Some aspects of queueing and storage processes : a thesis in partial fulfilment of the requirements for the degree of Master of Science in Statistics at Massey University

In this study the nature of systems consisting of a single queue are first considered. Attention is then drawn to an analogy between such systems and storage systems. A development of the single queue viz queues with feedback is considered after first considering feedback processes in general. The behaviour of queues, some with feedback loops, combined into networks is then considered. Finally, the application of such networks to the analysis of interconnected reservoir systems is considered and the conclusion drawn that such analytic methods complement the more recently developed mathematical programming methods by providing analytic solutions for sub systems behaviour and thus guiding the development of a system model

Workloads and waiting times in single-server systems with multiple customer classes

One of the most fundamental properties that single-server multi-class service systems may possess is the property of work conservation. Under certain restrictions, the work conservation property gives rise to a conservation law for mean waiting times, i.e., a linear relation between the mean waiting times of the various classes of customers. This paper is devoted to single-server multi-class service systems in which work conservation is violated in the sense that the server's activities may be interrupted although work is still present. For a large class of such systems with interruptions, a decomposition of the amount of work into two independent components is obtained; one of these components is the amount of work in the corresponding systemwithout interruptions. The work decomposition gives rise to a (pseudo)conservation law for mean waiting times, just as work conservation did for the system without interruptions

Simple and explicit bounds for multi-server queues with 1/(1−ρ)1/(1 - \rho) (and sometimes better) scaling

We consider the FCFS $GI/GI/n$ queue, and prove the first simple and explicit bounds that scale as $\frac{1}{1-\rho}$ (and sometimes better). Here $\rho$ denotes the corresponding traffic intensity. Conceptually, our results can be viewed as a multi-server analogue of Kingman's bound. Our main results are bounds for the tail of the steady-state queue length and the steady-state probability of delay. The strength of our bounds (e.g. in the form of tail decay rate) is a function of how many moments of the inter-arrival and service distributions are assumed finite. More formally, suppose that the inter-arrival and service times (distributed as random variables $A$ and $S$ respectively) have finite $r$th moment for some $r > 2.$ Let $\mu_A$ (respectively $\mu_S$) denote $\frac{1}{\mathbb{E}[A]}$ (respectively $\frac{1}{\mathbb{E}[S]}$). Then our bounds (also for higher moments) are simple and explicit functions of $\mathbb{E}\big[(A \mu_A)^r\big], \mathbb{E}\big[(S \mu_S)^r\big], r$, and $\frac{1}{1-\rho}$ only. Our bounds scale gracefully even when the number of servers grows large and the traffic intensity converges to unity simultaneously, as in the Halfin-Whitt scaling regime. Some of our bounds scale better than $\frac{1}{1-\rho}$ in certain asymptotic regimes. More precisely, they scale as $\frac{1}{1-\rho}$ multiplied by an inverse polynomial in $n(1 - \rho)^2.$ These results formalize the intuition that bounds should be tighter in light traffic as well as certain heavy-traffic regimes (e.g. with $\rho$ fixed and $n$ large). In these same asymptotic regimes we also prove bounds for the tail of the steady-state number in service. Our main proofs proceed by explicitly analyzing the bounding process which arises in the stochastic comparison bounds of amarnik and Goldberg for multi-server queues. Along the way we derive several novel results for suprema of random walks and pooled renewal processes which may be of independent interest. We also prove several additional bounds using drift arguments (which have much smaller pre-factors), and make several conjectures which would imply further related bounds and generalizations

A multiple channel queueing model under an uncertain environment with multiclass arrivals for supplying demands in a cement industry

In recent years, cement consumption has increased in most Asian countries, including Malaysia. There are many factors which affect the supply of the increasing order demands in the cement industry, such as traffic congestion, logistics, weather and machine breakdowns. These factors hinder smooth and efficient supply, especially during periods of peak congestion at the main gate of the industry where queues occur as a result of inability to keep to the order deadlines. Basic elements, such as arrival and service rates, that cannot be predetermined must be considered under an uncertain environment. Solution approaches including conventional queueing techniques, scheduling models and simulations were unable to formulate the performance measures of the cement queueing system. Hence, a new procedure of fuzzy subset intervals is designed and embedded in a queuing model with the consideration of arrival and service rates. As a result, a multiple channel queueing model with multiclass arrivals, (M1, M2)/G/C/2Pr, under an uncertain environment is developed. The model is able to estimate the performance measures of arrival rates of bulk products for Class One and bag products for Class Two in the cement manufacturing queueing system. For the (M1, M2)/G/C/2Pr fuzzy queueing model, two defuzzification techniques, namely the Parametric Nonlinear Programming and Robust Ranking are used to convert fuzzy queues into crisp queues. This led to three proposed sub-models, which are sub-model 1, MCFQ-2Pr, sub-model 2, MCCQESR-2Pr and sub-model 3, MCCQ-GSR-2Pr. These models provide optimal crisp values for the performance measures. To estimate the performance of the whole system, an additional step is introduced through the TrMF-UF model utilizing a utility factor based on fuzzy subset intervals and the α-cut approach. Consequently, these models help decision-makers deal with order demands under an uncertain environment for the cement manufacturing industry and address the increasing quantities needed in future