Priority based scheduling disciplines are widely used by existing computer operating systems. However, the mathematical analysis and modelling of these systems present great difficulties since priority schedulling is not compatible with exact product form solutions of queueing network models (QNM's). It is therefore, necessary to employ credible approximate techniques for solving QNM's with priority classes. The principle of maximum entropy (ME) is a method of inference for estimating a probability distribution given prior information in the form of expected values. This principle is applied, based on marginal utilisation, mean queue length and idle state probability constraints, to characterise new product-form approximations for general open and closed QNM's with priority (preemptive-resume, non-preemtive head-of-line) and non-priority (first-come-first-served, processor-sharing, last-come-first-served with, or without preemtion) servers. The ME solutions are interpreted in terms of a decomposition of the original network into individual stable GIG11 queueing stations with assumed renewal arrival processes. These solutions are implemented by making use of the generalised exponential (GE) distributional model to approximate the interarrival-time and service-time distributions in the network. As a consequence the ME queue length distribution of the stable GE/GEzl priority queue, subject to mean value constraints obtained via classical queueing theory on bulk queues, is used as a 'building block' together with corresponding universal approximate flow formulae for the analysis of general QNM's with priorities. The credibility of the ME method is demonstrated with illustrative numerical examples and favourable comparisons against exact, simulation and other approximate methods are made. / Algerian government

Analytical modelling and characterization of Weighted Fair Queueing (WFQ) have recently received considerable attention by several researches since WFQ offers the minimum delay and optimal fairness guarantee. However, all previous work on WFQ has focused on developing approximations of the scheduler with an infinite buffer because of supposed scalability problems in the WFQ computation. The main aims of this thesis are to study WFQ system, by providing an analytical WFQ model which is a theoretical construct based on a form of processor sharing for finite capacity. Furthermore, the solutions for classes with Poisson arrivals and exponential service are derived and verified against global balance solution. This thesis shows that the analytical models proposed can give very good results under particular conditions which are very close to WFQ algorithms, where accuracy of the models is verified by simulations of WFQ model. Simulations were performed with QNAP-2 simulator. In addition, the thesis presents several performance studies signifying the power of the proposed analytical model in providing an accurate delay bounds to a large number of classes. These results are not able to cover all unsolved issues in the WFQ system. They represent a starting point for the research activities that the Author will conduct in the future. The author believes that the most promising research activities exist in the scheduler method to provide statistical guarantees to multi-class services. The author is convinced that alternative software, for example, on the three class model buffer case, is able to satisfy the large number of buffer because of the software limitation in this thesis. While they can be a good topic for long-term research, the short-medium term will show an increasing interest in the modification of the WFQ models to provide differentiated services. / Ministry of Higher Education

