Multidimensional Residues, Generating Functions, and Their Application to Queueing Networks

SIAM Review ◽  
1993 ◽  
Vol 35 (2) ◽  
pp. 239-268 ◽  
Author(s):  
Andrea Bertozzi ◽  
James Mckenna
Keyword(s):  
Generating Functions ◽  
Queueing Networks ◽  
Multidimensional Residues

A sample path analysis of the M/M/1 queue

1989 ◽  
Vol 26 (02) ◽  
pp. 418-422 ◽  
Author(s):  
Francois Baccelli ◽  
William A. Massey
Keyword(s):  
Generating Functions ◽  
Queueing Networks ◽  
Transient Analysis ◽  
Bessel Functions ◽  
Sample Path ◽  
Laplace Transforms ◽  
Sample Path Analysis ◽  
Transient Distribution ◽  
The Past ◽  
Novel Approach

The exact solution for the transient distribution of the queue length and busy period of the M/M/1 queue in terms of modified Bessel functions has been proved in a variety of ways. Methods of the past range from spectral analysis (Lederman and Reuter (1954)), combinatorial arguments (Champernowne (1956)), to generating functions coupled with Laplace transforms (Clarke (1956)). In this paper, we present a novel approach that ties the computation of these transient distributions directly to the random sample path behavior of the M/M/1 queue. The use of Laplace transforms is minimized, and the use of generating functions is eliminated completely. This is a method that could prove to be useful in developing a similar transient analysis for queueing networks.

A sample path analysis of the M/M/1 queue

1989 ◽  
Vol 26 (2) ◽  
pp. 418-422 ◽  
Author(s):  
Francois Baccelli ◽  
William A. Massey
Keyword(s):  
Generating Functions ◽  
Queueing Networks ◽  
Transient Analysis ◽  
Bessel Functions ◽  
Sample Path ◽  
Laplace Transforms ◽  
Sample Path Analysis ◽  
Transient Distribution ◽  
The Past ◽  
Novel Approach

The exact solution for the transient distribution of the queue length and busy period of the M/M/1 queue in terms of modified Bessel functions has been proved in a variety of ways. Methods of the past range from spectral analysis (Lederman and Reuter (1954)), combinatorial arguments (Champernowne (1956)), to generating functions coupled with Laplace transforms (Clarke (1956)). In this paper, we present a novel approach that ties the computation of these transient distributions directly to the random sample path behavior of the M/M/1 queue. The use of Laplace transforms is minimized, and the use of generating functions is eliminated completely. This is a method that could prove to be useful in developing a similar transient analysis for queueing networks.

An algorithm to compute blocking probabilities in multi-rate multi-class multi-resource loss models

1995 ◽  
Vol 27 (4) ◽  
pp. 1104-1143 ◽  
Author(s):  
Gagan L. Choudhury ◽  
Kin K. Leung ◽  
Ward Whitt
Keyword(s):  
Steady State ◽  
Generating Functions ◽  
Queueing Networks ◽  
Special Structure ◽  
Product Form ◽  
Direct Algorithm ◽  
Inversion Algorithm ◽  
Closed Queueing Networks ◽  
Loss Models ◽  
Small Models

