Some analyses on the control of queues using level crossings of regenerative processes

1980 ◽  
Vol 17 (3) ◽  
pp. 814-821 ◽  
Author(s):  
J. G. Shanthikumar
Keyword(s):  
Waiting Time ◽  
Time Distribution ◽  
Queueing Systems ◽  
Density Functions ◽  
Waiting Time Distribution ◽  
Regenerative Processes ◽  
Stieltjes Transform ◽  
Expected Number ◽  
Control Of Queues ◽  
Special Case

Some properties of the number of up- and downcrossings over level u, in a special case of regenerative processes are discussed. Two basic relations between the density functions and the expected number of upcrossings of this process are derived. Using these reults, two examples of controlled M/G/1 queueing systems are solved. Simple relations are derived for the waiting time distribution conditioned on the phase of control encountered by an arriving customer. The Laplace-Stieltjes transform of the distribution function of the waiting time of an arbitrary customer is also derived for each of these two examples.

Some analyses on the control of queues using level crossings of regenerative processes

1980 ◽  
Vol 17 (03) ◽  
pp. 814-821 ◽  
Author(s):  
J. G. Shanthikumar
Keyword(s):  
Waiting Time ◽  
Time Distribution ◽  
Queueing Systems ◽  
Density Functions ◽  
Waiting Time Distribution ◽  
Regenerative Processes ◽  
Stieltjes Transform ◽  
Expected Number ◽  
Control Of Queues ◽  
Special Case

Some properties of the number of up- and downcrossings over level u, in a special case of regenerative processes are discussed. Two basic relations between the density functions and the expected number of upcrossings of this process are derived. Using these reults, two examples of controlled M/G/1 queueing systems are solved. Simple relations are derived for the waiting time distribution conditioned on the phase of control encountered by an arriving customer. The Laplace-Stieltjes transform of the distribution function of the waiting time of an arbitrary customer is also derived for each of these two examples.

On the waiting time distribution in a generalized queueing system with uniformly bounded sojourn times

1972 ◽  
Vol 9 (3) ◽  
pp. 642-649 ◽  
Author(s):  
Jacqueline Loris-Teghem
Keyword(s):  
Generating Function ◽  
Waiting Time ◽  
Service Time ◽  
Time Distribution ◽  
Queueing System ◽  
Waiting Time Distribution ◽  
Stieltjes Transform ◽  
Sojourn Times ◽  
Stieltjes Transforms ◽  
Uniformly Bounded

A generalized queueing system with (N + 2) types of triplets (delay, service time, probability of joining the queue) and with uniformly bounded sojourn times is considered. An expression for the generating function of the Laplace-Stieltjes transforms of the waiting time distributions is derived analytically, in a case where some of the random variables defining the model have a rational Laplace-Stieltjes transform.The standard Kl/Km/1 queueing system with uniformly bounded sojourn times is considered in particular.

A laplace transform representation in a class of renewal queueing and risk processes

1999 ◽  
Vol 36 (02) ◽  
pp. 570-584 ◽  
Author(s):  
Gordon E. Willmot
Keyword(s):  
Laplace Transform ◽  
Waiting Time ◽  
Time Distribution ◽  
Waiting Time Distribution ◽  
Stieltjes Transform ◽  
Ruin Theory ◽  
Occurrence Time ◽  
Queueing Processes ◽  
Risk Processes

For a class of renewal queueing processes characterized by a rational Laplace–Stieltjes transform of the arrival inter-occurrence time distribution, the Laplace–Stieltjes transform of the equilibrium (actual) waiting time distribution is re-expressed in a manner which facilitates explicit inversion under certain conditions. The results are of interest in other contexts as well, as for example in insurance ruin theory. Various analytic properties of these quantities are then obtained as a result.

scholarly journals M/G/1/K system with push-out scheme under vacation policy

1996 ◽  
Vol 9 (2) ◽  
pp. 143-157 ◽  
Author(s):  
Shoji Kasahara ◽  
Hideaki Takagi ◽  
Yutaka Takahashi ◽  
Toshiharu Hasegawa
Keyword(s):  
Waiting Time ◽  
Time Distribution ◽  
Waiting Times ◽  
Numerical Results ◽  
The Other ◽  
Waiting Time Distribution ◽  
Stieltjes Transform ◽  
Multiple Vacations ◽  
Push Out

We consider an M/G/1/K system with push-out scheme and multiple vacations. This model is particularly important in situations where it is essential to provide short waiting times to messages which are selected for service. We analyze the behavior of two types of messages: one that succeeds in transmission and the other that fails. We derive the Laplace-Stieltjes transform of the waiting time distribution for the message which is eventually served. Finally, we show some numerical results including the comparisons between the push-out and the ordinary blocking models.