Most used security mechanisms in high-speed networks have been adopted without adequate quantification of their impact on performance degradation. Appropriate quantitative network models may be employed for the evaluation and prediction of ¿optimal¿ performance vs. security trade-offs. Several quantitative models introduced in the literature are based on queueing networks (QNs) and generalised stochastic Petri nets (GSPNs). However, these models do not take into consideration Performance Engineering Principles (PEPs) and the adverse impact of traffic burstiness and security protocols on performance. The contributions of this thesis are based on the development of an effective quantitative methodology for the analysis of arbitrary QN models and GSPNs through discrete-event simulation (DES) and extended applications into performance vs. security trade-offs involving infrastructure and infrastructure-less high-speed networks under bursty traffic conditions. Specifically, investigations are carried out focusing, for illustration purposes, on high-speed network routers subject to Access Control List (ACL) and also Robotic Ad Hoc Networks (RANETs) with Wired Equivalent Privacy (WEP) and Selective Security (SS) protocols, respectively. The Generalised Exponential (GE) distribution is used to model inter-arrival and service times at each node in order to capture the traffic burstiness of the network and predict pessimistic ¿upper bounds¿ of network performance. In the context of a router with ACL mechanism representing an infrastructure network node, performance degradation is caused due to high-speed incoming traffic in conjunction with ACL security computations making the router a bottleneck in the network. To quantify and predict the trade-off of this degradation, the proposed quantitative methodology employs a suitable QN model consisting of two queues connected in a tandem configuration. These queues have single or quad-core CPUs with multiple-classes and correspond to a security processing node and a transmission forwarding node. First-Come-First-Served (FCFS) and Head-of-the-Line (HoL) are the adopted service disciplines together with Complete Buffer Sharing (CBS) and Partial Buffer Sharing (PBS) buffer management schemes. The mean response time and packet loss probability at each queue are employed as typical performance metrics. Numerical experiments are carried out, based on DES, in order to establish a balanced trade-off between security and performance towards the design and development of efficient router architectures under bursty traffic conditions. The proposed methodology is also applied into the evaluation of performance vs. security trade-offs of robotic ad hoc networks (RANETs) with mobility subject to Wired Equivalent Privacy (WEP) and Selective Security (SS) protocols. WEP protocol is engaged to provide confidentiality and integrity to exchanged data amongst robotic nodes of a RANET and thus, to prevent data capturing by unauthorised users. WEP security mechanisms in RANETs, as infrastructure-less networks, are performed at each individual robotic node subject to traffic burstiness as well as nodal mobility. In this context, the proposed quantitative methodology is extended to incorporate an open QN model of a RANET with Gated queues (G-Queues), arbitrary topology and multiple classes of data packets with FCFS and HoL disciplines under bursty arrival traffic flows characterised by an Interrupted Compound Poisson Process (ICPP). SS is included in the Gated-QN (G-QN) model in order to establish an ¿optimal¿ performance vs. security trade-off. For this purpose, PEPs, such as the provision of multiple classes with HoL priorities and the availability of dual CPUs, are complemented by the inclusion of robot¿s mobility, enabling realistic decisions in mitigating the performance of mobile robotic nodes in the presence of security. The mean marginal end-to-end delay was adopted as the performance metric that gives indication on the security improvement. The proposed quantitative methodology is further enhanced by formulating an advanced hybrid framework for capturing ¿optimal¿ performance vs. security trade-offs for each node of a RANET by taking more explicitly into consideration security control and battery life. Specifically, each robotic node is represented by a hybrid Gated GSPN (G-GSPN) and a QN model. In this context, the G-GSPN incorporates bursty multiple class traffic flows, nodal mobility, security processing and control whilst the QN model has, generally, an arbitrary configuration with finite capacity channel queues reflecting ¿intra¿-robot (component-to-component) communication and ¿inter¿-robot transmissions. Two theoretical case studies from the literature are adapted to illustrate the utility of the QN towards modelling ¿intra¿ and ¿inter¿ robot communications. Extensions of the combined performance and security metrics (CPSMs) proposed in the literature are suggested to facilitate investigating and optimising RANET¿s performance vs. security trade-offs. This framework has a promising potential modelling more meaningfully and explicitly the behaviour of security processing and control mechanisms as well as capturing the robot¿s heterogeneity (in terms of the robot architecture and application/task context) in the near future (c.f. [1]. Moreover, this framework should enable testing robot¿s configurations during design and development stages of RANETs as well as modifying and tuning existing configurations of RANETs towards enhanced ¿optimal¿ performance and security trade-offs. / Ministry of Higher Education in Libya and the Libyan Cultural Attaché bureau in London

I detta examensarbete inom tillämpad matematik och industriell ekonomi undersöks bemanningsstrategin för Mat AB för att sänka den totala leveranstiden. Mat AB är ett startup som arbetar med leverans av mat- och apoteksvaror. Precis som många andra företag i branschen fick Mat AB ett uppsving under våren 2020 och möter precis som många andra snabbt växande företag organisatoriska utmaningar. Arbetet undersöker leverans- och försäljningsdata från våren 2021. Utifrån detta fastställs om dagens bemanningsstrategi är optimal eller om det finns bättre sätt att organisera personalen på för att sänka leveranstiden och på så sätt öka värdeskapandet och sänka personalkostnader. Modelleringen och undersökningen av verksamheten genomfördes med multipel linjär regressionsanalys och köteori. Genom att modellera verksamheten som olika kösystem kunde olika bemanningsstrategier teoretiskt prövas för att fastställa om mer optimala strategier finns att tillgå. Arbetet presenterar utifrån datan den befintliga leveranstiden och ger förslag på alternativa bemanningsstrategier genom modellering. Resultatet visar att det finns alternativa bemanningsstrategier att tillgå som skulle sänka den totala leveranstiden för Mat AB. I rapporten diskuteras även faktorer som inte finns med i modelleringen som kan påverka leveranstiden för vidare analys. / This thesis combines applied mathematics with industrial economics to investigate the staffing strategy for Mat AB in order to reduce the total delivery time. Mat AB is a startup that works with delivery of food and pharmacy goods. Like many other companies in the industry, Mat AB received a boost in the spring of 2020 and, like many other fast-growing companies, faces organizational challenges. The thesis examines delivery and sales data from the spring of 2021. Based on this, it is determined whether the current staffing strategy is optimal or whether there are better ways to organize the staff to reduce the delivery time and thus increase value creation and reduce staff costs. The modeling and investigation of the operation was carried out with multiple linear regression analysis and queue theory. By modeling the business as a queuing system, different staffing strategies could theoretically be tested to determine whether more optimal strategies were available. Based on the data, the thesis presents the existing delivery time and suggests alternative staffing strategies through modeling. The results show that there are alternative staffing strategies that would reduce the total delivery time for Mat AB. The report also discusses factors that are not included in the modeling that may affect the delivery time for further analyses.

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