# Modeling and Analysis: An Introduction to System Performance Evaluation Methodology

... D. G. Kendall [8] extended Feller's birth-and-death (BD) process by considering the birth and death rates as any specified functions of the time t, λ(t) and µ(t). The BD process is a special class of time-continuous discrete-state Markov process, and has found applications in many scientific and engineering fields, including population biology [9], teletraffic and queueing theory [10], [11], [12], system modeling [13] [14], pp. 63-94, [15], pp. ...

... 5 The time-asynchronous event scheduling approach is a more time efficient and accurate simulation method than a time-synchronous approach. See e.g., [13] pp. 230-234, or [14], pp. ...

... 59-62). 13 This definition can be extended for a complex number z, with (z) > 0. 14 It will be worth noting that Ren and Kobayashi [23], [24] discuss this type of PDE in the analysis of multiple on-off sources in traffic characterization of a data network. 4.1 When the system is initially empty, i.e., I(0) = 0 ...

Why are the epidemic patterns of COVID-19 so different among different cities or countries which are similar in their populations, medical infrastructures, and people's behavior? Why are forecasts or predictions made by so-called experts often grossly wrong, concerning the numbers of people who get infected or die? The purpose of this study is to better understand the stochastic nature of an epidemic disease, and answer the above questions. Much of the work on infectious diseases has been based on "SIR deterministic models," (Kermack and McKendrick:1927.) We will explore stochastic models that can capture the essence of the seemingly erratic behavior of an infectious disease. A stochastic model, in its formulation, takes into account the random nature of an infectious disease. The stochastic model we study here is based on the "birth-and-death process with immigration" (BDI for short), which was proposed in the study of population growth or extinction of some biological species. The BDI process model ,however, has not been investigated by the epidemiology community. The BDI process is one of a few birth-and-death processes, which we can solve analytically. Its time-dependent probability distribution function is a "negative binomial distribution" with its parameter $r$ less than $1$. The "coefficient of variation" of the process is larger than $\sqrt{1/r} > 1$. Furthermore, it has a long tail like the zeta distribution. These properties explain why infection patterns exhibit enormously large variations. The number of infected predicted by a deterministic model is much greater than the median of the distribution. This explains why any forecast based on a deterministic model will fail more often than not.

... In general, the mean response time R for tasks arriving at a processing element is given from the Little's law defined in [8], as follows: ...

... where N is the mean number of tasks at that processing element. In a system of n processing elements, the mean response time is given as follows [8]: ...

... where l k is the mean arrival rate and is the mean service rate at the processing element k. It follows that the mean response time for the whole system is given as follows [8]: ...

Masalah penjadualan kerja ialah satu masalah pengoptimuman kombinatorik yang diketahui mempunyai darjah kebebasan berinteraksi yang besar. Oleh itu, masalah ini selalunya dikategorikan sebagai NP-lengkap. Kebanyakan penyelesaian bagi masalah ini menggunakan heuristik. Penyelesaiannya termasuk pendekatan berasaskan penjadualan tersenarai, teori giliran, teori graf dan pencaraian berenumerasi. Dalam kertas ini, satu kaedah penjadualan dinamik dicadangkan bagi memeta satu set kerja kepada satu set pemproses dalam rangkaian jaring boleh-konfigurasi. Model kami dipanggil Dynamic Scheduler on Reconfigurable Mesh (DSRM). Model ini berasaskan sistem giliran m/m/c di mana kerja-kerja yang tiba menunggu giliran masing-masing mengikut taburan Poisson, dan diservis mengikut taburan eksponen. Objektif utama dalam kajian ini ialah untuk menghasilkan jadual yang mempunyai taburan kerja seimbang pada pemproses-pemproses. Objektif kedua ialah untuk mempertingkat kadar pengagihan kerja ke tahap maksimum untuk memastikan kejayaan. Kedua-dua objektif ini merupakan satu masalah dikenali sebagai maksimum-minimum, di mana kejayaan dalam satu objektif boleh menyebabkan kemerosotan dalam objektif yang satu lagi. Keberkesanan pendekatan ini dikaji melalui model simulasi DSRM. Kata kunci: Jaring boleh-konfigurasi; penjadualan kerja; pengseimbangan beban dan perkomputeran selari Task scheduling is a combinatorial optimisation problem that is known to have large interacting degrees of freedom and is gerally classified as NP-complete. Most solutions to the problem have been proposed in the form of heuristics. These include approaches using list scheduling, queueing theory, graph theoretic and enumerated search. In this paper, we present a dynamic scheduling method for mapping tasks onto a set of processing elements (PEs) on the reconfigurable mesh parallel computing model. Our model called the Dynamic Scheduler on Reconfigurable Mesh (DSRM) is based on the Markovian m/m/c queueing system, where tasks arrive and form a queue according to Poisson distribution, and are serviced according to the exponential distribution. The main objective in our strudy is to produce a schedule that distributes the tasks fairly by balancing the load on all PEs. The second objective is to produce a high rate of sucessfully assigned tasks on the PEs. These two requirements tend to conflict and they constitute the maximum-minimum problem in optimisation, where the maximum of one causes the other to be minimum. We study the effectiveness of our approach in dealing with these two requirements in DSRM. Key words: Reconfigurable mesh; task scheduling; load balancing; and parallel computing

