The optimization of the performance of a constant Work-in-Progress (CONWIP) production system through the sequencing of its backlog list is investigated. The performance measures considered are throughput, optimum WIP level (m*), and flow time. Analysis of the effects of sequence dependent bottlenecks on system performance is provided. A procedure is presented to determine a lower bound for (m*) given the product mix. A method that determines (m*) given the sequence of jobs is provided. A heuristic algorithm is provided for the purpose of determining a sequence to minimize (m*). The algorithm attempts to sequence the jobs to achieve the "best fit" between consecutive jobs so that machine and job idle times are minimized. The algorithm is tested through computer implementation to reveal its proficiency. / Master of Science

We define a service factory to be a network of service-related-workstations, at which assigned workers process work-in-progress that flows through the workstations. Examples of service factory work include mail processing and sorting, check processing and telephoned order processing. Exogenous work may enter the factory at any workstation according to any time-of-day profile. Work-in-progress flows though the factory in discrete time according to Markovian routings. Workers, who in general are cross trained, may work part time or full time shifts, may start work only at designated shift starting times, and may change job assignments at midshift. In order to smooth the flow of work-in-progress through the service factory, work-in-progress may be temporarily inventoried (in buffers) at work stations. The objective is to schedule the workers (and correspondingly, the workflow) in a manner that minimizes labor costs subject to a variety of service-level, contractural and physical constraints. Motivated in part by analysis techniques of discrete time linear time-invariant (LTI) systems, an object-oriented linear programming (OOLP) model is developed. Using exogenous input work profiles typical of large U. S. mail processingfacilities, illustrative computational results are included.

Analyses are addressed for a number of problems in queueing systems and stochastic modeling that arose due to an investigation into techniques that could be used to approximate general closed networks. In Chapter II, a method is presented to calculate the system size distribution at an arbitrary point in time and at departures for a (n)/G/1/N queue. The analysis is carried out using an embedded Markov chain approach. An algorithm is also developed that combines our analysis with the recursive method of Gupta and Rao. This algorithm compares favorably with that of Gupta and Rao and will solve some situations when Gupta and Rao's method fails or becomes intractable. In Chapter III, an approach is developed for generating exact solutions of the time-dependent conditional joint probability distributions for a phase-type renewal process. Closed-form expressions are derived when a class of Coxian distributions are used for the inter-renewal distribution. The class of Coxian distributions was chosen so that solutions could be obtained for any mean and variance desired in the inter-renewal times. In Chapter IV, an algorithm is developed to generate numerical solutions for the steady-state system size probabilities and waiting time distribution functions of the SM/PH/1/N queue by using the matrix-analytic method. Closed form results are also obtained for particular situations of the preceding queue. In addition, it is demonstrated that the SM/PH/1/N model can be implemented to the analysis of a sequential two-queue system. This is an extension to the work by Neuts and Chakravarthy. In Chapter V, principal results developed in the preceding chapters are employed for approximate analysis of the closed network of queues with arbitrary service times. Specifically, the (n)/G/1/N queue is applied to closed networks of a general topology, and a sequential two-queue model consisting of the (n)/G/1/N and SM/PH/1/N queues is proposed for tandem queueing networks.

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.

The goal of this thesis was to understand Queueing Theory and use it for analysis of queueing system of hypermarket company PZV. Theoretical part is consists of basic iformations which is necessary to know for application of queueing theory. First chapter is about random variables, random event, random proces, stochastic and Poisson processes and Markov chains. Then was possible to describe queueing theory basic characteristics, parameters, analy-sis, classification and basic models. In the practical part was from the beginning analyze development of custo-mers going through cashier zone. On the base of analysis were customers divided into 4 groups. For these groups were counted characteristics and parameters, next was analyzed the system, but because data were too inaccurate, it was necessary to use different values for the groups and to count analysis for these values. The last step was to find number of cashiers to make system optimal. The result, it was 4 ca-shiers, was unfortunately not accurate. For better results it would be necessary to have more detail data.

