An explicit upper bound for the mean busy period in a GI/G/1 queue

1978 ◽  
Vol 15 (02) ◽  
pp. 452-455 ◽  
Author(s):  
Richard Loulou
Keyword(s):  
Random Walk ◽  
Upper Bound ◽  
Busy Period ◽  
Random Variables ◽  
Cycle Duration ◽  
Traffic Intensity ◽  
The Mean ◽  
Random Walk Theory ◽  
Busy Cycle

In this paper, an upper bound is derived for the mean busy cycle duration in GI/G/1 queues. The bound is of the form A/(1 – ρ), where ρ is the traffic intensity and A involves three moments of the basic random variables of the queue. The proof uses a well-known result of random walk theory.

An explicit upper bound for the mean busy period in a GI/G/1 queue

1978 ◽  
Vol 15 (2) ◽  
pp. 452-455 ◽  
Author(s):  
Richard Loulou
Keyword(s):  
Random Walk ◽  
Upper Bound ◽  
Busy Period ◽  
Random Variables ◽  
Cycle Duration ◽  
Traffic Intensity ◽  
The Mean ◽  
Random Walk Theory ◽  
Busy Cycle

In this paper, an upper bound is derived for the mean busy cycle duration in GI/G/1 queues. The bound is of the form A/(1 – ρ), where ρ is the traffic intensity and A involves three moments of the basic random variables of the queue. The proof uses a well-known result of random walk theory.

A simple construction of an upper bound for the mean of the maximum of n identically distributed random variables

1985 ◽  
Vol 22 (04) ◽  
pp. 844-851
Author(s):  
A. Gravey
Keyword(s):  
Upper Bound ◽  
Random Variables ◽  
Simple Construction ◽  
The Mean

This note gives a method of finding an upper bound for the mean of the maximum of n identically distributed non-negative random variables. The bound is explicitly given and numerically compared with the exact value of the mean of the maximum for some classical distributions (geometric, Poisson, Erlang, hyperexponential).

A simple construction of an upper bound for the mean of the maximum of n identically distributed random variables

1985 ◽  
Vol 22 (4) ◽  
pp. 844-851 ◽  
Author(s):  
A. Gravey
Keyword(s):  
Upper Bound ◽  
Random Variables ◽  
Simple Construction ◽  
The Mean

This note gives a method of finding an upper bound for the mean of the maximum of n identically distributed non-negative random variables. The bound is explicitly given and numerically compared with the exact value of the mean of the maximum for some classical distributions (geometric, Poisson, Erlang, hyperexponential).

Busy period in GIX/G/∞

1996 ◽  
Vol 33 (3) ◽  
pp. 815-829 ◽  
Author(s):  
Liming Liu ◽  
Ding-Hua Shi
Keyword(s):  
Service Time ◽  
Busy Period ◽  
Random Variables ◽  
Batch Service ◽  
Idle Period ◽  
Special Cases ◽  
Infinite Server Queues ◽  
General Relations ◽  
Number Of Customers ◽  
Busy Cycle

Busy period problems in infinite server queues are studied systematically, starting from the batch service time. General relations are given for the lengths of the busy cycle, busy period and idle period, and for the number of customers served in a busy period. These relations show that the idle period is the most difficult while the busy cycle is the simplest of the four random variables. Renewal arguments are used to derive explicit results for both general and special cases.

A new informative embedded Markov renewal process for the PH/G/1 queue

1986 ◽  
Vol 18 (02) ◽  
pp. 533-557 ◽  
Author(s):  
Marcel F. Neuts
Keyword(s):  
Markov Chain ◽  
Statistical Analysis ◽  
Queue Length ◽  
Busy Period ◽  
Waiting Times ◽  
Arrival Process ◽  
Embedded Markov Chain ◽  
Markov Renewal Process ◽  
The Mean ◽  
Busy Cycle

We consider a new embedded Markov chain for the PH/G/1 queue by recording the queue length, the phase of the arrival process and the number of services completed during the current busy period at the successive departure epochs. Algorithmically tractable matrix formulas are obtained which permit the analysis of the fluctuations of the queue length and waiting times during a typical busy cycle. These are useful in the computation of certain profile curves arising in the statistical analysis of queues. In addition, informative expressions for the mean waiting times in the stable GI/G/1 queue and a simple new algorithm to evaluate the waiting-time distributions for the stationary PH/PH/1 queue are obtained.

scholarly journals An improved approximation for assessing the statistical significance of molecular sequence features

2003 ◽  
Vol 40 (02) ◽  
pp. 427-441 ◽  
Author(s):  
S. Mercier ◽  
D. Cellier ◽  
D. Charlot
Keyword(s):  
Random Walk ◽  
Asymptotic Expression ◽  
Statistical Significance ◽  
Random Variables ◽  
Exact Distribution ◽  
Molecular Sequence ◽  
Sequence Features ◽  
Random Walk Theory ◽  
The One ◽  
New Formula