... Essa técnica é a que fornece resultados mais precisos, entretanto sua aplicação requer que o sistema em questão esteja fisicamente disponível. Cuidado especial deve ser tomado para que as próprias instruções que compõem essa técnica não interfiram de maneira significativa nos resultados da análise [KOB78,SAN90a,FER92]. O tempo e o custo consumidos no emprego dessa técnica são determinados pelo tipo e complexidade da coleta de dados a ser efetuada. ...

... Como exemplo da Figura 3.5 pode-se considerar uma agência bancária onde os vários caixas atendem a clientes que formam duas filas: uma para os clientes em geral e outra para idosos, gestantes e portadores de deficiências físicas. Os clientes em geral só são atendidos quando não houver ninguém na fila especial Nestes quatro casos existem parâmetros que devem ser estimados nos elementos de um modelo baseado em redes de filas, como padrão de chegada, mecanismo de serviço e disciplina da fila [KOB78]. ...

... No entanto, a chegada regular é irreal em muitas situações. O outro modo são as chegadas completamente aleatórias [KOB78]. Neste caso, os dados podem ser obtidos através de tabelas ou pela utilização de uma distribuição de probabilidade. ...

... The Poisson distribution of mean λ has the variance equal to the mean: σ 2 = λ; and for large λ, its CDF (cumulative distribution function) converges to that of the normal (or Gaussian distribution); Thus, 68 percent and 95 percent confidence levels could be well approximated by the λ ± √ λ, and the λ ± 2 √ λ, respectively. 6 log 10 e at = log 10 e × 0.2t = 0.087t. 7 In this analogy between the infection process and the individual's wealth growth, we interpret the infection rate λ as the ROI (return on investment), and the recovery/death rate µ as the rate of expenditures. ...

... As for detailed discussion of simulation techniques, see e.g.,[6], Chapter 4,[7], Chapter 16.3 Note that [n(λ + µ) + ν] is the coefficient of the Pn(t) term in the RHS of differential equation of Pn(t)(5). ...

In Part 1, we introduced a stochastic model of an infectious disease, based on the BDI (birth and death with immigration) process. We showed that random processes defined by this model can capture the essence of the stochastic, often erratic, behavior of the infection process. The most significant finding was that it is negative binomial distributed with small r, hence it is geometrically distributed with an exceedingly long tail. This leads to a much larger disparity in the epidemic patterns than has been known to the modeling community. In Part 2 we conduct simulation experiments by implementing an event-driven simulator. Several independent runs are presented to verify the findings reported in Part 1. The enormous variations among the sample paths obtained from several consecutive and independent runs with statistically identical conditions confirm our analysis. Note that the epidemic pattern that we observe is merely one sample path taken out of infinitely many paths. Thus, one such path cannot represent the ensemble of the underlying random process. By plotting the simulation runs in the semi-log scale, we can see that the probabilistic chances in the early phase of the infection process determine the behavior of the process. Once the process has reached a considerable number, the weak law of large numbers sets in, and the process behaves less erratically than in the early phase. One important implication of our findings is that it would be a futile effort to attempt to identify all plausible causes of the epidemic patterns. Mere luck may play a more significant role than most people believe. It would be worth remarking that the BDI process might be applicable to explain the disparity in wealth among individuals of similar earning power, expenditure, and investment portfolio.

... The process ( ) is never negative, therefore ( ) should also be limited to values ≥ 0. It is done by placing at = 0 a barrier which prevents the process from moving into negative part of axis. One choice is to place at = 0 a reflecting barrier [43] that limits the process to a positive -axis and is equivalent to the condition: ...

... The solution of Eq. (3) with boundary conditions defined by Eq. (8) gives us: [43] ( , ...