The purpose of this research is to investigate and analyse some batch arrival queueing systems with Bernoulli scheduled vacation process and single server providing service. The study aims to explore and extend the work done on vacation and unreliable queues with a combination of assumptions like balking and re-service, reneging during vacations, time homogeneous random breakdowns and fluctuating modes of service. We study the steady state properties, and also transient behaviour of such queueing systems. Due to vacations the arriving units already in the system may abandon the system without receiving any service (reneging). Customers may decide not to join the queue when the server is in either working or vacation state (balking). We study this phenomenon in the framework of two models; a single server with two types of parallel services and two stages of service. The model is further extended with re-service offered instantaneously. Units which join the queue but leave without service upon the absence of the server; especially due to vacation is quite a natural phenomenon. We study this reneging behaviour in a queueing process with a single server in the context of Markovian and non-Markovian service time distribution. Arrivals are in batches while each customer can take the decision to renege independently. The non-Markovian model is further extended considering service time to follow a Gamma distribution and arrivals are due to Geometric distribution. The closed-form solutions are derived in all the cases. Among other causes of service interruptions, one prime cause is breakdowns. We consider breakdowns to occur both in idle and working state of the server. In this queueing system the transient and steady state analysis are both investigated. Applying the supplementary variable technique, we obtain the probability generating function of queue size at random epoch for the different states of the system and also derive some performance measures like probability of server‟s idle time, utilization factor, mean queue length and mean waiting time. The effect of the parameters on some of the main performance measures is illustrated by numerical examples to validate the analytical results obtained in the study. The Mathematica 10 software has been used to provide the numerical results and presentation of the effects of some performance measures through plots and graphs.

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.

In this thesis, we study the tradeoff of average delay with average service cost and average utility for both continuous time and discrete time single server queueing models without and with admission control. The continuous time and discrete time queueing models that we consider are motivated by cross-layer models for point-to-point links with random packet arrivals and fading at slow and fast time scales. Our studies are motivated by the need to optimally tradeoff the average delay of the packets (a network layer performance measure) with the average service cost of transmitting the packets, e.g. the average power required for transmission (a physical layer performance measure) under a lower bound constraint on the average throughput, in various point-to-point communication scenarios. The tradeoff problems are studied for a class of monotone and stationary scheduling policies and under the assumption that the service cost rate and utility rate are respectively convex and concave functions of the service rate and arrival rate. We also consider the problem of optimally trading off the average delay and average error rate of randomly arriving message symbols which are transmitted over a noisy point-to-point link, in which case the service cost function is non-convex. The solutions to the tradeoff problems that we address in the thesis are asymptotic in nature, and are similar in spirit to the Berry-Gallager asymptotic bounds. It is intuitive that to keep a queue stable under a lower bound constraint on the average utility a minimum number of customers have to be served per unit time. This in turn implies that queue stability requires a minimum average service cost expenditure. In the thesis we obtain an asymptotic characterization of the minimum average delay for monotone stationary policies subject to an upper bound constraint on the average service cost and a lower bound constraint on the average utility, in the asymptotic regime where the average service cost constraint is made arbitrarily close to the above minimum average service cost. In the thesis, we obtain asymptotic lower bounds on the minimum average delay for the cases for which lower bounds were previously not known. The asymptotic characterization of the minimum average delay for monotone stationary policies, for both continuous time and discrete time models, is obtained via geometric bounds on the stationary probability of the queue length, in the above asymptotic regime. The restriction to monotone stationary policies enables us to obtain an intuitive explanation for the behaviour of the asymptotic lower bounds using the above geometric bounds on the stationary probability distribution of the queue length. The geometric bounds on the stationary probability of the queue length also lead to a partial asymptotic characterization of the structure of any optimal monotone stationary policy, in the above asymptotic regime, which was not available in previous work. Furthermore, the geometric bounds on the stationary probability can be extended to analyse the tradeoff problem in other scenarios, such as for other continuous time queueing models, multiple user communication models, queueing models with service time control, and queueing models with general holding costs. Usually, queueing models with integer valued queue evolution, are approximated by queueing models with real valued queue evolution and strictly convex service cost functions for analytical tractability. Using the asymptotic bounds, we show that for some cases the average delay does not grow to infinity in the asymptotic regime, although the approximate model suggests that the average delay does grow to infinity. In other cases where the average delay does grow to infinity in the asymptotic regime, our results illustrate that the tradeoff behaviour of the approximate model is different from that of the original integer valued queueing model unless the service cost function is modelled as the piecewise linear lower convex envelope of the service cost function for the original model.