On the waiting time distribution in a generalized queueing system with uniformly bounded sojourn times

1972 ◽  
Vol 9 (03) ◽  
pp. 642-649
Author(s):  
Jacqueline Loris-Teghem
Keyword(s):  
Generating Function ◽  
Waiting Time ◽  
Service Time ◽  
Time Distribution ◽  
Queueing System ◽  
Waiting Time Distribution ◽  
Stieltjes Transform ◽  
Sojourn Times ◽  
Stieltjes Transforms ◽  
Uniformly Bounded

A generalized queueing system with (N+ 2) types of triplets (delay, service time, probability of joining the queue) and with uniformly bounded sojourn times is considered. An expression for the generating function of the Laplace-Stieltjes transforms of the waiting time distributions is derived analytically, in a case where some of the random variables defining the model have a rational Laplace-Stieltjes transform.The standardKl/Km/1 queueing system with uniformly bounded sojourn times is considered in particular.

scholarly journals Single-server queueing systems with uniformly limited queueing time

1964 ◽  
Vol 4 (4) ◽  
pp. 489-505 ◽  
Author(s):  
D. J. Daley
Keyword(s):  
Integral Equation ◽  
Waiting Time ◽  
Time Distribution ◽  
Busy Period ◽  
Queueing System ◽  
Queueing Systems ◽  
Single Server ◽  
Waiting Time Distribution ◽  
Loss Distribution ◽  
Queueing Time

SummaryThe paper considers the queueing system GI/G/1 with a type of customer impatience, namely, that the total queueing-time is uniformly limited. Using Lindiley's approach [10], an integral equation for the limiting waiting- time distribution is derived, and this is solved explicitly for M/G/1 using an expansion of the Pollaczek-Khintchine formula. It is also solved, in principle for Ej/G/l, and explicitly for Ej/Ek/l. A duality noted between GIA(x)/GB(x)/l and GIB(x)/GA(x)/l relates solutions for GI/Ek/l to Ek/G/l. Finally the equation for the busy period in GI/G/l is derived and related to the no-customer-loss distribution and dual distributions.

A laplace transform representation in a class of renewal queueing and risk processes

1999 ◽  
Vol 36 (2) ◽  
pp. 570-584 ◽  
Author(s):  
Gordon E. Willmot
Keyword(s):  
Laplace Transform ◽  
Waiting Time ◽  
Time Distribution ◽  
Waiting Time Distribution ◽  
Stieltjes Transform ◽  
Ruin Theory ◽  
Occurrence Time ◽  
Queueing Processes ◽  
Risk Processes

For a class of renewal queueing processes characterized by a rational Laplace–Stieltjes transform of the arrival inter-occurrence time distribution, the Laplace–Stieltjes transform of the equilibrium (actual) waiting time distribution is re-expressed in a manner which facilitates explicit inversion under certain conditions. The results are of interest in other contexts as well, as for example in insurance ruin theory. Various analytic properties of these quantities are then obtained as a result.

Asymptotic correlation in a queue

1969 ◽  
Vol 6 (03) ◽  
pp. 573-583 ◽  
Author(s):  
B. D. Craven
Keyword(s):  
Characteristic Function ◽  
Stationary State ◽  
Waiting Time ◽  
Time Distribution ◽  
Queueing Systems ◽  
Traffic Intensity ◽  
Waiting Time Distribution ◽  
General Applicability ◽  
The Difference

Let Xt denote the waiting time of customer t in a stationary GI/G/1 queue, with traffic intensity τ; let ρn denote the correlation between Xt and Xt+n. For a rational GI/G/1 queue, in which the distribution of the difference between arrival and service intervals has a rational characteristic function, it is shown that, for large n, ρn is asymptotically proportional to n– 3/2 e –βn , where β and the factor of proportionality are calculable. The asymptotic law n –3/2 e–βn applies also to the approach of the waiting-time distribution to the stationary state in an initially empty rational GI/G/1 queue, and to the correlations in the queueing systems recently analysed by Cohen [1]. Its more general applicability is discussed.

Waiting time distribution in M/D/1 queueing systems

1999 ◽  
Vol 35 (25) ◽  
pp. 2184 ◽  
Author(s):  
V.B. Iversen ◽  
L. Staalhagen
Keyword(s):  
Waiting Time ◽  
Time Distribution ◽  
Queueing Systems ◽  
Waiting Time Distribution

Asymptotic correlation in a queue

1969 ◽  
Vol 6 (3) ◽  
pp. 573-583 ◽  
Author(s):  
B. D. Craven
Keyword(s):  
Characteristic Function ◽  
Stationary State ◽  
Waiting Time ◽  
Time Distribution ◽  
Queueing Systems ◽  
Traffic Intensity ◽  
Waiting Time Distribution ◽  
General Applicability ◽  
The Difference

Let Xt denote the waiting time of customer t in a stationary GI/G/1 queue, with traffic intensity τ; let ρn denote the correlation between Xt and Xt+n. For a rational GI/G/1 queue, in which the distribution of the difference between arrival and service intervals has a rational characteristic function, it is shown that, for large n, ρn is asymptotically proportional to n–3/2e–βn, where β and the factor of proportionality are calculable. The asymptotic law n–3/2e–βn applies also to the approach of the waiting-time distribution to the stationary state in an initially empty rational GI/G/1 queue, and to the correlations in the queueing systems recently analysed by Cohen [1]. Its more general applicability is discussed.

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