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

Performance analysis of time-dependent queueing systems: survey and classification

Many queueing systems are subject to time-dependent changes in system parameters, such as the arrival rate or number of servers. Examples include time-dependent call volumes and agents at inbound call centers, time-varying air traffic at airports, time-dependent truck arrival rates at seaports, and cyclic message volumes in computer systems.There are several approaches for the performance analysis of queueing systems with deterministic parameter changes over time. In this survey, we develop a classification scheme that groups these approaches according to their underlying key ideas into (i) numerical and analytical solutions,(ii)approaches based on models with piecewise constant parameters, and (iii) approaches based on mod-ified system characteristics. Additionally, we identify links between the different approaches and provide a survey of applications that are categorized into service, road and air traffic, and IT systems

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

Non-Equilibrium Statistical Physics of Currents in Queuing Networks

We consider a stable open queuing network as a steady non-equilibrium system of interacting particles. The network is completely specified by its underlying graphical structure, type of interaction at each node, and the Markovian transition rates between nodes. For such systems, we ask the question ``What is the most likely way for large currents to accumulate over time in a network ?'', where time is large compared to the system correlation time scale. We identify two interesting regimes. In the first regime, in which the accumulation of currents over time exceeds the expected value by a small to moderate amount (moderate large deviation), we find that the large-deviation distribution of currents is universal (independent of the interaction details), and there is no long-time and averaged over time accumulation of particles (condensation) at any nodes. In the second regime, in which the accumulation of currents over time exceeds the expected value by a large amount (severe large deviation), we find that the large-deviation current distribution is sensitive to interaction details, and there is a long-time accumulation of particles (condensation) at some nodes. The transition between the two regimes can be described as a dynamical second order phase transition. We illustrate these ideas using the simple, yet non-trivial, example of a single node with feedback.Comment: 26 pages, 5 figure

Markovian arrivals in stochastic modelling: a survey and some new results

This paper aims to provide a comprehensive review on Markovian arrival processes (MAPs), which constitute a rich class of point processes used extensively in stochastic modelling. Our starting point is the versatile process introduced by Neuts (1979) which, under some simplified notation, was coined as the batch Markovian arrival process (BMAP). On the one hand, a general point process can be approximated by appropriate MAPs and, on the other hand, the MAPs provide a versatile, yet tractable option for modelling a bursty flow by preserving the Markovian formalism. While a number of well-known arrival processes are subsumed under a BMAP as special cases, the literature also shows generalizations to model arrival streams with marks, nonhomogeneous settings or even spatial arrivals. We survey on the main aspects of the BMAP, discuss on some of its variants and generalizations, and give a few new results in the context of a recent state-dependent extension.Peer Reviewe

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

Usefulness of Cardiac Biomarker Score for Risk Stratification in Stable Patients Undergoing Elective Cardiac Evaluation Across Glycemic Status

Several clinically available cardiac biomarkers have established their prognostic value in patients with acute coronary syndromes. However, their relative prognostic significance in stable subjects has not been prospectively validated, either individually or in combination. The aim of this study was to evaluate the extent to which B-type natriuretic peptide, myeloperoxidase, and high-sensitivity C-reactive protein alone or together could be prognostic biomarkers in 3,635 consecutive stable patients without acute coronary syndrome who underwent elective diagnostic coronary angiography. After adjusting for traditional risk factors and renal function, each of the markers monitored was a significant predictor of incident major adverse cardiovascular events (death, nonfatal myocardial infarction, and stroke) over 3 years. A cardiac biomarker score based on the sum total of “positive” biomarkers provided independent prediction of future risk for incident major adverse cardiovascular events at 3 years (hazard ratio [HR] 7.61, 95% confidence interval [CI] 4.98 to 11.65, p \u3c0.001), even after adjusted for traditional risk factors (HR 6.11, 95% CI 3.98 to 9.38, p \u3c0.001). A positive cardiac biomarker score remained a strong and independent predictor of 3-year risk for major adverse cardiovascular events among those with normal glycemic control (HR 4.24, 95% CI 1.96 to 9.18, p \u3c0.001), those with prediabetes (HR 7.62, 95% CI 3.87 to 15.01, p \u3c0.001), and those with diabetes (HR 5.61, 95% CI 2.55 to 12.33, p \u3c0.001), as well as within subjects without significant angiographic evidence of coronary artery disease (HR 10.82, 95% CI 3.82 to 30.6, p \u3c0.001). In conclusion, an integrated assessment of cardiac biomarkers may provide independent prognostic value for long-term adverse clinical events in stable cardiac patients

Stochastic Systems ACHIEVING RAPID RECOVERY IN AN OVERLOAD CONTROL FOR LARGE-SCALE SERVICE SYSTEMS

We consider an automatic overload control for two large service systems modeled as multi-server queues, such as call centers. We assume that the two systems are designed to operate independently, but want to help each other respond to unexpected overloads. The proposed overload control automatically activates sharing (sending some customers from one system to the other) once a ratio of the queue lengths in the two systems crosses an activation threshold (with ratio and activation threshold parameters for each direction). To prevent harmful sharing, sharing is allowed in only one direction at any time. In this paper, we are primarily concerned with ensuring that the system recovers rapidly after the overload is over, either (i) because the two systems return to normal loading or (ii) because the direction of the overload suddenly shifts in the opposite direction. To achieve rapid recovery, we introduce lower thresholds for the queue ratios, below which one-way sharing is released. As a basis for studyin