Distributed Discrete Event Simulation (DDES) has received much attention in recent years, owing to the fact that uniprocessor based serial simulations may require excessive amount of simulation time and computational resources. It is therefore natural to attempt to use multiple processors to exploit the inherent parallelism in discrete event simulations in order to speed up the simulation process. In this dissertation we study the performance of distributed simulation of queueing networks, by analysing queueing models of message flows in distributed discrete event simulators. Most of the prior work in distributed discrete event simulation can be catego­rized as either empirical studies or analytic (or formal) models. In the empirical studies, specific experiments are run on both conservative and optimistic simulators to see which strategy results in a faster simulation. There has also been increasing activity in analytic models either to better understand a single strategy or to compare two strategies. Little attention seems to have been paid to the behaviour of the interprocessor message queues in distributed discrete event simulators. To begin with, we study how to model distributed simulators of queueing networks. We view each logical process in a distributed simulation as comprising a message sequencer with associated message queues, followed by an event processor. A major contribution in this dissertation is the introduction of the maximum lookahead sequencing protocol. In maximum lookahead sequencing, the sequencer knows the time-stamp of the next message to arrive in the empty queue. Maximum lookahead is an unachievable algorithm, but is expected to yield the best throughput compared to any realisable sequencing technique. The analysis of maximum lookahead, therefore, should lead to fundamental limits on the performance of any sequencing algorithm We show that, for feed forward type simulators, with standard stochastic assump-tions for message arrival and time-stamp processes, the message queues are unstable for conservative sequencing, and for conservative sequencing with maximum lookahead and hence for optimistic resequencing, and for any resequencing algorithm that does not employ interprocessor "flow control". It follows that the resequencing problem is fundamentally unstable and some form of interprocessor flow control is necessary in order to make the message queues stable (without message loss). We obtain some generalizations of the insta­bility results to time-stamped message arrival processes with certain ergodicity properties. For feedforward type distributed simulators, we study the throughput of the event sequencer without any interprocessor flow control. We then incorporate flow control and study the throughput of the event sequencer. We analyse various flow control mechanisms. For example, we can bound the buffers of the message queues, or various logical processes can be prevented from getting too far apart in virtual time by means of a mechanism like Moving Time Windows or Bounded Lag. While such mechanisms will serve to stabilize buffers, our approach, of modelling and analysing the message flow processes in the simulator, points towards certain fundamental limits of efficiency of distributed simulation, imposed by the synchronization mechanism. Next we turn to the distributed simulation of more general queueing networks. We find an upper bound to the throughput of distributed simulators of open and closed queueing networks. The upper bound is derived by using flow balance relations in the queueing network and in the simulator, processing speed constraints, and synchronization constraints in the simulator. The upper bound is in terms of parameters of the queueing network, the simulator processor speeds, and the way the queueing network is partitioned or mapped over the simulator processors. We consider the problem of choosing a mapping that maximizes the upper bound. We then study good solutions o! this problem as possible heuristics for the problem of partitioning the queueing network over the simulator processors. We also derive a lower bound to the throughput of the distributed simulator for a simple queueing network with feedback. We then study various properties of the maximum lookahead algorithm. We show that the maximum lookahead algorithm does not deadlock. Further, since there are no syn­chronization overheads, maximum lookahead is a simple algorithm to study. We prove that maximum lookahead sequencing (though unrealisable) yields the best throughput compared to any realisable sequencing technique. These properties make maximum lookahead a very useful algorithm in the study of distributed simulators of queueing networks. To investigate the efficacy of the partitioning heuristic, we perform a study of queue­ing network simulators. Since it is important to study the benefits of distributed simula­tion, we characterise the speedup in distributed simulation, and find an upper bound to the speedup for a given mapping of the queues to the simulator processors. We simulate distributed simulation with maximum lookahead sequencing, with various mappings of the queues to the processors. We also present throughput results foT the same mappings but using a distributed simulation with the optimistic sequencing algorithm. We present a num­ber of sufficiently complex examples of queueing networks, and compare the throughputs obtained from simulations with the upper bounds obtained analytically. Finally, we study message flow processes in distributed simulators of open queueing networks with feedback. We develop and study queueing models for distributed simulators with maximum lookahead sequencing. We characterize the "external" arrival process, and the message feedback process in the simulator of a simple queueing network with feedback. We show that a certain "natural" modelling construct for the arrival process is exactly correct, whereas an "obvious" model for the feedback process is wrong; we then show how to develop the correct model. Our analysis throws light on the stability of distributed simulators of queueing networks with feedback. We show how the stability of such simulators depends on the parameters of the queueing network.

Databases form the common component of many software systems, including mission critical transaction processing systems and multi-tier Internet applications. There is a large body of research in the performance of database management system components, while studies of overall database system performance have been limited. Moreover, performance models specifically targeted at the database design have not been extensively studied. This thesis attempts to address this concern by proposing a performance evaluation method for database designs based on queueing network models. The method is targeted at designs of large databases in which I/O is the dominant cost factor. The database design queueing network performance model is suitable in providing what if comparisons of database designs before database system implementation. A formal specification that captures the essential database design features while keeping the performance model sufficiently simple is presented. Furthermore, the simplicity of the modelling algorithms permits the direct mapping between database design entities and queueing network models. This affords for a more applicable performance model that provides relevant feedback to database designers and can be straightforwardly integrated into early database design development phases. The accuracy of the modelling technique is validated by modelling an open source implementation of the TPC-C benchmark. The contribution of this thesis is considered to be significant in that the majority of performance evaluation models for database systems target capacity planning or overall system properties, with limited work in detailed database transaction processing and behaviour. In addition, this work is deemed to be an improvement over previous methodologies in that the transaction is modelled at a finer granularity, and that the database design queueing network model provides for the explicit representation of active database rules and referential integrity constraints.