The article introduces an approach combining diffusion approximation and simulation ones. Furthermore, it describes how it can be used to evaluate active queue management (AQM) mechanisms. Based on the obtained queue distributions, the simulation part of the model decides on package losses and modifies the flow intensity sent by the transmitter. The diffusion is used to estimate queue distributions and the goal of the simulation part of the model is to represent the AQM mechanism. On the one hand, the use of the diffusion part considerably accelerates the performance of the whole model. On the other hand, the simulation increases the accuracy of the diffusion part. We apply the model to compare the performance of fractional order PIη controller used in AQM with the performance of RED, a well known active queue management mechanism.

... For simplification the phase distribution r in proposed model equals two. The type of using model is [7,9] the proposed queueing model allows realize realistic approach; take into account more than one flows between data plane (communication equipment) and feedbacks. The scheme of proposed queueing model represents on Fig. 2. ...

... Service waiting time depends on service discipline. The / M H m can be obtained by transforming the Little's low [9,12]. A function that determines the probability of service waiting time in the queuing system can be defined as: ...

The analytical approach for analysis the quality of service characteristics of Software-Defined Networking is proposed in the paper. Software-Defined Networking is promising concept that aimed to improving the performance and efficiency of computing networks. Current complexity of networks based this concept requires define the adequate methods for the calculation work characteristics. The model characterized by a Poisson arrival process with parameter and service time characterized by hyper-exponential distribution with r phases proposed to use for analysis QoS. Performance analysis using the proposed queuing model indicates that the model is accurate. Such QoS characteristics as maximum flow intensity that can be serving on switch and on controller, average packet sojourn time on single serve elements, delay and CP utilization of serve elements are calculated.

... For service stations in equilibrium, with < 1, similar theorems are unknown and we should rely on heuristic confirmation of the utility of this approximation. The process N (t) is never negative, hence X(t) should be also restrained to x ≥ 0. A simple solution is to put a reflecting barrier at x = 0, [43,44]. In this case ∂f (x, t; x 0 ) ∂t dx = 0 . ...

... in the similar way as it has been done in Eqs. (43)(44)(45), replacing r ij by λ ...

Diffusion theory is already a vast domain of modelling and performance evaluation. This tutorial does not cover all results but it presents in a coherent way an approach we have adopted and used in analysis of a series of models concerning evoluation of some traffic control mech-anisms in computer and communication networks. Diffusion approximation is here presented from an engineer's point of view, stressing its utility and commenting numerical problems of its implementation. Diffusion approximation is a method to model the behavior of a single queue-ing station or a network of stations. It allows us to include in a queueing model general sevice times, general (also self-similar) input streams and to investigate transient states, which are of interest in presence of bursty and constantly changing traffic. Let A(x), B(x) denote the interarrival and service time distributions at a service station. The distributions are general but not specified, the method requires only the knowledge of their two first moments. The means are denoted as E[A] = 1/λ, E[B] = 1/µ and variances are Var[A] = σ 2 A , Var[B] = σ 2 B .

... O número de processos em uma aplicação e sua demanda cumulativa são gerados com o auxHio de uma distribuição hiperexponencial bifásica (detalhes em [5]). Os parâmetros de entrada desta distribuição são a média m, o coeficiente de variação Cz e um valor o, neste caso igual a 0.95. ...

Sistemas paralelos de grande escala apresentam um custo muito alto, que aliado ao fato de serem geralmente monoprogramados, inviabiliza sua aquisição por muitas classes de usuários que deles poderiam se beneficiar. Por este motivo vem crescendo o uso de tais sistemas de forma multiprogramada. Multiprogramação eficiente implica no uso de políticas de escalonamento eficientes. Este trabalho, usando um modelo de simulação, investiga o comportamento de diversas políticas de escalonamento de processadores em diferentes configurações hipotéticas de sistemas paralelos multiprogramados. A partir dos resultados das simulações o artigo analisa o impacto das varias politicas de escalonamento no desempenho dos sistemas.

... The method of diffusion approximation was introduced by Gelenbe [19] and Kobayashi [20]; then the authors of [21] studied its numerical side. In this method, the number of customers in a queuing system is approximated by the value of the diffusion process. ...

The paper addresses two issues: (i) modeling dynamic flows transmitted in vast TCP/IP networks and (ii) modeling the impact of energy-saving algorithms. The approach is based on the fluid-flow approximation, which applies first-order differential equations to analyze the evolution of queues and flows. We demonstrate that the effective implementation of this method overcomes the constraints of storing large data in numerical solutions of transient problems in vast network topologies. The model is implemented and executed directly in a database system. It can analyze transient states in topologies of more than 100,000 nodes, i.e., the size which was not considered until now. We use it to investigate the impact of an energy-saving algorithm on the performance of a vast network. We find that it reduces network congestion and save energy costs but significantly lower network throughput.

