J.P.C. Blanc

On a numerical method for calculating state probabilities for queueing systems with more than one waiting line

Abstract Keane, Hooghiemstra and Van de Ree have proposed a new numerical method for calculating state probabilities for queueing systems with more than one waiting line in parallel. The method is based on power series expansions of state probabilities as functions of the traffic intensity of a system. The coefficients of these power series can be recursively calculated. The coefficients of the power series expansions of moments of queue length distributions can be derived from those of the state probabilities in a straightforward manner. The above method is discussed for a rather general class of exponential queueing systems. The asymptotic behaviour of moments in heavy traffic is used to obtain extrapolations of the coefficients of their power series expansions at the origin. The calculation of moments is strongly improved by means of these extrapolations.

Performance evaluation of polling systems by means of the power-series algorithm

Polling systems are widely used to model communication networks with several classes of messages, a single transmission channel and a collision-free access prolocol. However, they can only be analysed exactly for some special service disciplines. The power-series algorithm provides a means for the numerical analysis of polling systems with a moderate number of stations, for a wide variety of access protocols. This paper contains a general description of the power-series algorithm, with emphasis on the application to a general class of polling systems with Poisson arrival streams, with Coxian service and switching time distributions, with infinite buffers, with a fixed periodic visit order, and with a Bernoulli schedule for each visit to a station. The applicability and the complexity of the algorithm are discussed for several more service disciplines for polling systems.

The relaxation time of two queueing systems in series

This paper deals with the time-dependent behaviour of two queueing systems in series, with a Posson arrival stream and exponential service times. The Laplace transform of the probability p0(t) that the tandem system is empty at time t given that it was empty at time 0 is obtained by reducing the functional equation for the generating function of the joint queue length distribution to a Riemann-Hilbert boundary value problem. From this Laplace transform the relaxation time of p 0(t) is determined for all parameter values, and the first term of the asymptotic expansion of p 0(t)-p 0(∞) as t →∞ is found in the ergodic and in the null recurrent cases.

Optimization of polling systems with Bernoulli schedules

Abstract Many computer-communication networks in which the transmission right is circulated among the nodes have been modeled as polling systems. This paper concerns optimization of cyclic polling systems with respect to the service disciplines at the nodes. The service disciplines are chosen to be Bernoulli schedules. Because the optimization problem is not analytically tractable, a numerical approach to determine the optimal schedule, based on the power-series algorithm, is discussed. Light- and heavy-traffic asymptotes of the optimal schedule are presented; they are based on light-traffic asymptotes of the mean waiting times and the stability condition, respectively. A partial solution of the optimization problem is given; this follows directly from the μc-rule for priority systems. The influence of system parameters on the optimal Bernoulli schedule is examined. Finally, a fast approach to approximate the optimal schedule is presented and tested.

An algorithmic solution of polling models with limited service disciplines

The power-series algorithm, an iterative numerical technique for the evaluation of the joint queue length distributions for a broad class of multiqueue systems, is extended to polling systems with limited service disciplines. Some examples are provided to validate the algorithm. >

Performance Analysis and Optimization with the Power-Series Algorithm

The power-series algorithm (PSA) is a flexible device for computing performance measures for systems which can be modeled as multi-queue/multi-server systems with a quasi-birth-and-death structure. An overview of this technique is provided, including a motivation of the principles of the PSA, the derivation of recursive computation schemes, discussions of efficient implementation of the PSA, of methods for improving the convergence of the power series, of the numerical complexity of the PSA, and of the computation of derivatives with respect to system parameters, and examples of application of the PSA.

Bad luck when joining the shortest queue

A frequent observation in service systems with queues in parallel is that customers in other queues tend to be served faster than those in ones own queue. This paper quantifies the probability that ones service would have started earlier if one had joined another queue than the queue that was actually chosen, for exponential multiserver systems with queues in parallel in which customers join one of the shortest queues upon arrival and in which jockeying is not possible.

Analysis of the M/GI/1:2WZ./M/1 queueing model

An M/GI/1 queueing system is in series with a unit with negative exponential service times and infinite waiting room capacity. We determine a closed form expression for the generating function of the joint queue length distribution in steady state. This result is obtained via the solution of a new type of functional equation in two variables.

Analysis of communication systems with timed token protocols using the power-series algorithm

The IEEE 802.4 and FDDI (Fibre Distributed Data Interface) standards are high speed MAC (Medium Access Control) protocols for LAN/MANs employing a timer-controlled token passing mechanism, the so-called Timed Token Protocol, to control station access to the shared media. These protocols support synchronous and real-time (i.e., time-critical) applications, and provide priority among asynchronous (i.e., non time-critical) applications. During the last few years, much research has focused on the study of timed token protocols, to obtain performance measures such as throughputs or mean waiting times. The recent development of the Power-Series Algorithm (PSA) has opened new perspectives in the analysis of this class of protocols. This paper shows the versatility of the PSA technique when evaluating the station buffer occupancy and delay distributions of a very general model that can be used to represent the behavior of several LAN/MANs MAC protocols, among which the timed token MAC protocols. Specifically, the focus of the paper is on the solution of an almost exact model of the IEEE 802.4 MAC protocol. Since the model we propose and solve numerically by exploiting the PSA technique, is an approximate model of the FDDI MAC protocol, the paper also reports on a comparison between performance measures obtained for this model and simulation results for the corresponding exact model of FDDI

Computation of Derivatives by means of the Power-Series Algorithm

The power-series algorithm (PSA) is a flexible tool for computing performance measures for moderately-sized queueing systems for which the underlying process has a multidimensional quasi birth-and-death structure. In the present paper the PSA is extended to the computation of derivatives of system performance measures with respect to a general class of system parameters. This extension is useful for analyzing the sensitivity of the system performance with respect to the system parameters and for solving a wide variety of optimization problems in queueing systems.