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In this thesis, a model is developed for a finite population queueing system deployed to meet a constant demand for the situation where failed units require a single spare part to initiate repair action. A supporting inventory of spare parts is included in the model operating under a one-for-one ordering policy. The system was modeled as a Markov process, and an algorithm is presented that numerically evaluates the steady state probabilities. Cost was chosen as the measure of effectiveness of the system. Total system cost consists of shortage costs for not being able to meet the demand for units, population and repair facility annual equivalent costs, holding costs for keeping spare parts in inventory, and procurement and spare part costs related to procurement and purchase activities. A computer program in BASIC, designed for a microcomputer, enables the decision maker to interactively find the design that optimizes the effectiveness measure of the system. The decision variables considered are the number of units in the population, the number of repair channels, and the maximum level of spare parts. One specific design problem is presented. / M.S.

The thesis of this paper is to investigate how the dependency in the arrival process affects the queueing performance measures. The Markov renewal arrival process (MRAP) was chosen as the arrival process. This choice was made because many of the typical arrival processes can be obtained as special cases of the MRAP. But the main reason behind this choice is that the interarrival times of the MRAP are dependent. We assume that the queue is a single server queue with exponential service time and the investigation was carried out numerically because no analytical solution was available. There are 5 parameters of the arrival process used in this investigation: the traffic intensity (ρ), the squared coefficient of variation (scν), the serial correlation defined by the lag-1 correlation (corr) plus the rate ξ and the coefficient of skewness (𝛾). Here are the performance measures of the MR/M/1 queue we investigate: the expected queue length at arbitrary times (L^𝓽), the standard deviation (σ) of the queue length at arbitrary times and the caudal characteristic η. The other performance measures such as: the expected queue length at arrival time, the waiting time, the sojourn time, etc. can be easily obtained from L^𝓽. We compare these performance measures against those of the corresponding GI/M/1 queue. When the lag-1 correlation of the arrival process is negative (this means that the lags of the serial correlation alternate in signs), the L^t of the MR/M/1 queue is smaller (but not by much) than the L^𝓽 of the GI/M/1 queue. Therefore, we focus our attention to the MR/M/1 queue with positive serial correlation. The results are presented using graphs. We find that the coefficient of skewness of the arrival process (𝛾) plays an important role. The L^𝓽 curve decreases rapidly as 𝛾 increases and after certain values of 𝛾 called the turning region, the L^𝓽 curves Hatten. This important observation indicates that to the left of the turning region, the L^𝓽 is almost insensitive to the dependency in the arrival process. However, to the right of the turning region, the L^𝓽 is sensitive to the positive serial correlation in the arrival process. Highly correlated arrival process (large corr and ξ) can cause the L^𝓽 to be significantly larger than the L^𝓽 for the uncorrelated queue. For the MR/M/1 queue, the magnitude of the standard deviation σ is larger than the corresponding L^𝓽. However, the shapes of the σ curves are similar to those of the L^𝓽 curves. So, all of the conclusions drawn for the L^𝓽 also apply to the standard deviation σ. For the M/M/1 queue, the caudal characteristic η equals to the traffic intensity ρ (η=ρ). For the uncorrelated Gl/M/1 queue, one would expect that when scν<1.0, η<ρ (i.e., the queue would behave like a H/M/1 queue) and when scν>1.0, η>ρ (i.e., the queue would behave like a H/M/1 queue). Our results indicates that this is not necessarily true. We found again that the coefficient of skewness (𝛾) plays an important role. For the uncorrelated GI/M/1 queue with scν>1.0, η can be smaller than ρ when 𝛾 is large enough. For the correlated MR/M/1 queue, even for scν<1.0, a low 𝛾 value combined with the positive serial correlation can cause η to be larger than ρ. On the other hand, scν>1.0 does not necessarily results in η>ρ. A large value of 𝛾 can cause η to be smaller than ρ, even for the queue with highly correlated interarrival times. / Ph. D.

