One of causes of increased latency service over the whole IMS network can be unbalanced SIP (Session Initiation Protocol) signalling traffic through CSCF (Call Session Control Function). This thesis is devoted to the proposal of weight-based load balancing algorithm which can be used for the S-CSCF assignment performed by I-CSCF during the initial registration procedure of subscribers over the IMS architecture. The designed mechanism is implemented and evaluated in the mathematical model of IMS subsystem based on single servers with FIFO queues with the unlimited capacity in the numerical computing environment - Matlab. Two test-cases with different performance conditions of available S-CSCFs are described. The influence of measured latency affected by performance of other nodes (e.g. P-CSCF, I-CSCF, HSS, etc.) is minimized. Each of simulated test-cases is measured with various service times of SCSCFs and inter-arrival times. The obtained results of designed algorithm are compared with well-known selection algorithm – the round-robin algorithm. It is shown that new selection mechanism improved the service latency of whole IMS network. The possible weakness of the designed weight-based algorithm is sensitivity to traffic model over the modelled IMS architecture.

Cognitive radio networks (CRN) have been tipped as one of the most promising paradigms for next generation wireless communication, due primarily to its huge promise of mitigating the spectrum scarcity challenge. To help achieve this promise, CRN develop mechanisms that permit spectrum spaces to be allocated to, and used by more than one user, either simultaneously or opportunistically, under certain preconditions. However, because of various limitations associated with CRN, spectrum and other resources available for use in CRN are usually very scarce. Developing appropriate models that can efficiently utilise the scarce resources in a manner that is fair, among its numerous and diverse users, is required in order to achieve the utmost for CRN. 'Resource allocation (RA) in CRN' describes how such models can be developed and analysed. In developing appropriate RA models for CRN, factors that can limit the realisation of optimal solutions have to be identified and addressed; otherwise, the promised improvement in spectrum/resource utilisation would be seriously undermined. In this thesis, by a careful examination of relevant literature, the most critical limitations to RA optimisation in CRN are identified and studied, and appropriate solution models that address such limitations are investigated and proffered. One such problem, identified as a potential limitation to achieving optimality in its RA solutions, is the problem of heterogeneity in CRN. Although it is indeed the more realistic consideration, introducing heterogeneity into RA in CRN exacerbates the complex nature of RA problems. In the study, three broad classifications of heterogeneity, applicable to CRN, are identified; heterogeneous networks, channels and users. RA models that incorporate these heterogeneous considerations are then developed and analysed. By studying their structures, the complex RA problems are smartly reformulated as integer linear programming problems and solved using classical optimisation. This smart move makes it possible to achieve optimality in the RA solutions for heterogeneous CRN. Another serious limitation to achieving optimality in RA for CRN is the strictness in the level of permissible interference to the primary users (PUs) due to the activities of the secondary users (SUs). To mitigate this problem, the concept of cooperative diversity is investigated and employed. In the cooperative model, the SUs, by assisting each other in relaying their data, reduce their level of interference to PUs significantly, thus achieving greater results in the RA solutions. Furthermore, an iterative-based heuristic is developed that solves the RA optimisation problem timeously and efficiently, thereby minimising network complexity. Although results obtained from the heuristic are only suboptimal, the gains in terms of reduction in computations and time make the idea worthwhile, especially when considering large networks. The final problem identified and addressed is the limiting effect of long waiting time (delay) on the RA and overall productivity of CRN. To address this problem, queueing theory is investigated and employed. The queueing model developed and analysed helps to improve both the blocking probability as well as the system throughput, thus achieving significant improvement in the RA solutions for CRN. Since RA is an essential pivot on which the CRN's productivity revolves, this thesis, by providing viable solutions to the most debilitating problems in RA for CRN, stands out as an indispensable contribution to helping CRN realise its much-proclaimed promises. / Thesis (PhD)--University of Pretoria, 2017. / Electrical, Electronic and Computer Engineering / PhD / Unrestricted