... When packets arrive in the buffer, the number of bytes in the buffer increases and the buffer content continues to grow until the maximum defined size threshold or the waiting time threshold is reached. We represent the growth process of the content of the buffer by a diffusion approximation [32,33,52] process. Suppose that a diffusion approximation process X(t) represents the number of bytes stored in the buffer at time t, then the dynamic changes in the number of bytes accumulated in the buffer can be modelled by diffusion equation (which is a parabolic partial differential equation describing Brownian motion of tiny particles) [51] ...

The transmission of massive amounts of small packets generated by access networks
through high-speed Internet core networks to other access networks or cloud computing data centres
has introduced several challenges such as poor throughput, underutilisation of network resources,
and higher energy consumption. Therefore, it is essential to develop strategies to deal with these
challenges. One of them is to aggregate smaller packets into a larger payload packet, and these
groups of aggregated packets will share the same header, hence increasing throughput, improved
resource utilisation, and reduction in energy consumption. This paper presents a review of packet
aggregation applications in access networks (e.g., IoT and 4G/5G mobile networks), optical core
networks, and cloud computing data centre networks. Then we propose new analytical models based
on diffusion approximation for the evaluation of the performance of packet aggregation mechanisms.
We demonstrate the use of measured traffic from real networks to evaluate the performance of packet
aggregation mechanisms analytically. The use of diffusion approximation allows us to consider
time-dependent queueing models with general interarrival and service time distributions. Therefore
these models are more general than others presented till now.

... We approximate the size of the content of the buffer by a diffusion process. Diffusion approximation, since its very beginning [12], [13], [14] is applied in queueing problems and performance evaluation, appears also in a patent [15]. The models are often related to G/G/1, G/G/1/N or G/G/m/M+K stations and their networks, cf. ...

Today’s telecommunication network infrastructure and services are dramatically changing due partially to the rapid increase in the amount of traffic generation and its transportation. This rapid change is also caused by the increased demand for a high quality of services and the recent interest in green networking strengthened by cutting down carbon emission and operation cost. Access networks generate short electronic packets of different sizes, which are aggregated into larger optical packets at the ingress edge nodes of the optical backbone network. It is transported transparently in the optical domain, reconverted into the electronic domain at the egress edge nodes, and delivered to the destination access networks. Packet aggregation provides many benefits at the level of MAN, and core networks such as, increased spectral efficiency, energy efficiency, optimal resource utilisation, simplified traffic management which significantly reduces protocol and signalling overhead. However, packet aggregation introduces performance bottleneck at the edge node as the packets from the access networks are temporarily stored in the aggregation buffers during the packet aggregation process. In this article, we apply the diffusion approximation model and other stochastic modelling methods to analytically evaluate the performance of a new packet aggregation mechanism which was developed specifically for an N-GREEN (Next Generation of Routers for Energy Efficiency) metro network. We obtain the distribution of the packets’ queue in the aggregation buffer, which influences the distribution of the waiting time (delay) experienced by packets in the aggregation buffer. We then, demonstrate the influence of the probability p of successfully inserting the packet data units from the aggregation queue to the optical ring within a defined timeslot \(\varDelta \). We also discuss the performance evaluation of the complete ring by deriving the utilisation of each link.

... In comparison, a textbook used by the authors in their college years has an entire section dedicated to "Markov Chains and Their Properties" and Markovian techniques are seen throughout this earlier textbook 12 . While the authors may be dinosaurs clinging to the past, the importance of integrating Markov Chains into an M&S course has already been stressed. ...

... There are some "special" queueing networks in the simple models: thus, M/M/c/K (J. Medhi, 2003;Hisashi Kobayashi, 1978) The steps taken to create a queueing network are the same as those for a single queueing model: the first is to apply the corresponding NewInput function; then, optionally, the CheckInput function is called, to help with the input parameters of the model, and finally the model is built with the QueueingModel function. ...

... System approach is used as a way which gives the best results, and model as an investigation medium which contributes to observing complex reality. During the research numerous references have been used, but only few are emphasized, as are: Bauer and Wegener 1975, House 1975, Jililemann and Kuhn 1987, Kobayashi 1978, Sage 1977, Zrnic 1979, Zrnic 1996aand Zrnic 1997 ...