Consider a Markov jump process, X(t), with a nonnegative state space as a model for a queueing system. The motivation of this study is about useful estimates of system performance. For example, in a system with finite queues, the probability of the system of queues going from an empty state to a state in which the population of at least one queue reaches a large number before becoming empty again is one and the typical sample trajectory of this event is another. To answer these questions, we establish the large deviation principle (LDP) for an appropriate class of queueing processes. The model of our concern is the Jackson network which has a tree-type topological structure. Under carefully designed conditions, the LDP for a time homogeneous Markov process has been well established by Wentzel. However, mainly due to the nonnegativity constraint, the queue length process, X(t), of our model does not satisfy the assumed conditions. As a detour, we define the “potential process”, Y(t), which allows the negativity in state space in the way that even if a queue is empty, the server in the empty queue is working with a same rate as if the queue is not empty. Therefore, each Y_i(t) can be expressed as the difference of the accumulated number of customers who came to station i and the accumulated number of services, done in station, i, up to time t. Then the scaled processes, Y^∊(t) = ∊Y(t/∊), obeys LDP with a certain rate function, I_[0,T](x,Φ), i.e. P(Y^∊(.)∈ B|Y^∊(0) = x| ≈ exp[-1/∊ inf_Φ∊B I_[0,T](x,Φ)], (UTLE) for some B ⊂ D^r[0,T] = { right continuous R^r — valued function which has a left limit at every point on [0,T]}. UTLE stands for ‘up to logarithmic equivalence’. By defining an appropriate Skorohod problem, we obtain a continuous mapping θ from D^r to D^r₍₊₎,) such that θ(Y)(t) is a version of X(t). Then we “push the LDP of potential process through” θ so that LDP of the queue length process can be achieved. The procedure of ‘pushing through’ is another principle of the large deviation theory. It is called “contraction principle” [3]. The contraction principle provides the rate function J_[0,T] of the LDP for the queue length process and J_[0,T](Φ) = inf_ψ|θ(ψ)=Φ J_[0,T](ψ). That is, when X^∊ ≡ ∊X(t/∊), for an appropriate set B ⊂ D^r₍₊₎, P(X^∊(.)∈ B|X^∊(0) = x| ≈ exp[-1/∊ inf_Φ∊B I_[0,T](x,Φ)], (UTLE) The rate function, J_[0,T], is expressed in a closed form. / Ph. D.

A new implementation of the calendar queue is described in this thesis. The calendar queue as previously implemented depended upon the use of multiple linked lists for the control of queue discipline. In the calendar heap implementation, the heap has been used to replace the previous functions of the linked list. Testing of the claim of O(1) execution time for the calendar queue was done. Comparisons of execution times of the calendar queue and the calendar heap have been made. Descriptions of the implementation as well as the complete C code for the calendar heap are included.

This dissertation describes queueing-theoretic analysis of shared service systems that require that customers leave the system in the sequence in which they arrived. This requirement makes it necessary to resequence customers before they leave the system. Resequencing adds new complications to the analysis of queueing systems. While waiting time is still important, resequencing results in a new type of "non-working" delay of a customer called the resequencing delay. This dissertation presents primarily analytical and numerical methods to determine the distribution and mean value of resequencing delay, and of total delay. In the simplest models closed form analytical expressions have been obtained, but in more complex models numerical methods have been developed to compute the distribution and mean of resequencing delay, and of total delay. This enables us to study the behavior of resequencing and total delay as system parameters are changed. For several composite server models we present expressions for the distribution and mean of resequencing delay, and of total delay. In particular we consider the M/M/∞ composite server model, the M/H(K)/∞ composite server model, the G/M/∞ composite server model, the M/M/m composite server model, and the G/M/m composite server model. The formulas are interpreted using asymptotic approximation or bounding techniques. For more general composite server models, it is difficult to obtain closed form expressions for resequencing and total delay. We develop numerical methods based on matrix-geometric methods to compute resequencing and total delay. In particular, we develop numerical methods for the computation of the mean resequencing delay, and mean total delay for the M/H₂/m composite server model, and the M/Hypo₂/m composite server model.