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Developing equations to compute congestion measures for the general G/G/s/c queueing model and networks of such nodes has always been a challenge. One approach to analyzing such systems is to approximate the model-specified general input processes and distributions by processes and distributions from the more computationally friendly family of phase-type processes and distributions. We develop numerical approximation methods for analysis of general time-dependent queueing nodes by introducing new approximations for the time-dependent first two moments of the number-in-system and departure-count processes. / Ph. D.

The determination of the right number of servers in a multi-server queueing system is one of the most important problems in applied queueing theory. The problem becomes more complex in a system that consists of both cross-trained and specialized servers. Such queueing systems are readily found in the call centres (also called contact centres) of financial institutions, telemarketing companies and other organizations that provide services to customers in multiple languages. They are also found in computer network systems where some servers are dedicated and others are flexible enough to handle various clients' requests. Over-staffing of these systems causes increased labour costs for the underutilized pool of agents on duty, while under-staffing results in reduced revenue from lost customers and an increase in queue times. The efficient design and analysis of these systems helps management in making better staffing decisions. This thesis aims to develop models for establishing agent staffing levels in organizations with cross-trained and specialized staff with a view to minimizing cost and maintaining a desirable customer satisfaction. The work investigates the effect of various traffic loads on the number of agents required and the cost. It also considers how using specialized agents, flexible agents and a combination of both categories of agents affects the system. It uses a contact centre that has agents with monolingual, bilingual and trilingual (English, French and Spanish) capabilities to do the study.

The determination of the right number of servers in a multi-server queueing system is one of the most important problems in applied queueing theory. The problem becomes more complex in a system that consists of both cross-trained and specialized servers. Such queueing systems are readily found in the call centres (also called contact centres) of financial institutions, telemarketing companies and other organizations that provide services to customers in multiple languages. They are also found in computer network systems where some servers are dedicated and others are flexible enough to handle various clients' requests. Over-staffing of these systems causes increased labour costs for the underutilized pool of agents on duty, while under-staffing results in reduced revenue from lost customers and an increase in queue times. The efficient design and analysis of these systems helps management in making better staffing decisions. This thesis aims to develop models for establishing agent staffing levels in organizations with cross-trained and specialized staff with a view to minimizing cost and maintaining a desirable customer satisfaction. The work investigates the effect of various traffic loads on the number of agents required and the cost. It also considers how using specialized agents, flexible agents and a combination of both categories of agents affects the system. It uses a contact centre that has agents with monolingual, bilingual and trilingual (English, French and Spanish) capabilities to do the study.

A Generalized Cambridge Ring is a queueing system that can be used as an approximate model of some material handling systems used in modern factories. It consists of one or more vehicles that carry cargo from origins to destinations around a loop, with queues forming when cargo temporarily exceeds the capacity of the system. For some Generalized Cambridge Rings that satisfy the usual traffic conditions for stability, it is demonstrated that some nonidling scheduling polices are unstable. A good scheduling policy will increase the efficiency of these systems by reducing waiting times and by therefore also reducing work in process (WIP). Simple heuristic policies are developed which provide substantial improvements over the commonly used first-in-first-out (FIFO) policy. Variances are incorporated into previously developed fluid models that used only means to produce a more accurate partially discrete fluid mean-variance model, which is used to further reduce waiting times. Optimal policies are obtained for some simple special cases, and simulations are used to compare policies in more general cases. The methods developed may be applicable to other queueing systems. / text

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.

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We consider a class of queueing systems that consist of server pools in parallel and multiple customer classes. Customer service times are assumed to be exponentially distributed. We study the asymptotic behavior of these queueing systems in a heavy traffic regime that is known as the Halfin and Whitt many-server asymptotic regime. Our main contribution is a general framework for establishing state space collapse results in the Halfin and Whitt many-server asymptotic regime for parallel server systems having multiple customer classes. In our work, state space collapse refers to a decrease in the dimension of the processes tracking the number of customers in each class waiting for service and the number of customers in each class being served by various server pools. We define and introduce a state space collapse function, which governs the exact details of the state space collapse. Our methodology is similar in spirit to that in Bramson (1998); however, Bramson studies an asymptotic regime in which the number of servers is fixed and Bramson does not require a state space collapse function. We illustrate the applications of our results in three different parallel server systems. The first system is a distributed parallel server system under the minimum-expected-delay faster-server-first (MED-FSF) or minimumexpected- delay load-balancing (MED-LB) policies. We prove that the MED-FSF policy minimizes the stationary distribution of total number of customers in the system. However, under the MED-FSF policy all the servers in our distributed system except those with the lowest service rate experience 100% utilization but under the MED-LB policy, on the other hand, the utilizations of all the server pools are equal. The second system we consider is known as the N-model. We show that when the service times only depend on the server pool providing service a static priority rule is asymptotically optimal. Finally, we study two results conjectured in the literature for V-systems. We show for all of these systems that the conditions on the hydrodynamic limits can easily be checked using the standard tools that have been developed in the literature to analyze fluid models.