<p>Queueing systems are everywhere: in transportation networks, service centers, communication systems, clinics, manufacturing systems, etc. In this dissertation, we contribute to the theory of queueing in two aspects. In the first part, we dilate the interplay between retrials and strategic arrival behavior in single-class queueing networks. Specifically, we study a variation of the ‘Network Concert Queueing Game,’ wherein a fixed but large number of strategic users arrive at a network of queues where they can be routed to other queues in the network following a fixed routing matrix, or potentially fedback to the end of the queue they arrive at. Working in a non-atomic setting, we prove the existence of Nash equilibrium arrival and routing profiles in three simple, but non-trivial, network topologies/architectures. In two of them, we also prove the uniqueness of the equilibrium. Our results prove that Nash equilibrium decisions on when to arrive and which queue to join in a network are substantially impacted by routing, inducing ‘herding’ behavior under certain conditions on the network architecture. Our theory raises important design implications for capacity-sharing in systems with strategic users, such as ride-sharing and crowdsourcing platforms.</p> <p><br></p> <p>In the second part, we develop a new method of data-driven model calibration or estimation for queueing models. Statistical and theoretical analyses of traffic traces show that the doubly stochastic Poisson processes are appropriate models of high intensity traffic arriving at an array of service systems. On the other hand, the statistical estimation of the underlying latent stochastic intensity process driving the traffic model involves a rather complicated nonlinear filtering problem. In this thesis we use deep neural networks to ‘parameterize’ the path measures induced by the stochastic intensity process, and solve this nonlinear filtering problem by maximizing a tight surrogate objective called the evidence lower bound (ELBO). This framework is flexible in the sense that we can also estimate other stochastic processes (e.g., the queue length process) and their related parameters (e.g., the service time distribution). We demonstrate the effectiveness of our results through extensive simulations. We also provide approximation guarantees for the estimation/calibration problem. Working with the Markov chain induced by the Euler-Maruyama discretization of the latent diffusion, we show that (1) there exists a sequence of approximate data generating distributions that converges to the “ground truth” distribution in total variation distance; (2) the variational gap is strictly positive for the optimal solution to the ELBO. Extending to the non-Markov setting, we identify the variational gap minimizing approximate posterior for an arbitrary (known) posterior and further, prove a lower bound on the optimal ELBO. Recent theoretical results on optimizing the ELBO for related (but ultimately different) models show that when the data generating distribution equals the ground truth distribution and the variational gap is zero, the probability measures that achieve these conditions also maximize the ELBO. Our results show that this may not be true in all problem settings.</p>

Peer-to-peer swarming, as used by BitTorrent, is one of the de facto solutions for content dissemination in today’s Internet. By leveraging resources provided by users, peer-to-peer swarming is a simple, scalable and efficient mechanism for content distribution. Although peer-to-peer swarming has been widely studied for a decade, prior work has focused on the dissemination of one commodity (a single file). This thesis focuses on the multi-commodity case. We have discovered through measurements that a vast number of publishers currently disseminate multiple files in a single swarm (bundle). The first contribution of this thesis is a model for content availability. We use the model to show that, when publishers are intermittent, bundling K files increases content availability exponentially as function of K. When there is a stable publisher, we consider content availability among peers (excluding the publisher). Our second contribution is the estimate of the dependency of peers on the stable publisher, which is useful for provisioning purposes as well as in deciding how to bundle. To this goal, we propose a new metric, swarm self-sustainability, and present a model that yields swarm self-sustainability as a function of the file size, popularity and service capacity of peers. Then, we investigate reciprocity and the use of barter that occurs among peers. As our third contribution, we prove that the loss of efficiency due to the download of unrequested content to enforce direct reciprocity, as opposed to indirect reciprocity, is at most two in a class of networks without relays. Finally, we study algorithmic and economic problems faced by enterprises who leverage swarming systems and who control prices and bundling strategies. As our fourth contribution, we present two formulations of the optimal bundling problem, and prove that one is NP hard whereas the other is solvable by a greedy strategy. From an economic standpoint, we present conditions for the existence and uniqueness of an equilibrium between publishers and peers.