In this paper we consider a family of product-form loss models, including loss networks (or circuit-switched communication networks) and a class of resource-sharing models. There can be multiple classes of requests for multiple resources. Requests arrive according to independent Poisson processes. The requests can be for multiple units in each resource (the multi-rate case, e.g. several circuits on a trunk). There can be upper-limit and guaranteed-minimum sharing policies as well as the standard complete-sharing policy. If all the requirements of a request cannot be met upon arrival, then the request is blocked and lost. We develop an algorithm for computing the (exact) steady-state blocking probability of each class and other steady state descriptions in these loss models. The algorithm is based on numerically inverting generating functions of the normalization constants. In a previous paper we introduced this approach to product-form models and developed a full algorithm for a class of closed queueing networks. The inversion algorithm promises to be even more useful for loss models than for closed queueing networks because fewer alternative algorithms are available for loss models. Indeed, for many loss models with sharing policies other than traditional complete sharing, our algorithm is the first effective algorithm. Unlike some recursive algorithms, our algorithm has a low storage requirement. To treat the loss models here, we derive the generating functions of the normalization constants and develop a new scaling algorithm especially tailored to the loss models. In general, the computational complexity grows exponentially in the number of resources, but the computation can often be reduced dramatically by exploiting conditional decomposition based on special structure and by appropriately truncating large finite sums. We illustrate our numerical inversion algorithm by applying it to several examples. To validate our algorithm on small models, we also develop a direct algorithm. The direct algorithm itself is of interest, because it tends to be more efficient when the number of resources is large, but the number of request classes is small. Furthermore, it also allows a form of conditional decomposition based on special structure.

An algorithm to compute blocking probabilities in multi-rate multi-class multi-resource loss models

1995 ◽  
Vol 27 (04) ◽  
pp. 1104-1143 ◽  
Author(s):  
Gagan L. Choudhury ◽  
Kin K. Leung ◽  
Ward Whitt
Keyword(s):  
Steady State ◽  
Generating Functions ◽  
Queueing Networks ◽  
Special Structure ◽  
Product Form ◽  
Direct Algorithm ◽  
Inversion Algorithm ◽  
Closed Queueing Networks ◽  
Loss Models ◽  
Small Models

In this paper we consider a family of product-form loss models, including loss networks (or circuit-switched communication networks) and a class of resource-sharing models. There can be multiple classes of requests for multiple resources. Requests arrive according to independent Poisson processes. The requests can be for multiple units in each resource (the multi-rate case, e.g. several circuits on a trunk). There can be upper-limit and guaranteed-minimum sharing policies as well as the standard complete-sharing policy. If all the requirements of a request cannot be met upon arrival, then the request is blocked and lost. We develop an algorithm for computing the (exact) steady-state blocking probability of each class and other steady state descriptions in these loss models. The algorithm is based on numerically inverting generating functions of the normalization constants. In a previous paper we introduced this approach to product-form models and developed a full algorithm for a class of closed queueing networks. The inversion algorithm promises to be even more useful for loss models than for closed queueing networks because fewer alternative algorithms are available for loss models. Indeed, for many loss models with sharing policies other than traditional complete sharing, our algorithm is the first effective algorithm. Unlike some recursive algorithms, our algorithm has a low storage requirement. To treat the loss models here, we derive the generating functions of the normalization constants and develop a new scaling algorithm especially tailored to the loss models. In general, the computational complexity grows exponentially in the number of resources, but the computation can often be reduced dramatically by exploiting conditional decomposition based on special structure and by appropriately truncating large finite sums. We illustrate our numerical inversion algorithm by applying it to several examples. To validate our algorithm on small models, we also develop a direct algorithm. The direct algorithm itself is of interest, because it tends to be more efficient when the number of resources is large, but the number of request classes is small. Furthermore, it also allows a form of conditional decomposition based on special structure.

Short Generating Functions for some Semigroup Algebras

10.37236/1729 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Graham Denham
Keyword(s):  
Rational Function ◽  
Generating Functions ◽  
Hilbert Series ◽  
Simple Expression ◽  
Arbitrary Field ◽  
Graded Algebra ◽  
Semigroup Algebras ◽  
Positive Integers

Let $a_1,\ldots,a_n$ be distinct, positive integers with $(a_1,\ldots,a_n)=1$, and let k be an arbitrary field. Let $H(a_1,\ldots,a_n;z)$ denote the Hilbert series of the graded algebra k$[t^{a_1},t^{a_2},\ldots,t^{a_n}]$. We show that, when $n=3$, this rational function has a simple expression in terms of $a_1,a_2,a_3$; in particular, the numerator has at most six terms. By way of contrast, it is known that no such expression exists for any $n\geq4$.

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