Wiener-Hopf techniques for the analysis of the time-dependent behavior of queues

Summary In this thesis we study the time-dependent behavior of queueing systems. The study is focused on the queueing systems: 1. the $GI/G/1$ system, 2. the $GI/H_m/s$ system, 3. the Markovian Fluid Flow Model, the fluid flow model that is modulated by a Markov process, 4. the Semi-Markovian Fluid Flow Model, a generalization of the Markovian Fluid Flow Model. In general, the time-dependent behavior of queueing systems is much influenced by the initial server(s)’s work load. This leads us to consider the queueing systems with non-zero initial server(s)’s work load. In the $GI/G/1$ system and the $GI/H_m/s$ system this means that in the beginning there exist a number of (special) customers to serve. In the last two systems, initially the buffer has non-zero content. The technique that is used to analyze the behavior of the queueing systems studied in this thesis is based on the Wiener-Hopf factorization. A brief discussion on the Wiener-Hopf factorization is given in chapter 2, where we also give the conditions on the existence of uniqueness of the factorization. In this chapter we also give some liminaries that we need for the analysis in the rest chapters. The first major step in the analysis is the derivation of the (system of) transformed Wiener-Hopf equation(s). Wiener-Hopf factorization is then applied to its symbol. Since the queueing systems we consider have a non-zero initial working load, the Wiener-Hopf factorization should be followed by a decomposition on a certain (matrix) function. The Wiener-Hopf factorization and the decomposition yields a (formal) solution of the (system of) equation(s). If the stability condition is fulfilled, then the steady-state distributions of interest can be determined by applying Abel’s limit theorem to the solution of the (system of) equation(s). In chapter 3 we study the system $GI/G/1$ with non-zero initial number of customers. We get the explicit factorizations for two special systems, the systems $GI/K_n/1$ and $K_m/G/1$. These results give explicit expressions for the Lapace-Stieltjes transform of actual waiting times and virtual waiting times. Then, by applying a contour integration, we get the expectation of number of customers at arrival epochs and in continuous times as well. At the end of this chapter we give numerical results to illustrate the behavior of the system as it tends to the steady-state. In chapter 4 we study the system $GI/H_m/s$ with non-zero initial number of customers. As in chapter 3, the Wiener-Hopf factorization gives explicit expressions for the Lapace-Stieltjes transform of actual waiting times and virtual waiting times. Then, the distributions of the queue length and the number of customers in the system are derived, both at arrival epochs and in continuous time. At the end of this chapter we again give numerical results. In chapter 5 we study the Markovian Fluid Flow Model, the fluid flow model in which the rate of data that flow into the buffer depends on the state of a Markov process. The Wiener-Hopf factorization gives us an explicit expression for the Laplace-Stieltjes transform of buffer content at transition epochs of the underlying Markov process. From this we can derive the distribution of the buffer content in continuous times. We conclude this chapter with some numerical results. In chapter 6 we study a generalization of the model of chapter 5. The times between transitions of the underlying Markov process are not assumed to be exponentially distributed anymore but are assumed to be either hyper-exponentially distributed or hypoexponentially distributed. With this assumption, the symbol of Wiener-Hopf-type equations is still a rational matrix in $\phi$, and each element of this matrix has only simple poles. This matrix can be factorized by the Wiener-Hopf factorization technique as we apply in chapter 5. We have obtained the distribution of the buffer content and the corresponding numerical results.