Alan G. Konheim

Queueing Models for Computer Communications System Analysis

Modeling and performance prediction are becoming increasingly important issues in the design and operation of computer communications systems. Complexities in their configuration and sophistications in resource sharing found in todays computer communications demand our intensive effort to enhance the modeling capability. The present paper is intended to review the state of affairs of analytic methods, queueing analysis techniques in particular, which are essential to modeling of computer communication systems. First we review basic properties of exponential queueing systems, and then give an overview of recent progress made in the areas of queueing network models and discrete-time queueing systems. A unified treatment of buffer storage overflow problems will be discussed as an application example, in which we call attention to the analogy between buffer behavior and waiting time in the GI/G/1 queue. Another application deals with the analysis of various multiplexing techniques and network configuration. An extensive reference list of the subject fields is also provided.

Waiting Lines and Times in a System with Polling

A communication system consisting of a number of buffered input terminals connected to a computer by a single channel is analyzed. The terminals are polled in sequence and the data is removed from the terminals buffer. When the buffer has been emptied, the channel, for an interval of randomly determined length, is used for system overhead and/or to transmit data to the terminals. The system then continues with a poll of the next terminal. The stationary distributions of the length of the waiting line and the queueing delay are calculated for the case of identically distributed input processes.

A Queueing Model with Finite Waiting Room and Blocking

A two-stage queueing network with feedback and a finite intermediate waiting room is studied. The first-stage server is blocked whenever M requests are enqueued in the second stage. The analysis of this system under exponential assumptions is carried out. An algorithm to calculate the stationary state probabilities is given and some special cases are considered.

Approximate analysis of exponential queueing systems with blocking

SummaryA network of service stations Q0Q1,...,QM is studied. Requests arrive at the centers according to independent Poisson processes; they travel through (part of) the network demanding amounts of service, with independent and negative exponentially distributed lengths, from those centers which they enter, and finally depart from the network. The waiting rooms or buffers at each service station in this exponential service system are finite. When the capacity at Qi is reached, service at all nodes which are currently processing a request destined next for Qi is instantaneously interrupted. The interruption lasts until the service of the request in the saturated node Qi is. completed. This blocking phenomenon makes an exact analysis intractable and a numerical solution computationally infeasible for most exponential systems. We introduce an approximation procedure for a class of exponential systems with blocking and show that it leads to accurate approximations for the marginal equilibrium queue length distributions. The applicability of the approximation method may not be limited to blocking systems.

Service in a Loop System

The s ta t i s t ica l behavior of a loop service sys tem is s tudied. The sys tem consists of a main s ta t ion , a server and N s ta t ions ar ranged on a loop. Customers arr ive at each s ta t ion according to a r andom process. The server makes successive tours along the loop br inging customers from the N s ta t ions to the main s ta t ion. Two related measures of the grade of service are considered: the average queue length and the average v i r tua l wai t ing t ime at each s ta t ion.

Descendant set: an efficient approach for the analysis of polling systems

Polling systems have been used to model a large variety of applications and much research has been devoted to the derivation of efficient algorithms for computing the delay measures in these systems. Recent research efforts in this area, which have focused on the optimization of these systems, have raised the need for very efficient such algorithms. This work develops the descendant set approach as a general efficient algorithm for deriving all moments of customer delay (in particular, mean delay) in these systems. The method is applied to a very large variety of model variations, including: 1) The exhaustive and gated service policies, 2) Fractional service policies, 3) The cyclic visit order, 4) Arbitrary periodic visit orders (polling tables), and 5) Customer routing. For most of these variations the method significantly outperforms the algorithms commonly used today. >

The Product Form for Sojourn Time Distributions in Cyclic Exponential Queues

Consider a closed cyclic queuing system consisting of M exponential queues. The Laplace-Stieltjes transform of the joint distribution of the consecutive sojourn times of a customer at the M queues is determined and shown to have a product form. The proof is based on a reversibility argument.

Analysis of Integrated Voice/Data Multiplexing

-A model of a moving-boundary, fixed frame length, integrated multiplexor is proposed and analyzed. The assignment of slots within a frame to the voice and data sources is made by an allocation function. The joint distributions of queue length and expected waiting time are derived.

Finite Capacity Queuing Systems with Applications in Computer Modeling

A queuing system with a buffer of unlimited capacity in front of a cyclic arrangement of two exponential server queues is analyzed. The main feature of the system is blocking, i.e., when the population in the two queues attains a maximum value M, say, new arrivals are held back in the buffer. The solution is given in form of polynomial equations which require the roots of a characteristic equation. A solution algorithm is provided. The stability condition is given in terms of these roots and also in explicit form. Limiting cases which are of practical interest are discussed. These limiting cases lead to a better understanding of some popular approximation techniques.

An Elementary Solution of the Queuing System G/G/1

In this note we give an elementary method for calculating the stationary distribution of waiting time in a G/G/1 queue.