scholarly journals Transient solution to the time-dependent multiserver Poisson queue

2005 ◽  
Vol 42 (03) ◽  
pp. 766-777 ◽  
Author(s):  
B. H. Margolius
Keyword(s):  
Random Walk ◽  
Transition Probabilities ◽  
Arrival Rate ◽  
Time Dependent ◽  
Time Varying ◽  
Expected Number ◽  
Straightforward Application ◽  
Transient Probabilities ◽  
Poisson Arrivals ◽  
The Right

We derive an integral equation for the transient probabilities and expected number in the queue for the multiserver queue with Poisson arrivals, exponential service for time-varying arrival and departure rates, and a time-varying number of servers. The method is a straightforward application of generating functions. We can express p ĉ−1(t), the probability that ĉ − 1 customers are in the queue or being served, in terms of a Volterra equation of the second kind, where ĉ is the maximum number of servers working during the day. Each of the other transient probabilities is expressed in terms of integral equations in p ĉ−1(t) and the transition probabilities of a certain time-dependent random walk. In this random walk, the rate of steps to the right equals the arrival rate of the queue and the rate of steps to the left equals the departure rate of the queue when all servers are busy.

scholarly journals Transient solution to the time-dependent multiserver Poisson queue

2005 ◽  
Vol 42 (3) ◽  
pp. 766-777 ◽  
Author(s):  
B. H. Margolius
Keyword(s):  
Random Walk ◽  
Transition Probabilities ◽  
Arrival Rate ◽  
Time Dependent ◽  
Time Varying ◽  
Expected Number ◽  
Straightforward Application ◽  
Transient Probabilities ◽  
Poisson Arrivals ◽  
The Right

We derive an integral equation for the transient probabilities and expected number in the queue for the multiserver queue with Poisson arrivals, exponential service for time-varying arrival and departure rates, and a time-varying number of servers. The method is a straightforward application of generating functions. We can express pĉ−1(t), the probability that ĉ − 1 customers are in the queue or being served, in terms of a Volterra equation of the second kind, where ĉ is the maximum number of servers working during the day. Each of the other transient probabilities is expressed in terms of integral equations in pĉ−1(t) and the transition probabilities of a certain time-dependent random walk. In this random walk, the rate of steps to the right equals the arrival rate of the queue and the rate of steps to the left equals the departure rate of the queue when all servers are busy.

Dynamics of Elastic Objects under Movable Inertial Loading - Some Features of the Mathematical Models and Analogies

2019 ◽  
Vol 968 ◽  
pp. 427-436
Author(s):  
Anatoliy G. Demianenko
Keyword(s):  
Mathematical Models ◽  
Mathematical Physics ◽  
Time Dependent ◽  
Time Varying ◽  
Varying Length ◽  
Inertial Loading ◽  
The Right ◽  
Elastic Elements

This paper describes some features and analogies of the mathematical models for the elastic elements with movable load and for the elastic elements of changeable length. In these systems two forms of own oscillations - the own component and the accompanying one, displaced in phase to the right angle correspond to every frequency of the system. The accompanying component is caused by the mobile inertia load or by the changeable length and they are not trivial only when this factor exists. As for objects with time-varying length, these problems lie in outside of the scope classical problems of mathematical physics due to that the eigenfrequencies and eigenforms become time-dependent functions. This non-classical section of the mathematical physics is waiting for its development, new researches and generalizations.

On the Expected Number of Visits of a Particle before Absorption in a Correlated Random Walk

1973 ◽  
Vol 16 (3) ◽  
pp. 389-395 ◽  
Author(s):  
G. C. Jain
Keyword(s):  
Random Walk ◽  
Correlated Random Walk ◽  
Expected Number ◽  
Unit Distance ◽  
Straight Line ◽  
Particle Move ◽  
Number Of Visits ◽  
The Right ◽  
Previous Step

Let a particle move along a straight line a unit distance during every interval of time τ. During the first interval τ it moves to the right with probability ρ1 and to the left with probability ρ2 = 1 - ρ1. Thereafter at the end of each interval τ, the particle with probability p continues its motion in the same direction as in the previous step and with probability q = l - p reverses it.

Symmetrizable Difference Dchemes

2005 ◽  
Vol 5 (1) ◽  
pp. 3-50 ◽  
Author(s):  
Alexei A. Gulin
Keyword(s):  
Initial Data ◽  
Difference Scheme ◽  
Difference Schemes ◽  
Time Dependent ◽  
Stability Boundaries ◽  
Typical Representative ◽  
Variable Weight ◽  
The Stability ◽  
Conditions Of Stability ◽  
The Right

AbstractA review of the stability theory of symmetrizable time-dependent difference schemes is represented. The notion of the operator-difference scheme is introduced and general ideas about stability in the sense of the initial data and in the sense of the right hand side are formulated. Further, the so-called symmetrizable difference schemes are considered in detail for which we manage to formulate the unimprovable necessary and su±cient conditions of stability in the sense of the initial data. The schemes with variable weight multipliers are a typical representative of symmetrizable difference schemes. For such schemes a numerical algorithm is proposed and realized for constructing stability boundaries.

The solution of an infinite set of differential-difference equations occurring in polymerization and queueing problems

1963 ◽  
Vol 59 (1) ◽  
pp. 117-124 ◽  
Author(s):  
A. Wragg
Keyword(s):  
Difference Equations ◽  
Queueing Theory ◽  
Bessel Functions ◽  
Formal Solution ◽  
Monomer Concentration ◽  
Arrival Rate ◽  
Time Dependent ◽  
Type Integral ◽  
Dependent Parameter ◽  
Infinite Set

AbstractThe time-dependent solutions of an infinite set of differential-difference equations arising from queueing theory and models of ‘living’ polymer are expressed in terms of modified Bessel functions. Explicit solutions are available for constant values of a parameter describing the arrival rate or monomer concentration; for time-dependent parameter a formal solution is obtained in terms of a function which satisfies a Volterra type integral equation of the second kind. These results are used as the basis of a numerical method of solving the infinite set of differential equations when the time-dependent parameter itself satisfies a differential equation.

Stochastic inflation as a time-dependent random walk

1989 ◽  
Vol 39 (8) ◽  
pp. 2245-2252 ◽  
Author(s):  
Henry E. Kandrup
Keyword(s):  
Random Walk ◽  
Time Dependent
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