Server sharing with a limited number of service positions and symmetric queues

1987 ◽  
Vol 24 (04) ◽  
pp. 990-1000 ◽  
Author(s):  
Benjamin Avi-Itzhak ◽  
Shlomo Halfin
Keyword(s):  
Finite Number ◽  
Generating Functions ◽  
Delay Time ◽  
Time Distribution ◽  
Equilibrium Distribution ◽  
Laplace Transforms ◽  
Processor Sharing ◽  
Time Sharing ◽  
Time Variance ◽  
Special Case

A class of M/G/1 time-sharing queues with a finite number of service positions and unlimited waiting space is described. The equilibrium distribution of symmetric queues belonging to this class is invariant under arbitrary service-independent reordering of the customers at instants of arrivals and departures. The delay time distribution, in the special case of one service position where preempted customers join the end of the line, is provided in terms of Laplace transforms and generating functions. It is shown that placing preempted customers at the end of the line rather than at the beginning of the line results in a reduction of the delay time variance. Comparisons with the delay time variance of the case of unlimited number of service positions (processor sharing system) are presented.

Server sharing with a limited number of service positions and symmetric queues

1987 ◽  
Vol 24 (4) ◽  
pp. 990-1000 ◽  
Author(s):  
Benjamin Avi-Itzhak ◽  
Shlomo Halfin
Keyword(s):  
Finite Number ◽  
Generating Functions ◽  
Delay Time ◽  
Time Distribution ◽  
Equilibrium Distribution ◽  
Laplace Transforms ◽  
Processor Sharing ◽  
Time Sharing ◽  
Time Variance ◽  
Special Case

A class of M/G/1 time-sharing queues with a finite number of service positions and unlimited waiting space is described. The equilibrium distribution of symmetric queues belonging to this class is invariant under arbitrary service-independent reordering of the customers at instants of arrivals and departures. The delay time distribution, in the special case of one service position where preempted customers join the end of the line, is provided in terms of Laplace transforms and generating functions. It is shown that placing preempted customers at the end of the line rather than at the beginning of the line results in a reduction of the delay time variance. Comparisons with the delay time variance of the case of unlimited number of service positions (processor sharing system) are presented.

THE MX/D/c BATCH ARRIVAL QUEUE

2005 ◽  
Vol 19 (3) ◽  
pp. 345-349 ◽  
Author(s):  
Geert Jan Franx
Keyword(s):  
Explicit Expression ◽  
Waiting Time ◽  
Probabilistic Analysis ◽  
Generating Functions ◽  
Time Distribution ◽  
Laplace Transforms ◽  
Batch Arrival ◽  
Waiting Time Distribution

A surprisingly simple and explicit expression for the waiting time distribution of the MX/D/c batch arrival queue is derived by a full probabilistic analysis, requiring neither generating functions nor Laplace transforms. Unlike the solutions known so far, this expression presents no numerical complications, not even for high traffic intensities.

Bounds on the extinction time distribution of a branching process

1974 ◽  
Vol 6 (2) ◽  
pp. 322-335 ◽  
Author(s):  
Alan Agresti
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Time Distribution ◽  
Branching Process ◽  
Probability Generating Function ◽  
Extinction Time ◽  
Expected Time ◽  
Time To Extinction ◽  
Special Case ◽  
Better Than

The class of fractional linear generating functions, one of the few known classes of probability generating functions whose iterates can be explicitly stated, is examined. The method of bounding a probability generating function g (satisfying g″(1) < ∞) by two fractional linear generating functions is used to derive bounds for the extinction time distribution of the Galton-Watson branching process with offspring probability distribution represented by g. For the special case of the Poisson probability generating function, the best possible bounding fractional linear generating functions are obtained, and the bounds for the expected time to extinction of the corresponding Poisson branching process are better than any previously published.

Response times in M/M/1 time-sharing schemes with limited number of service positions

1988 ◽  
Vol 25 (3) ◽  
pp. 579-595 ◽  
Author(s):  
Benjamin Avi-Itzhak ◽  
Shlomo Halfin
Keyword(s):  
Response Time ◽  
Response Times ◽  
Laplace Transforms ◽  
Time Sharing ◽  
Time Variance ◽  
Waiting Line ◽  
Time Sharing System

Two service schemes for an M/M/1 time-sharing system with a limited number of service positions are studied. Both schemes possess the equilibrium properties of symmetric queues; however, in the first one, a preempted job is placed at the end of the waiting line; while in the second one, it is placed at the head of the line. Methods for calculating the Laplace transforms and moments of the response times are presented. The variances of the response times are then compared numerically to indicate that the first scheme is superior to the second scheme. It is also indicated that in both cases the response time variance decreases when the number of service positions increases.