The paper has the aim to present the possibility of application of the procedure TPD in designing of a complex systems for transportation of copper ore in underground mining. Special attention is paid to evaluation of performances of the system and its components, identification of knot points (bottlenecks of the system) and their optimisation, as a means for improvement of performance of the whole system. For the complex system of transportation of copper ore in underground mining the algorythm is presented for optimisation of performances, i.e. for defining final design variables of the whole system through optimization of performances and design variables of subsystems and knot points. In this paper, a pragmatic approach is presented which combines simulation model with an multiattributive evaluation model (based on the multi utility theory).

... The P-value is determined from the F-value and the DF, and is a way to validate the prediction of the response. It is assumed that the model has a significant influence on the results if the P-value is less than 0.05 (Kobayashi, 1978;Devore, 1999). ...

Asphaltene deposition had always been a challenge for the flow assurance issues in oil industry due to both high removal costs and complex behavior in various temperature and composition conditions. These conditions could occur in any enhanced oil recovery (EOR) in the reservoir, and production of oil in tubing or surface facilities. The central composite experimental design of orthogonal type and response surface methodology (RSM) were used to develop reliable models to investigate the effects of input variables on asphaltene onset pressure (AOP). The considered input variables were the temperature and the gas mole% in CO2 and N2 gas injection scenarios. Regression analysis showed agreement of the experimental data with the determination coefficient (R²) values of 0.991 and 0.987 for CO2 and N2 gas injection models; respectively. Analysis of variance (ANOVA) was employed to test the significance of the RSM polynomial models. ANOVA study showed, in CO2 injection model, the effect of temperature was greater than the gas mole% while in N2 injection model, the gas mole% effect was dominant over the temperature effect. By investigating the models, it was seen that in N2 gas injection, any amount of gas increased the AOP at different temperatures. But in CO2 gas injection, depend on the temperature, AOP might rise or fall upon adding gas.

... Spending1/Ptime units at each of N spots, the time in the system will be T = N(1/P), or equivalently N = PT in accord with Little law. Further more, Kobayashi has shown that Little's law is true not only for First-Come-First-Serve (FCFS) and Last-Come-First-Serve (LCFS), but also for any queue discipline (Kobayashi, 1978). ...

In this study, we consider a queuing model extension for a production system composed of several parallel machines and the same number of transporters. To obtain the minimum waiting time of the jobs in the queue, we present an exact solution for the proposed queuing model. The solution integrates M/M/C system with M/M/1 system. We obtain explicit expressions for its steady-state behavior under M/M/C and M/M/1 assumptions. Further, in order to illustrate the usefulness of the proposed methods, numerical examples are solved. On the basis of the results of these examples, some important conclusions are drawn.

... (4) Una interpretación biológica de los parámetros de la distribución Erlang se muestra en la Fig. 21.1. Un organismo que pasa por K estados con un tiempo promedio de residencia, media = 1/ a, tiene una distribución Erlang(a, K) si, dentro de cada estado, el tiempo promedio de residencia es Y / K y la tasa de transferencia de un compartimiento (estado) a otro es K/ Y = aK, es decir, se considera que el tiempo de residencia dentro de cada compartimiento tiene una distribución exponencial(aK), por lo que el tiempo total de residencia, o tiempo de vida, es la suma de K variables exponenciales: Exp(i), i = 1, 2, ..., K. De esto se deduce que la distribución Erlang representa la suma de K variables exponenciales aleatorias (Kobayashi, 1978;Gordon, 1978;Smerage, 1992;López-Collado, 1994). ...

... Traditionally, analytical models were used more commonly since simulation models took a relatively longer time to build and execute. In 1978, the prevailing perception of simulations was lead by Kobayashi: It is quite often found, however, that a simulation model takes much longer to construct, requires much more time to execute, and yet provides much less information than the model writer expected [Kob78]. However, as the studied systems became larger and more complex, analytical models require making unrealistic assumptions and approximations [BGdMK06]. ...

p>Military organisations today operate small fleets of unique aircraft and need to be sure that the spares purchased to support operations meet the organisations needs whilst remaining the minimum necessary to minimise unnecessary government expenditure. Historically this task was undertaken by using historical consumption as the basis of the calculation. This is not seen as appropriate today and a range of deterministic models are used to produce the spares lists. However, their failure to apply a particular flying programme means that the output is viewed with some scepticism by military staffs. Simulation provides the means to apply that flying programme and, moreover, allows a series of what if evaluations to be undertaken. This thesis covers the work undertaken by myself to design and produce a suitable simulation application to meet the above requirement. Whilst data was available it was of a simple form without sufficient fidelity to allow the underlying distributions to be derived. Consequently, the opportunity to examine the effect of applying different distributions for both failure and repair times was taken allowing the scope of the work to broaden. Having produced the simulation a number of alternative flying programmes were simulated to identity their impact on the overall achievement and aircraft availability. This work has allowed me to not only provide a model which can be used with a deterministic application to assess the validity of the spares list but, has also allowed investigation into the effect of applying different distributions to both failure and repair times.</p

