We use a mathematical model of an open queueing network in heavy traffic. The law of the iterated logarithm for the extreme virtual waiting time of a customer in heavy traffic in open queueing networks has been presented. Finally, we present an application of the theorem a reliability model from computer network practice. Key–Words: mathematical models of technical systems, performance evaluati...

Proof. Let X(t) denote the number of customers in the system at time t. Since M/M/s process is a birth and death process, it follows from the previous proposition that {X(t), t ≥ 0} is time reversible. Now going forward in time, the time instants at which X(t) increases by 1 are the arrival instants of a Poisson process. Hence, by time reversibility, the time Points at which X(t) increases by 1...

We consider a class of stochastic processing networks. Assume that the networks satisfy a complete resource pooling condition. We prove that each maximum pressure policy asymptotically minimizes the workload process in a stochastic processing network in heavy traffic. We also show that, under each quadratic holding cost structure, there is a maximum pressure policy that asymptotically minimizes...

Corollary 1.1. Consider an M/M/s queue with Poisson(λ ) arrivals and each server having exponential service time exp(μ) service. If λ > sμ , then the output process in steady state is Poisson(λ ). Proof. Let X(t) denote the number of customers in the system at time t. Since M/M/s process is a birth and death process, it follows from the previous proposition that {X(t), t ≥ 0} is time reversible...

This paper describes an extension to Layered Queueing Networks (LQN), a form of an extended queueing network used to investigate performance problems, to model passive resources such as counting semaphores and buffers. Layered queueing networks can be constructed directly, or from UML design models which incorporate the MARTE profile, either directly or via the Core Scenario Model. Layered Queu...

In this paper, we derive worst case bounds on delay and backlog for non-preemptive priority queueing systems using network calculus. There are known results for the average behaviour of non-preemptive priority queueing systems from traditional queueing theory. By use of numerical investigations, we compare our worst case bounds to those average behaviour results in order to give a feel as to ho...

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Mobile networks are intriguing in recent years due to their practical implications. Previous routing strategies for improving transport efficiency have little attention what order should the packets be forwarded, just simply used first-in-first-out queue discipline. Here we apply priority queueing discipline to shortest distance routing strategy on mobile networks. Numerical experiments show it...

The tra c equations are a set of linear equations, which are the basis for the exact analysis of product form queueing networks, and the approximate analysis of non-product form queueing networks. Conditions characterising the structure of the network that guarantees the existence of a solution for the tra c equations are therefore of great importance. This note provides a necessary and su cien...

A new class of queueing networks with \negative and positive" customers was introduced and shown to have a non-standard product form 4]; since then this model has undergone several generalizations 5, 6, 7, 8]. Positive customers are identical to the usual customers of a queueing network, while a negative customer which arrives to a queue simply destroys a positive customer. We call these genera...