Using random walk theory, we first establish explicitly the exact distribution of the maximal partial sum of a sequence of independent and identically distributed random variables. This result allows us to obtain a new approximation of the distribution of the local score of one sequence. This approximation improves the one given by Karlin et al., which can be deduced from this new formula. We obtain a more accurate asymptotic expression with additional terms. Examples of application are given.

scholarly journals An improved approximation for assessing the statistical significance of molecular sequence features

2003 ◽  
Vol 40 (2) ◽  
pp. 427-441 ◽  
Author(s):  
S. Mercier ◽  
D. Cellier ◽  
D. Charlot
Keyword(s):  
Random Walk ◽  
Asymptotic Expression ◽  
Statistical Significance ◽  
Random Variables ◽  
Exact Distribution ◽  
Molecular Sequence ◽  
Sequence Features ◽  
Random Walk Theory ◽  
The One ◽  
New Formula

Using random walk theory, we first establish explicitly the exact distribution of the maximal partial sum of a sequence of independent and identically distributed random variables. This result allows us to obtain a new approximation of the distribution of the local score of one sequence. This approximation improves the one given by Karlin et al., which can be deduced from this new formula. We obtain a more accurate asymptotic expression with additional terms. Examples of application are given.

Busy period in GIX/G/∞

1996 ◽  
Vol 33 (03) ◽  
pp. 815-829
Author(s):  
Liming Liu ◽  
Ding-Hua Shi
Keyword(s):  
Service Time ◽  
Busy Period ◽  
Random Variables ◽  
Batch Service ◽  
Idle Period ◽  
Special Cases ◽  
Infinite Server Queues ◽  
General Relations ◽  
Number Of Customers ◽  
Busy Cycle

Busy period problems in infinite server queues are studied systematically, starting from the batch service time. General relations are given for the lengths of the busy cycle, busy period and idle period, and for the number of customers served in a busy period. These relations show that the idle period is the most difficult while the busy cycle is the simplest of the four random variables. Renewal arguments are used to derive explicit results for both general and special cases.

VII.—Some Contributions to the Theory of Random Walk and Multiple Scattering of Particles

1971 ◽  
Vol 69 (2) ◽  
pp. 115-124
Author(s):  
A. J. Allnutt ◽  
R. Fürth
Keyword(s):  
Distribution Function ◽  
Random Walk ◽  
Multiple Scattering ◽  
Three Dimensional ◽  
Mean Square ◽  
Cubic Lattice ◽  
First Approximation ◽  
The Mean ◽  
Random Walk Theory ◽  
Lateral Displacements

SynopsisA theory of the random walk with “persistence” of movement of a point in a three-dimensional cubic lattice is presented from which explicit expressions for the moments of the distribution function for the displacements of an ensemble of points after N steps for any arbitrary initial average velocity are derived. The results are applied to the problem of small angle multiple scattering of particles on their passage through a material medium, and formulae for the mean square of the lateral displacements are obtained which, in first approximation, have the form of the expressions, generally used for evaluating the experimental results but, in higher approximation, indicate a deviation from this relationship for greater thickness of matter.Another approach to the same problem of multiple scattering is further presented which is based on Kramer's stochastic differential equation for the distribution function for the position and velocities of an ensemble of particles in phase space. By this method formulae for the mean square of the scatter angles, the lateral displacements and the correlation products between these are derived. The first of these expressions shows again characteristic deviations from the usual ones for greater thickness of matter, the second coincides essentially with the expression obtained from the random walk theory.

A new informative embedded Markov renewal process for the PH/G/1 queue

1986 ◽  
Vol 18 (2) ◽  
pp. 533-557 ◽  
Author(s):  
Marcel F. Neuts
Keyword(s):  
Markov Chain ◽  
Statistical Analysis ◽  
Queue Length ◽  
Busy Period ◽  
Waiting Times ◽  
Arrival Process ◽  
Embedded Markov Chain ◽  
Markov Renewal Process ◽  
The Mean ◽  
Busy Cycle

We consider a new embedded Markov chain for the PH/G/1 queue by recording the queue length, the phase of the arrival process and the number of services completed during the current busy period at the successive departure epochs. Algorithmically tractable matrix formulas are obtained which permit the analysis of the fluctuations of the queue length and waiting times during a typical busy cycle. These are useful in the computation of certain profile curves arising in the statistical analysis of queues. In addition, informative expressions for the mean waiting times in the stable GI/G/1 queue and a simple new algorithm to evaluate the waiting-time distributions for the stationary PH/PH/1 queue are obtained.

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