Waiting Time ◽  
Initial Conditions ◽  
Probability Distributions ◽  
Stationary Sequence ◽  
Limiting Distribution ◽  
Stationary Representation

The paper deals with the asymptotic behaviour of queues for which the generic sequence is not necessarily stationary but is asymptotically stationary in some sense. The latter property is defined by an appropriate type of convergence of probability distributions of the sequences to the distribution of a stationary sequence We consider six types of convergence of to The main result is as follows: if the sequence of the distributions converges in one of six ways then the sequence of distributions of the sequences converges in the same way, independently of initial conditions. Furthermore the limiting distribution is the same as the limiting distribution obtained by the weak convergence of the distributions Here wk and w∗ k denote the waiting time of the kth unit in the queue generated by ( v, u ) and ( v 0, u 0) respectively.

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