In this study the maximum entropy formalism [JAYN 57] is suggested as an alternative theory for general queueing systems of computer performance analysis. The motivation is to overcome some of the problems arising in this field and to extend the scope of the results derived in the context of Markovian queueing theory. For the M/G/l model a unique maximum entropy solution., satisfying locALl balance is derived independent of any assumptions about the service time distribution. However, it is shown that this solution is identical to the steady state solution of the underlying Marko-v process when the service time distribution is of the generalised exponential (CE) type. (The GE-type distribution is a mixture of an exponential term and a unit impulse function at the origin). For the G/M/1 the maximum entropy solution is identical in form to that of the underlying Markov process, but a GE-type distribution still produces the maximum overall similar distributions. For the GIG11 model there are three main achievements: first, the spectral methods are extended to give exaft formulae for the average number of customers in the system for any G/G/l with rational Laplace transform. Previously, these results are obtainable only through simulation and approximation methods. (ii) secondly, a maximum entropy model is developed and used to obtain unique solutions for some types of the G/G/l. It is also discussed how these solutions can be related to the corresponding stochastic processes. (iii) the importance of the G/GE/l and the GE/GE/l for the analysis of general networks is discussed and some flow processes for these systems are characterised. For general queueing networks it is shown that the maximum entropy solution is a product of the maximum entropy solutions of the individual nodes. Accordingly, existing computational algorithms are extended to cover general networks with FCFS disciplines. Some implementations are suggested and a flow algorithm is derived. Finally, these results are iised to improve existing aggregation methods. In addition, the study includes a number of examples, comparisons, surveys, useful comments and conclusions.

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.

No description available.

Stochasticity is a ubiquitous feature of cellular processes such as gene expression that can give rise to phenotypic differences for genetically identical cells. Understanding how the underlying biochemical reactions give rise to variations in mRNA/protein levels is thus of fundamental importance to diverse cellular processes. Recent technological developments have enabled single-cell measurements of cellular macromolecules which can shed new light on processes underlying gene expression. Correspondingly, there is a need for the development of theoretical tools to quantitatively model stochastic gene expression and its consequences for cellular processes. In this dissertation, we address this need by developing general stochastic models of gene expression. By mapping the system to models analyzed in queueing theory, we derive analytical expressions for the noise in steady-state protein distributions. Furthermore, given that the underlying processes are intrinsically stochastic, cellular regulation must be designed to control the`noise' in order to adapt and respond to changing environments. Another focus of this dissertation is to develop and analyze stochastic models of post-transcription regulation. The analytical solutions of the models proposed provide insight into the effects of different mechanisms of regulation and the role of small RNAs in fine-tunning the noise in gene expression. The results derived can serve as building blocks for future studies focusing on regulation of stochastic gene expression. / Ph. D.

<p>In this project we illustrate how queueing simulation may be used to find the optimal interval for checkpointing problems and compare results with theoretical computations for simple systems that may be treated analytically.</p><p>We consider a relatively simple model for an internet banking facility. From time to time, the application server breaks down. The information at the time of the breakdown has to be passed onto the back up server before service may be resumed. To make the change over as efficient as possible, information of the state of user’s accounts is saved at regular intervals. This is known as checkpointing.</p><p>Firstly, we use GPSS (a queueing simulation tool) to find, by simulation, an optimal checkpointing interval, which maximises the efficiency of the server. Two measures of efficiency are considered; the availability of the server and the average time a customer spends in the system. Secondly, we investigate how far the queueing theory can go to providing an analytic solution to the problem and see whether or not this is in line with the results obtained through simulation.</p><p>The analysis shows that checkpointing is not necessary if breakdowns occur frequently and log reading after failure does not take much time. Otherwise, checkpointing is necessary and the analysis shows how GPSS may be used to obtain the optimal checkpointing interval. Relatively complicated systems may be simulated, where there are no analytic tools available. In simple cases, where theoretical methods may be used, the results from our simulations correspond with the theoretical calculations.</p>

In this project we illustrate how queueing simulation may be used to find the optimal interval for checkpointing problems and compare results with theoretical computations for simple systems that may be treated analytically. We consider a relatively simple model for an internet banking facility. From time to time, the application server breaks down. The information at the time of the breakdown has to be passed onto the back up server before service may be resumed. To make the change over as efficient as possible, information of the state of user’s accounts is saved at regular intervals. This is known as checkpointing. Firstly, we use GPSS (a queueing simulation tool) to find, by simulation, an optimal checkpointing interval, which maximises the efficiency of the server. Two measures of efficiency are considered; the availability of the server and the average time a customer spends in the system. Secondly, we investigate how far the queueing theory can go to providing an analytic solution to the problem and see whether or not this is in line with the results obtained through simulation. The analysis shows that checkpointing is not necessary if breakdowns occur frequently and log reading after failure does not take much time. Otherwise, checkpointing is necessary and the analysis shows how GPSS may be used to obtain the optimal checkpointing interval. Relatively complicated systems may be simulated, where there are no analytic tools available. In simple cases, where theoretical methods may be used, the results from our simulations correspond with the theoretical calculations.