Exponential Approximations for Tail Probabilities in Queues, I: Waiting Times

A batch Markov arrival process (BMAP) X* = (N, J) is a 2-dimensional Markov process with two components, one is the counting process N and the other one is the phase process J. It is proved that the phase process is a time-homogeneous Markov chain with a finite state-space, or for short, Markov chain. In this paper, a new and inverse problem is proposed firstly: given a Markov chain J, can we deploy a process N such that the 2-dimensional process X* = (N, J) is a BMAP? The process X* = (N, J) is said to be an adjoining BMAP for the Markov chain J. For a given Markov chain the adjoining processes exist and they are not unique. Two kinds of adjoining BMAPs have been constructed. One is the BMAPs with fixed constant batches, the other one is the BMAPs with independent and identically distributed (i.i.d) random batches. The method we used in this paper is not the usual matrix-analytic method of studying BMAP, it is a path-analytic method. We constructed directly sample paths of adjoining BMAPs. The expressions of characteristic (D k , k = 0, 1, 2 · · ·) and transition probabilities of the adjoining BMAP are obtained by the density matrix Q of the given Markov chain J. Moreover, we obtained two frontal Theorems. We present these expressions in the first time.

This paper considers the asymptotic tail distribution of the number of cells queued in a statistical multiplexer fed with homogeneous generalized binary Markov sources. As the asymptotic decay rate is easy to obtain, we focus our effort on bounding the asymptotic constant, which is dependent on the initial phase combination of the sources and is hard to compute even for a moderate number of sources. We derive upper and lower bounds for the asymptotic constant, taking the initial phase combination into account. Numerical experiments show the accuracy of these bounds. They also show that, while the asymptotic decay rates are the same, the variation of initial phase combination of the sources may significantly affect the asymptotic constants.

Tandem queues are good mathematical models for description of information transmission in various communication systems and networks. These queues play also an important role for the validation of different decomposition algorithms designed for investigating more general queueing networks. So, their investigation is interesting for theory and applications. In this paper, we consider tandem queue suitable for modeling the systems and networks where information flows are correlated and bursty what is typical for many modern telecommunication networks. Possible correlation of customers inter-arrival times and batch arrivals are taken into account via of consideration of the Batch Markovian Arrival Process (BMAP) as input stream to the system. The system consists of two stations. The service time at the station 1 is assumed to be generally distributed. There is no buffer at this station, and customers who meet the busy server repeat attempts to enter the system in random time intervals. The service process at the station 2 is assumed to be described by the continuous time Markov chain with a finite state space. This assumption holds good, e.g., if the station 2 has a finite buffer, consists of a finite number of identical or heterogeneous servers where the service time distribution is assumed to be of PH (PHase) type. Markov chain embedded at service completion epochs at the station 1 and the process of system states at arbitrary time are under study. Ergodicity condition and algorithms for computing the steady state probabilities are presented.

As various video services become popular, video streaming will dominate the mobile data traffic. The H.264 standard has been widely used for video compression. As the successor to H.264, H.265 can compress video streaming better, hence it is gradually gaining market share. However, in the short term H.264 will not be completely replaced, and will co-exist with H.265. Using H.264 and H.265 standards, three types of frames are generated, and among different types of frames exist dependencies. Since the radio resources are limited, using dependencies and quantities of frames in buffers, an appropriate time division transmission policy can be applied to transmit different types of frames sequentially, in order to avoid the occurrence of video carton or decoding failure. Polling models with batch Markovian arrival process (BMAP) and across-queue state-dependent service discipline are considered to be effective means in the design and optimization of appropriate time division transmission policies. However, the BMAP and across-queue state-dependent service discipline of the polling models lead to the large state space and several coupled state transition processes, which complicate the performance analysis. There have been very few researches in this regard. In this paper, a polling model of this type is analyzed. By constructing a supplementary embedded Markov chain and applying the matrix-analytic method based on the semi-regenerative process, the expressions of important performance measures including the joint queue length distribution, the customer blocking probability and the customer mean waiting time are obtained. The analysis will provide inspiration for analyzing the polling models with BMAP and across-queue state-dependent service discipline, to guide the design and optimization of time division transmission policies for transmitting the video compressed by H.264 and H.265.

Markovian open many-body quantum systems display complicated relaxation dynamics. The spectral gap of the Liouvillian characterizes the asymptotic decay rate towards the stationary state, but it has recently been pointed out that the spectral gap does not necessarily determine the overall relaxation time. Our understanding on the relaxation process before the asymptotically long-time regime is still limited. We here present a collective relaxation dynamics of autocorrelation functions in the stationary state. As a key quantity in the analysis, we introduce the instantaneous decay rate, which characterizes the transient relaxation and converges to the conventional asymptotic decay rate in the long-time limit. Our theory predicts that a bulk-dissipated system generically shows an accelerated decay before the asymptotic regime due to the scrambling of quantum information associated with the operator spreading.