Fifty years have passed since Performance Evaluation (PE) was recognized as a discipline in its own right even if closely linked to computer science. In this period, computer systems, networks, applications and services have changed dramatically. Modern systems are very complex, current workloads consist of hundreds of thousands requests, and user expectations for performance become even more stringent. The computer engineering curricula for undergrad/graduate students and the courses taught in universities are very different from those of a few years ago. With such a completely new scenario, the time has come to analyze the situation in order to identify the teaching techniques of performance analysis in university courses that are best suited to today's world.

This book constitutes the post proceedings of the 28th International Symposium on Modelling, Analysis, and Simulation of Computer and Telecommunication Systems, MASCOTS 2020, held online -due to COVID -19- in Nice, France, in November 2020. The 17 full papers presented were carefully reviewed and selected from 124 submissions.
The symposium collected the most relevant papers describing state-of-the-art research in the areas of the performance evaluation of computer systems and networks as well as in related areas.

We present a general performance model for communication architectures in Internet of Things scenarios. The architecture involves three processing layers, i.e., edge, fog, and cloud computing. According to the objectives of edge and fog computing, we assume that a certain percentage of the data is finally processed at the edge and fog layer. The rest is forwarded to the next higher layer for further processing. Data processing is modeled as a sequence of queueing stations. This allows us to compute performance parameters such as throughput and response times at all layers. The main contribution of this paper is an easy to use method for IoT application designers that provides a very fast and very flexible parametric support for evaluating design trade-offs in edge-fog-cloud configurations.

Queueing theory models things waiting in lines. Such things include packets, telephone calls or computer jobs. Continuous time and discrete time single queues are reviewed. This includes M/M/1, Geom/Geom/1 and M/G/1 results. Networks of Markovian queues along with the mean value analysis (MVA) computational algorithm are discussed. Negative customer networks are examined. Recursive solutions for certain non-product form networks are covered. Stochastic Petri networks (SPN) with product form solutions are also considered. General solution techniques for these models are outlined.

Stochastic extensions to Petri Nets (SPNs) are another example of the impact that the original work of Carl Adam Petri had on many application and research fields. Stochastic processes are the common mathematical tools used for the performance and reliability evaluation of discrete-event dynamic systems (DEDSs), which are however difficult to use because of the complexity of real cases. SPNs (and their many extensions) allow one to build models which represent in a natural manner synchronization, congestion, blocking, concurrency, and dependency. SPN models permit rigorous reasoning about the behavior of a system and represent the basis for automated analysis through powerful software tools. Despite their practical success, much work is still needed to make SPNs the formalism of choice to analyze real DEDSs. Thus, we must still refer to the ideas of Carl Adam Petri to address the challenges coming from systems of ever-increasing complexity.

The book includes lectures given by the plenary and key speakers at the 9th International ISAAC Congress held 2013 in Krakow, Poland. The contributions treat recent developments in analysis and surrounding areas, concerning topics from the theory of partial differential equations, function spaces, scattering, probability theory, and others, as well as applications to biomathematics, queueing models, fractured porous media and geomechanics.

A recirculation system (RS) with a single input function, heterogeneous subsystems,and a finite capacity is considered. Inside the RS the subsystems are placed in parallel and are accessed randomly.Each subsystem is considered as a GI/M/I/o single server Queueing loss system with renewal input, no waiting room, and negative exponentialIy distributed service times. The single input function is a stationary counting process.
Inside the RS, at any cycle the incoming stream of units entering the system, travel inside to reach the routing mechanism. There, they are assigned upon arrival to one of the subsystems. At each subsystem,the units whicn find it full, retry to receive service oy recirculating back to the entrance of the system.There, they merge with the incoming arrival strem to form a new input to the routing mechanism at the next cycle. The units which enter each subsystem after receiving service depart from it. the previous steps are then repeated until the steady state is achieve.
An open Queueing network is developed to study the asymptotic performance of the above RS. The flows of units inside the RS are approximated by a two parameter method. The performance of the RS is measured by approximating the congestion inside the RS and evaluating the efficiency of eacn of its subsystms. Finally, an example is introduced and the approximation outcomes are campared against those from a simulation study.

