On the Stochastic Matrices Associated with Certain Queuing Processes

Foster, F. G.

doi:10.1214/aoms/1177728976

Cited by 401 publications

(215 citation statements)

References 3 publications

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“…For that purpose, we now state the following result, that can be attributed to Foster [8]. It has strong connections with Martingale theory and can be seen as a consequence of very general results of (super) Martingale theory.…”

Section: Foster's Theoremmentioning

confidence: 73%

Proving Positive Almost-Sure Termination

Bournez

Garnier

2005

Lecture Notes in Computer Science

151

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In order to extend the modeling capabilities of rewriting systems, it is rather natural to consider that the firing of rules can be subject to some probabilistic laws. Considering rewrite rules subject to probabilities leads to numerous questions about the underlying notions and results. We focus here on the problem of termination of a set of probabilistic rewrite rules. A probabilistic rewrite system is said almost surely terminating if the probability that a derivation leads to a normal form is one. Such a system is said positively almost surely terminating if furthermore the mean length of a derivation is finite. We provide several results and techniques in order to prove positive almost sure termination of a given set of probabilistic rewrite rules. All these techniques subsume classical ones for non-probabilistic systems.

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Section: Foster's Theoremmentioning

confidence: 73%

Proving Positive Almost-Sure Termination

Bournez

Garnier

2005

Lecture Notes in Computer Science

151

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“…By applying the well known Foster criterion, cf. [21], it follows that the joint queue length process (with irreducible state space) is positive recurrent and further that there is only one solution, which satisfies (4.1) and the normalizing condition.…”

Section: Solution Of the Functional Equationsmentioning

confidence: 99%

The shorter queue polling model

et al. 2013

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We consider a two-queue polling model in which customers upon arrival join the shorter of two queues. Customers arrive according to a Poisson process and the service times in both queues are independent and identically distributed random variables having the exponential distribution. The two-dimensional process of the numbers of customers at the queue where the server is and at the other queue is a two-dimensional Markov process. We derive its equilibrium distribution using two methodologies: the compensation approach and a reduction to a boundary value problem.

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“…,~ ~,0 p0 (5) for large m, where C is a constant, then the knowledge about the behavior of the CP yields the choice for a0. Since the CP behaves like an MIM[1 queueing system with arrival intensity 2 and service rate /~, we know the marginal equilibrium distribution {p~,. }…”

Section: Pm N ~-" I-' L ~Mpnmentioning

confidence: 99%

A compensation procedure for multiprogramming queues

et al. 1993

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Summary. In this paper we study a multiprogramming system consisting of an input-output unit (IO unit) and a central processor (CP). This system can be represented by a continuous time Markov process with states (m, n), where m and n denote the number of jobs at the CP and the IO unit respectively. The computation of the equilibrium distribution {Pro,,} of this Markov process is the purpose of the analysis in this paper.The analysis consists of two parts. In the first part, we use a compensation procedure to show that the equilibrium distribution {Pro, n} in those states (m, n) for which m + n is not too small, can be expressed as an infinite linear combination of product forms. Explicit formulae are given for the product forms and the coefficients of this infinite linear combination. In the second part of the analysis, we pay attention to some numerical aspects of the computation of the equilibrium distribution.

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On the Stochastic Matrices Associated with Certain Queuing Processes

Cited by 401 publications

References 3 publications

Proving Positive Almost-Sure Termination

Proving Positive Almost-Sure Termination

The shorter queue polling model

A compensation procedure for multiprogramming queues

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