Periodic Solution to The Time-Inhomogeneous Multi-Server Poisson Queue

We derive the periodic family of asymptotic distributions and the periodic moments for number in the queue for the multi-server queue with Poisson arrivals and exponential service for time-varying periodic arrival and departure rates, and time-varying periodic number of servers. The method is a straight-forward application of generating functions

Two Classes of Time-Inhomogeneous Markov Chains: Analysis of The Periodic Case

We consider the M/G/1 and GI/M/1 types of Markov chains for which their one step transitions depend on the times of the transitions. These types of Markov chains are encountered in several stochastic models, including queueing systems, dams, inventory systems, insurance risk models, etc. We show that for the cases when the time parameters are periodic the systems can be analyzed using some extensions of known results in the matrix-analytic methods literature. We have limited our examples to those relating to queueing systems to allow us a focus. An example application of the model to a real life problem is presented

The Matrices R and G of Matrix Analytic Methods and The Time-Inhomogeneous Periodic Quasi-Birth-and-Death Process

We solve for the asymptotic periodic distribution of the continuous time quasi-birth-and-death process with time-varying periodic rates in terms of R^ and G^ matrix functions which are analogues of the R and G matrices of matrix analytic methods. We evaluate these QBDs numerically by solving for R^ numerically

Finite Memory Recursive Solutions in Stochastic Models: Equilibrium and Transient Analysis

G/M/1 and M/G/1-type Markov processes provide natural models for widely differing stochastic phenomena. Efficient recursive solutions for the equilibrium and transient analysis of these processes are therefore of considerable interest. In this direction, a new class of recursive solutions are proposed for the analysis of M/G/l and G/M/l type processes. In this report, the notion of when a process is LEDI-complete, which means it has complete Level Entrance Direction Information, is introduced for G/M/1-type Markov processes. This notion leads to a new class of recursive solutions, called finite-memory recursive solutions, for the equilibrium probabilities of a class of G/M/ 1-type Markov processes. A finite-memory recursive solution of order k has the form πn+k = π W1 +π n+1W2 + ••• +πn+k-1 Wk7 where πn is the vector of limiting probabilities of the states on level n of the process and Wi, 1 \u3c i \u3c k, are square matrices. It is also shown that the concept of LEDI- completeness leads to a finite- memory recursive solution for the transient behavior of this class of G/M/-1- type processes. Such a recursive solution has the form πn+k(s) = ^n(s)W1(s) +π n+i(s)W2(s) + • • • + πn+k-i(s)Wk(s). where π(s) is the Laplace transform of πn(t), the vector of state occupancy probabilities at time t for the states on level n of the process. The relationship between these finite-memory recursive solutions and matrix geometric solutions is also explored. The results are extended to the case where the transition rates are level dependent. It is also briefly explained how a finite memory recursion for the equilibrium and transient probabilities of M/G/l type Markov processes can be obtained

Using Hybrid Simulation/Analytical Queueing Networks to Capacitate USAF Air Mobility Command Passenger Terminals

The objective of this study is to model operations at an airport passenger terminal to determine the optimal service capacities at each station given estimated passenger flow patterns and service rates. The central formulation is an open Jackson queueing network that can be applied to any USAF Air Mobility Command (AMC) terminal regardless of passenger type mix and flow data. A complete methodology for analyzing passenger flows and queue performance of a single flight is produced and then embedded in a framework to analyze the same for multiple departing flights. Queueing network analysis (QNA) is used because no special software license or methodological training is required, results are obtained in a spreadsheet model with computational response times that are instantaneous, and data requirements are substantially reduced compared with discrete-event simulation (DES). However, because of the assumptions of QNA, additional research contributions were required. First, arrivals of passengers are time-dependent, not steady-state. Theoretical results for time-dependent queue networks in the literature are limited, so a method for using DES to adjust for arrival time-dependency in QNA is developed. Second, beyond quality of service in the network, a key performance measure is the percentage of passengers who do not clear the system by a fixed time. To populate the QNA mean value system sojourn time, DES is used to develop a generic sojourn time probability distribution. All DES computations have been pre-calculated off-line in this thesis and complete a hybrid DES/QNA analytical model. The model is exercised and validated through analysis of the facility at Hickam AFB, which is currently undergoing redesign. For larger flights, adding a server at the high-utilization queues, namely the USDA inspection and security screening stations, halve system congestion and dramatically increase throughput