An appropriate model for a number of systems is a network of queues in which the output of one queue is fed into another. Under a wide range of assumptions, these networks may be modeled and analyzed by means of multidimensional birth-death processes. The salient result of this work is the product form solution in which the joint distribution of queue occupancies is the product of functions of the number in the individual queues. Networks satisfying the proper set of assumptions are called Jackson networks after J. R. Jackson, who discovered the product form solution.(1) In this chapter the model is applied to store-and-forward message-switched networks. Using the theory of Jackson networks we shall find queue occupancy and delay in message-switched networks. These results enable us to allocate transmission capacity in an optimum fashion. In the next chapter these same ideas are extended in order to model flow control in a store-and-forward network.

In chapter 9, we formulated the multi-stage circuit switching problem as a multi-commodity resource allocation problem, for which a terminal pair requests the simultaneous allocation of bandwidth on a set of links In contrast, packet switching does not require simultaneous allocation but allocation on a link-by-link basis. Local conflicts in resource allocation result in queueing. Based on results for single stage queueing developed in the previous chapter, we approximate the queueing behavior in multi-stage packet networks in this chapter. The first two sections deal with more general results such as reversibility and product form solutions for networks. The last three sections discuss performance analysis methods for buffered banyan networks under different traffic assumptions.

Large industrial organizations strongly depend on the use of enterprise information systems for the application of their complex business processes. Typically, an enterprise information system (EIS) consists of a set of autonomous distributed components providing basic services. Business processes can be realized as workflows consisting of: (1) tasks combining basic services provided by EIS components and (2) synchronization dependencies among tasks. EIS users have ever-increasing nonfunctional requirements (e.g. performance, reliability, availability, etc.) on the quality of those systems. To satisfy those requirements, EIS engineers must perform quality analysis and evaluation, which involves analytically solving, or simulating quality models of the system (e.g. Markov chains, Queuing-nets, Petri-nets etc).
Good quality models are hard to build and require lots of experience and effort, which are not always available. A possible solution to the previous issue is to build automated procedures for quality model generation. Such procedures shall encapsulate previous existing knowledge on quality modeling and their use shall decrease the cost of developing quality models. In this paper, we concentrate on the performance and reliability of EISs and we investigate the automated generation of quality models from EIS architectural descriptions comprising additional information related to the aspects that affect the quality of the EIS.

In diesem Kapitel werden die Grundlagen der Verkehrs- und Bedientheorie vorgestellt. Zunächst werden Grundbegriffe wie Verkehrsaufkommen, Hauptverkehrsstunde, Anrufrate, Enderate, Erfolgs Wahrscheinlichkeit, Verlustwahrscheinlichkeit usw. definiert. Im nächsten Abschnitt werden Ankunfts- und Bedienprozesse behandelt und die für die Modellierung häufig verwendeten Verteilungen vorgestellt. Anschließend wird das Warte- und Verlustsystem M/M/1 behandelt. Hier werden zunächst die Systemgleichungen detailliert abgeleitet und dann gezeigt, wie diese direkt aus dem Zustandsdiagramm des Systems abgelesen werden können. Die stationären Lösungen der Systemgleichungen ergeben die Systemzustandswahrscheinlichkeiten. Hieraus können die mittlere Anzahl der Anforderungen im System bzw. in der Warteschlange, der Durchsatz usw. errechnet werden. Es folgt die Ableitung des Gesetzes von Little in der allgemeinen Form. Dies ermöglicht die Berechnung der Warte- und Verweildauer im System.

For the purposes of analysis and design, communication channels and networks are customarily represented by models which are, deliberately, abstractions from and simplifications of the real entities. We may variously consider a model to be an ideal which real systems should copy as closely as possible, or an approximate representation of a pre-existing real system. Whether we take the Platonic or the Aristotelian view, an important property of the model is that it shall correspond with the real system in respect of those features which are of dominant importance in defining qualitative aspects of behaviour and quantitative measures of performance. Another important property is that the model shall be sufficiently definite and sufficiently tractable to admit mathematical analysis, computation or simulation from which properties and performance may be estimated. In pursuit of clarity and tractability, it is likely to depart from reality in either or both of two ways.

The previous chapter dealt with one of the tools for performance analysis—queueing theory. This chapter concentrates on another tool—simulation. In this chapter, we provide an overview of simulation: its historical background, importance, characteristics, and stages of development.