The sojourn-time distribution in the M/G/1 queue by processor sharing

1984 ◽  
Vol 21 (02) ◽  
pp. 360-378
Author(s):  
Teunis J. Ott
Keyword(s):  
Generating Functions ◽  
Sojourn Time ◽  
Time Distribution ◽  
Joint Distribution ◽  
Processor Sharing ◽  
Explicit Formulas ◽  
Sojourn Time Distribution ◽  
Number Of Customers ◽  
Stieltjes Transforms

This paper gives, in the form of Laplace–Stieltjes transforms and generating functions, the joint distribution of the sojourn time and the number of customers in the system at departure for customers in the general M/G/1 queue with processor sharing (M/G/1/PS). Explicit formulas are given for a number of conditional and unconditional moments, including the variance of the sojourn time of an ‘arbitrary' customer.

Bounds on the extinction time distribution of a branching process

1974 ◽  
Vol 6 (02) ◽  
pp. 322-335 ◽  
Author(s):  
Alan Agresti
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Time Distribution ◽  
Branching Process ◽  
Probability Generating Function ◽  
Extinction Time ◽  
Expected Time ◽  
Time To Extinction ◽  
Special Case ◽  
Better Than

The class of fractional linear generating functions, one of the few known classes of probability generating functions whose iterates can be explicitly stated, is examined. The method of bounding a probability generating function g (satisfying g″(1) &lt; ∞) by two fractional linear generating functions is used to derive bounds for the extinction time distribution of the Galton-Watson branching process with offspring probability distribution represented by g. For the special case of the Poisson probability generating function, the best possible bounding fractional linear generating functions are obtained, and the bounds for the expected time to extinction of the corresponding Poisson branching process are better than any previously published.

Response times in M/M/1 time-sharing schemes with limited number of service positions

1988 ◽  
Vol 25 (03) ◽  
pp. 579-595 ◽  
Author(s):  
Benjamin Avi-Itzhak ◽  
Shlomo Halfin
Keyword(s):  
Response Time ◽  
Response Times ◽  
Laplace Transforms ◽  
Time Sharing ◽  
Time Variance ◽  
Waiting Line ◽  
Time Sharing System

Two service schemes for an M/M/1 time-sharing system with a limited number of service positions are studied. Both schemes possess the equilibrium properties of symmetric queues; however, in the first one, a preempted job is placed at the end of the waiting line; while in the second one, it is placed at the head of the line. Methods for calculating the Laplace transforms and moments of the response times are presented. The variances of the response times are then compared numerically to indicate that the first scheme is superior to the second scheme. It is also indicated that in both cases the response time variance decreases when the number of service positions increases.

The sojourn-time distribution in the M/G/1 queue by processor sharing

1984 ◽  
Vol 21 (2) ◽  
pp. 360-378 ◽  
Author(s):  
Teunis J. Ott
Keyword(s):  
Generating Functions ◽  
Sojourn Time ◽  
Time Distribution ◽  
Joint Distribution ◽  
Processor Sharing ◽  
Explicit Formulas ◽  
Sojourn Time Distribution ◽  
Number Of Customers ◽  
Stieltjes Transforms

This paper gives, in the form of Laplace–Stieltjes transforms and generating functions, the joint distribution of the sojourn time and the number of customers in the system at departure for customers in the general M/G/1 queue with processor sharing (M/G/1/PS).Explicit formulas are given for a number of conditional and unconditional moments, including the variance of the sojourn time of an ‘arbitrary' customer.

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.

Tauberian theorems in a Banach lattice, with applications to the LP spaces

1954 ◽  
Vol 50 (2) ◽  
pp. 242-249
Author(s):  
D. C. J. Burgess
Keyword(s):  
Banach Space ◽  
Banach Lattice ◽  
Laplace Transforms ◽  
Tauberian Theorems ◽  
General Argument ◽  
Arbitrary Banach Space ◽  
Lp Spaces ◽  
Special Case

In a previous paper (2) of the author, there was given a treatment of Tauberian theorems for Laplace transforms with values in an arbitrary Banach space. Now, in § 2 of the present paper, this kind of technique is applied to the more special case of Laplace transforms with values in a Banach lattice, and investigations are made on what additional results can be obtained by taking into account the existence of an ordering relation in the space. The general argument is again based on Widder (5) to which frequent references are made.

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