We show for the M/G/1 processor sharing queue that the service time distribution is regularly varying of index -ν, ν non-integer, iff the sojourn time distribution is regularly varying of index -ν. This result is derived from a new expression for the Laplace–Stieltjes transform of the sojourn time distribution. That expression also leads to other new properties for the sojourn time distribution. We show how the moments of the sojourn time can be calculated recursively and prove that the kth moment of the sojourn time is finite iff the kth moment of the service time is finite. In addition, we give a short proof of a heavy traffic theorem for the sojourn time distribution, prove a heavy traffic theorem for the moments of the sojourn time, and study the properties of the heavy traffic limiting sojourn time distribution when the service time distribution is regularly varying. Explicit formulas and multiterm expansions are provided for the case that the service time has a Pareto distribution.

Sojourn time asymptotics in the M / G / 1 processor sharing queue

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Processor-Sharing Models for Integrated-Services Networks

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Performance Evaluation of Computer and Communication Systems

During last few decades the Egalitarian Processor Sharing (EPS) has gained a prominent role in applied probability, especially, in queueing theory and its computer applications. While the EPS paradigm emerged in 1967 as an idealization of round-robin (RR) scheduling algorithm in time-sharing computer systems, it has recently capture renewed interest as a powerful concept for modeling WEB servers. This paper summarizes the most important results concerning the exact solutions for the M/GI/1 queue with egalitarian processor sharing. The material is drawn, mainly, from recent authors' papers which are supplemented, in small degree, by other related results. Many of the further results are established under the direct influence of our earlier papers. Our main purpose is to give a survey of state-of-the-art with regard to main achievements of the contemporary theory of the M/GI/1 queueing system with processor sharing. The focus is on the methods and techniques of exact and asymptotic analysis of this queueing system. In contrast to the standard surveys, the abridged proofs (or their ideas) of some key theorems and corollaries are included in the paper. We outline recent developments in exact analysis of the M/GI/1-EPS queue with further emphasis on time-dependent (transient) probability distributions of the main characteristics. In particular, the present paper includes the results on the joint time-dependent distribution of the sojourn time of a job arriving at time t with the service demand (length) u, and of the number of jobs at time t- in the M/GI/1 queue with egalitarian processor sharing, which obtained in form of multiple transforms. We also show how the non-stationary solutions can be used to obtain known and new results which allow to predict the behaviour of the EPS queue and to yield additional insights into its new unexpected properties. We also discuss a number of limit theorems arising under the study of the processor sharing queues.

Processor sharing: A survey of the mathematical theory

This article considers new approaches in queuing theory associated with mathematical modelling and analysis of contemporary computer systems and networks using shared processors. Emphasis is placed on results concerning exact determination of probability distributions for such characteristics as sojourn times and waiting times. Various methods of analyzing shared-processor systems are considered, their characteristics analyzed, and special attention is devoted to the broad class of M ¦GI¦1 queuing disciplines that result in Poisson throughput. Several limit theorems for heavyload conditions are described.

Mathematical problems in the theory of shared-processor systems

The central model of this paper is anM/M/1 queue with a general probabilistic feedback mechanism. When a customer completes his ith service, he departs from the system with probability 1−p(i) and he cycles back with probabilityp(i). The mean service time of each customer is the same for each cycle. We determine the joint distribution of the successive sojourn times of a tagged customer at his loops through the system. Subsequently we let the mean service time at each loop shrink to zero and the feedback probabilities approach one in such a way that the mean total required service time remains constant. The behaviour of the feedback queue then approaches that of anM/G/1 processor sharing queue, different choices of the feedback probabilities leading to different service time distributions in the processor sharing model. This is exploited to analyse the sojourn time distribution in theM/G/1 queue with processor sharing.

The M/G/ 1 queue with processor sharing and its relation to a feedback queue

Sojourn times in feedback and processor sharing queues

In many modern computer-communication systems, a job may be processed in several phases, or a job may generate new tasks. Such phenomena can be modeled by service systems with feedback. In the queueing literature, attention has been mainly devoted to single-service queues with so-called Bernoulli feedback: when a customer (task) completes his service, he departs from the system with probability l-p and is fed back with probability p. In the present study a more general feedback mechanism is allowed: when a customer completes his i-th service, he departs from the system with probability l-p(i) and is fed back with probability p(i). We mainly restrict ourselves to the case of a Poisson external arrival process and identically, negative exponentially, distributed service times at each service. The resulting queueing model has the property that the joint queue-length distribution of type-i customers, i=1,2,⋯, is of product-form type. This property is exploited to analyse the sojourn-time process.

/pdf/sojourn-times-in-feedback-queues-2vgeulz0v1.pdf

Sojourn times in feedback queues

In this paper we consider an M/G/1 queueing model, in which each customer is fed back a fixed number of times. For the case of negative exponentially distributed service times at each visit, we determine the Laplace-Stieltjes transform of the joint distribution of the sojourn times of the consecutive visits. As a by-result, we obtain the (transform of the) total sojourn time distribution; it can be related to the sojourn time distribution in the M/D/l queue with processor sharing. For the case of generally distributed service times at each visit, a set of linear equations is derived, from which the mean sojourn times per visit can be calculated.

/pdf/sojourn-times-in-the-m-g-1-queue-with-deterministic-feedback-4875em5h7b.pdf

van den Hans Berg

Papers

The M/G/ 1 queue with processor sharing and its relation to a feedback queue

Sojourn times in feedback and processor sharing queues

Sojourn times in feedback queues

Sojourn times in the m/g/1 queue with deterministic feedback

Sojourn times in feedback and processor sharing queues