The problem of simultaneous due-date setting and priority sequencing is analyzed in the setting of a multiclass M/G/1 queueing system. The objective is to minimize the weighted average due-date lead time (due-date minus arrival date) of jobs subject to a constraint on either the fraction of tardy jobs or the average job tardiness. Several parametric and nonparametric due-date setting policies are proposed that depend on the class of arriving job, the state of the queueing system at the time of the job's arrival, and the sequencing policy (the weighted shortest expected processing time rule) that is used. In a simulation experiment performed on a two-class M/M/1 system, these policies outperformed traditional due-date setting policies, and due-date setting had a larger impact on performance than priority sequencing.

Due-date setting and priority sequencing in a multiclass M/G/1 queue

The Laguerre method for numerically inverting Laplace transforms is an old established method based on the 1935 Tricomi–Widder theorem, which shows (under suitable regularity conditions) that the desired function can be represented as a weighted sum of Laguerre functions, where the weights are coefficients of a generating function constructed from the Laplace transform using a bilinear transformation. We present a new variant of the Laguerre method based on: (i) using our previously developed variant of the Fourier-series method to calculate the coefficients of the Laguerre generating function, (ii) developing systematic methods for scaling, and (iii) using Wynn's ϵ-algorithm to accelerate convergence of the Laguerre series when the Laguerre coefficients do not converge to zero geometrically fast. These contributions significantly expand the class of transforms that can be effectively inverted by the Laguerre method. We provide insight into the slow convergence of the Laguerre coefficients as well as prop...

On the Laguerre Method for Numerically Inverting Laplace Transforms

/pdf/theory-and-algorithms-of-4m5sb55iw0.pdf

Theory and algorithms of

An M/G/1 queue with server vacations and gated time-limited service is analyzed At each visit, the server serves the queue up to a fixed amount of time When the time expires or after all candidate customers have been served, whichever occurs first, the server takes a vacation The service policy is gated, since only those customers present at the beginning of a server visit (poling instant) are candidates for service during that visit; subsequent arrivals are deferred until the next visit A functional equation which characterizes the amount of work, U/sub p/, at a polling instant is derived To solve the equation, a numerical technique is utilized in which the complementary cumulative function for U/sub p/ is closely approximated by a weighted sum of Laguerre functions with unknown coefficients The equation is then transformed into a set of linear equations from which the coefficients can be computed By the stochastic decomposition and Poisson-arrivals-see-time-averages properties, the average customer response time can be related to the average amount of work found by an arrival Several numerical examples are included The model studied is applicable to communication and computer systems where timers are used to allocate service to customers >

A single-server queue with vacations and gated time-limited service

In this article we define a class of distributions called bilateral phase type (BPH), and study its closure and computational properties. The class of BPH distributions is closed under convolution, negative convolution, and mixtures. The one-sided version of BPH, called generalized phase type (GPH), is also defined. The class of GPH distributions is strictly larger than the class of phase-type distributions introduced by Neuts, and is closed under convolution, negative convolution with nonnegativity condition, mixtures, and formation of coherent systems. We give computational schemes to compute the resulting distributions from the above operations and extend them to analyze queueing processes. In particular, we present efficient algorithms to compute the steady-state and transient waiting times in GPH/GPH/1 queues and a simple algorithm to compute the steady-state waiting time in M/GPH/1 queues.

Bilateral phase-type distributions

The bilateral Laguerre transform

: A novel transform is presented which maps continuum functions (such as probability distributions) into discrete sequences and permits rapid numerical calculation of convolutions, multiple convolutions, and Neumann expansions for Volterra integral equations. The transform is based on the Laguerre polynomials, associated Laguerre functions, and their convolution properties. Part 1 of this paper deals with functions having support only on (0 infinity). The resulting unilateral Laguerre transform finds applications in convolution of such functions, invercsion of Laplace transform, and in solution to renewal and related Volterra integral equations. Part 2 of this paper deals with functions having support on (-infinity, infinity) via a bilateral Laguerre transform which is an extension of the unilateral transform. Applications of this technique include convolution of such functions and analysis of the Lindley process. Part 1 has been published in Applied Mathematics and Computation and part 2 has been submitted for publication in that journal. (Author)

The Laguerre Transform.

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The bilateral Laguerre transform

The Laguerre Transform.