In an operational context, designing sophisticated data processing systems involves a lot of difficulties such as choosing the good hardware components, dispatching the tasks on the various computers... But, above all, one of the most critical problem is to predict system’s performances; that means to evaluate, before construction, the architecture behavior under processes loading.

In this chapter we analyze a simple single server queue that is frequently used to model components of computer systems. This queue is termed the M/G/1 queue. This is standard “queueing notation,” first introduced by Kendall. Typically a queue is described by four variables $$
A/S/k/c,
$$which have the following interpretation:
A,S — The arrival (A) or service (S) process where M means Poisson arrivals and exponential service times, G means the process is generally distributed, E
ℓ
denotes an ℓ-stage Erlang distribution, and D denotes a deterministic distribution.
k — The number of servers.
c — The buffer size of the system. This is the total number of customers the system can hold. Arrivals to the system already containing c customers are assumed to be lost. If not specified, c is assumed to be infinite.

input-output analysis;aggregation levels;mass balance;technological coefficients;generic positive system

Bereits bei den Überlegungen zur Modellbildung wurde darauf hingewiesen, daß Probleme der Nachrichten- und Regelungstechnik im allgemeinen Lösungen erfordern, die nicht nur für bestimmte einzelne Signale, sondern für eine große Anzahl möglicher Signale mit gewissen gemeinsamen Eigenschaften gelten. Ein mathematisches Modell für eine derartige Schar von Signalen ist der Zujallsprozeß (oder stochastische Prozeß). Dieser soll im nun folgenden Kapitel betrachtet werden. Wir werden dabei feststellen, daß ein Zufallsprozeß als eine Schar von Zufallsvariablen definiert werden kann und somit aIle Gesetze für Zufallsvariablen auch hier anwendbar sind.

Queueing theory is a useful tool in design of computer networks and their performance evaluation. The literature concerning this subject is abundant. However, it is in general limited to the analysis of steady states. It means that flows of customers considered in models are constant and obtained solutions do not depend on time. It is in glaring contrast with the flows observed in real networks where the perpetual changes of traffic intensities are due to the nature of users, sending variable quantities of data, cf. multimedia traffic, and also due to the performance of traffic control algorithms which are trying to avoid congestion in networks, e.g. the algorithm of congestion window used in TCP protocol which is adapting the rate of the sent traffic to the observed losses or transmission delays. We discuss here the means used to analyse transient states in queueing models. In computer applications a mathematical model is useful only when it furnishes quantitative results. Therefore practical issues related to numerical side of models are of importance and are here discussed. We present three approaches-Markov models solved numerically, fluid flow approximation and diffusion approximation. A particular importance is given to the latter as the author has here over 20 year experience in development and application of this method. He is also convinced of the qualities of this approach- its flexibility to treat various variants of queueing models. Traffic intensity observed in computer networks have a complex stochastic nature that influences the network performances. We discuss also this side of implemented queueing models.

Ocean thermal energy conversion (OTEC) is a method of power generation which harnesses the deference of temperature between warm sea water at ocean surface and abyssal cold sea water. Recent studies on OTEC plants are being carried out through a pilot plant which substitutes a heating and cooling source system for a part of warm and cold sea water. However, the efficiency of the conventional water supply system such as one tank system is not good enough to use as a OTEC pilot plant, because it takes much time to supply the appropriate water. Further it is impossible to make the desired water temperature and flow-rate at a constant level during the experiment. In this paper, a new temperature flow-rate water supply system with closed cycle is proposed for solving the existing problems. Proposed system is rather convinient to use as an OTEC pilot plant due to two main advantages. First is the ability of quickly supplying the objective water with an appropriate temperature and flow-rate. And the other is the possibility of conducting an experiment for a long time by adopting the closed cycle water supply system. In order to construct the procedure for improving the controllability of this system theoreticaly, controllable range of the pumps was also investigated and evaluated.

Response Time Preservation (RTP) is introduced as a general technique for developing approximate analysis procedures for queueing networks. The underlying idea is to replace a subsystem by an equivalent server whose response time in isolation equals that of the entire subsystem in isolation. The RTP based approximations, which belong to the class of decomposition approximations, can be viewed as a dual of the Norton's Theorem approach for solving queueing networks since it matches response times rather than throughputs. The generality of the RTP technique is illustrated by developing solution procedures for several important queueing systems which violate product form assumptions. Examples include FCFS servers with general service times, FCFS servers with different service times for multiple classes, priority scheduling, and distributed systems.

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