In this paper we present algorithms for the perfect sampling of singleserver time-varying queues with periodic Poisson arrivals under the first come first served (FCFS) discipline. The service durations have periodically time-dependent exponential (M t /M t /1) or homogeneous general (M t /G/1) distributions. Assuming a cycle length of 1, we construct discrete dominating processes at the integer instants n ∈ {0, ±1, . . .}. Perfect sampling of the M t /M t /1 queue is obtained using dominated CFTP (Kendall and Møller 2000) when the system is relatively lightly loaded or with the regenerative method (Sigman 2012) in the general case. For the M t /G/1 queue, perfect sampling is achieved with dominated CFTP.

We present sampling-based methods to treat work-conserving queueing systems. A variety of models are studied. Besides the First Come First Served (FCFS) queues, many efforts are putted on the accumulating priority queue (APQ), where a customer accumulates priority linearly while waiting. APQs have Poisson arrivals, multi-class customers with corresponding service durations, and single or multiple servers.Perfect sampling is an approach to draw a sample directly from the steady-state distribution of a Markov chain without explicitly solving for it. Statistical inference can be conducted without initialization bias. If an error can be tolerated within some limit, i.e. the total variation distance between the simulated draw and the stationary distribution can be bounded by a specified number, then we get a so called "nearly" perfect sampling.Coupling from the past (CFTP) is one approach to perfect sampling, but it usually requires a bounded state space. One strategy for perfect sampling on unbounded state spaces relies on construction of a reversible dominating process. If only the dominating property is guaranteed, then regenerative method (RM) becomes an alternative choice. In the case where neither the reversibility nor dominance hold, a nearly perfect sampling method will be proposed. It is a variant of dominated CFTP that we call the CFTP Block Absorption (CFTP-BA) method.Time-varying queues with periodic Poisson arrivals are being considered in this thesis. It has been shown that a particular limiting distribution can be obtained for each point in time in the periodic cycle. Because there are no analytical solutions in closed forms, we explore perfect (or nearly perfect) sampling of these systems.