Perfect and Nearly Perfect Sampling of Work-conserving Queues

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

Non-stationary service curves : model and estimation method with application to cellular sleep scheduling

In today’s computer networks, short-lived flows are predominant. Consequently, transient start-up effects such as the connection establishment in cellular networks have a significant impact on the performance. Although various solutions are derived in the fields of queuing theory, available bandwidths, and network calculus, the focus is, e.g., about the mean wake-up times, estimates of the available bandwidth, which consist either out of a single value or a stationary function and steady-state solutions for backlog and delay. Contrary, the analysis during transient phases presents fundamental challenges that have only been partially solved and is therefore understood to a much lesser extent. To better comprehend systems with transient characteristics and to explain their behavior, this thesis contributes a concept of non-stationary service curves that belong to the framework of stochastic network calculus. Thereby, we derive models of sleep scheduling including time-variant performance bounds for backlog and delay. We investigate the impact of arrival rates and different duration of wake-up times, where the metrics of interest are the transient overshoot and relaxation time. We compare a time-variant and a time-invariant description of the service with an exact solution. To avoid probabilistic and maybe unpredictable effects from random services, we first choose a deterministic description of the service and present results that illustrate that only the time-variant service curve can follow the progression of the exact solution. In contrast, the time-invariant service curve remains in the worst-case value. Since in real cellular networks, it is well known that the service and sleep scheduling procedure is random, we extend the theory to the stochastic case and derive a model with a non-stationary service curve based on regenerative processes. Further, the estimation of cellular network’s capacity/ available bandwidth from measurements is an important topic that attracts research, and several works exist that obtain an estimate from measurements. Assuming a system without any knowledge about its internals, we investigate existing measurement methods such as the prevalent rate scanning and the burst response method. We find fundamental limitations to estimate the service accurately in a time-variant way, which can be explained by the non-convexity of transient services and their super-additive network processes. In order to overcome these limitations, we derive a novel two-phase probing technique. In the first step, the shape of a minimal probe is identified, which we then use to obtain an accurate estimate of the unknown service. To demonstrate the minimal probing method’s applicability, we perform a comprehensive measurement campaign in cellular networks with sleep scheduling (2G, 3G, and 4G). Here, we observe significant transient backlogs and delay overshoots that persist for long relaxation times by sending constant-bit-rate traffic, which matches the findings from our theoretical model. Contrary, the minimal probing method shows another strength: sending the minimal probe eliminates the transient overshoots and relaxation times

Model for estimating and comparing the risk of occurrence accidents and incidents on the level crossings

U radu je određen matematički model sa ciljem proračuna maksimalnog rizika i utvrđivanja pouzdanosti putno-pružnih prelaza. U definisanom inženjerskom okviru, od determinističke do stohastičke granice apstraktnog broja nesreća i nezgoda, nalazi se  proporcija za estimaciju i komparaciju rizika od nastanka nezgoda i nesreća na putno-pružnim prelazima zasnovana na realnim događajima. Moguće je utvrđivanje nivoa bezbednosti za svaki putni prelaz pojedinačno, što predstavlja originalni doprinos u istraživanjima bezbednosti na mestima ukrštaja železničkog i drumskog saobraćaja.This paper is determined a mathematical model to calculate the maximum risk and determine the reliability of level crossings. In the strongly defined engineering framework, from the deterministic to the stochastic limit of the number of theoretical accidents and incidents, there is a proportion for estimating and comparing the risk of accidents and incidents at level crossings based on real occurrences. It is possible to determine the level of safety for each railroad crossing individually, which is an original contribution to safety research at the intersections of railway and road traffi