For a sparse non-singular matrix A, generally A −1 is a dense matrix. However, for a class of matrices, A −1 can be a matrix with off-diagonal decay properties, i.e., |A ij −1| decays fast to 0 with respect to the increase of a properly defined distance between i and j. Here we consider the off-diagonal decay properties of discretized Green’s functions for Schrodinger type operators. We provide decay estimates for discretized Green’s functions obtained from the finite difference discretization, and from a variant of the pseudo-spectral discretization. The asymptotic decay rate in our estimate is independent of the domain size and of the discretization parameter. We verify the decay estimate with numerical results for one-dimensional Schrodinger type operators.

We study the late-time dynamics of two particles confined in one spatial dimension and subject to two-body losses. The dynamics is exactly described by a non-Hermitian Hamiltonian that can be analytically studied both in the continuum and on a lattice. The asymptotic decay rate and the universal power-law form of the decay of the number of particles are exactly computed in the whole parameter space of the problem. When in the initial state the two particles are far apart, the average number of particles in the setup decays with time $t$ as ${t}^{\ensuremath{-}1/2}$; a different power law, ${t}^{\ensuremath{-}3/2}$, is found when the two particles overlap in the initial state. These results are valid both in the continuum and on a lattice, but in the latter case a logarithmic correction appears.

Let Z be a two-dimensional Brownian motion confined to the non-negative quadrant by oblique reflection at the boundary. Such processes arise in applied probability as diffusion approximations for two-station queueing networks. The parameters of Z are a drift vector, a covariance matrix, and a "direction of reflection" for each of the quadrant's two boundary rays. Necessary and sufficient conditions are known for Z to be a positive recurrent semimartingale, and they are the only restrictions imposed on the process data in our study. Under those assumptions, a large deviations principle (LDP) is conjectured for the stationary distribution of Z, and we recapitulate the cases for which it has been rigorously justified. For sufficiently regular sets B, the LDP says that the stationary probability of xB decays exponentially as x??, and the asymptotic decay rate is the minimum value achieved by a certain function I(?) over the set B. Avram, Dai and Hasenbein (Queueing Syst.: Theory Appl. 37, 259---289, 2001) provided a complete and explicit solution for the large deviations rate function I(?). In this paper we re-express their solution in a simplified form, showing along the way that the computation of I(?) reduces to a relatively simple problem of least-cost travel between a point and a line.

We present a direct numerical solution of a particular inertial oscillation, the so-called 'spin-over’ mode, in spherical geometry. This mode is particularly relevant to the fluid flow within a precessing oblate spheroid. We demonstrate that the oscillatory Ekman layer breaks down at ±30° latitude, and that this breakdown spawns internal shear layers. We show that the structure of these shear layers is different for a full sphere and a spherical shell, as noted in the preceding paper (Kerswell 1995). Despite the existence of these shear layers, however, the numerical decay rates agree to within 1 % with the asymptotic decay rates, which neglect any possible shear layers. Finally, we consider the nonlinear mean flow profiles driven by this mode, and demonstrate that our numerical results agree reasonably well with experimental results.

This chapter presents the Enhanced-real time Polling System (E-rtPS) with RT- Batch Markovian Arrival Process and proactive resource allocation framework to resolve the real-time network traffic issues and solve the interference problem optimistically. It proposes proactive resource allocation framework offers less computational overhead and yet again well-nigh performance in terms of resource allocation to maximize the transmission rate of each user. The chapter examines real-time results shows that the proposed framework outperforms the existing Institute of Electrical and Electronics Engineers (IEEE) 802.16 services in terms of throughput, session setup delay, and packet reception ratio. In IEEE 802.16, there are different Medium Access Control scheduling services, such as Unsolicited Grant Service, rtPS, and non-real-time Polling Service to provide better quality of services. The chapter presents an adaptive framework for dynamic speed management in smart cities. Since smart phone usage has risen to half a billion phones, mobile devices and connections account for 89 percent of the usage of mobile data traffic.

Queueing analysis and optimal control of [formula omitted] and [formula omitted] systems

We consider the GI/G/1 queue where customers are served in random order and the service time distribution has a finite exponential moment. We derive the large deviations result for the waiting time distribution by showing that the asymptotic decay rate of the waiting time distribution is the same as that of the busy period distribution.

To improve the service quality and save the system cost of the cloud computing center, this paper studies the joint optimization problem of energy consumption and performance of cloud computing centers with a batch Markovian arrival process. The system has multiple parallel processors and the processing time of each processor follows phase type distribution. The system has finite buffer. We construct the system as a BMAP/PH/N/M queueing system and analyze the performance of the system based on queuing theory. An optimization model is established by combing the energy consumption and system performance measures. The optimal number of servers is analyzed. Some managerial insights are given by numerical analysis.

A retrial queueing system with the batch Markovian arrival process and semi-Markovian service is investigated. We suppose that the intensity of retrials linearly depends on the number of repeated calls. The distribution of the number of calls in the system is the subject of research. Asymptotically quasi-Toeplitz 2-dimensional Markov chains are introduced into consideration and applied for solving the problem.

In this paper, we study a multi-server queueing system with retrials and an infinite orbit. The arrival of primary customers is described by a batch Markovian arrival process (BMAP), and the service times have a phase-type (PH) distribution. Previously, in the literature, such a system was mainly considered under the strict assumption that the intervals between the repeated attempts from the orbit have an exponential distribution. Only a few publications dealt with retrial queueing systems with non-exponential inter-retrial times. These publications assumed either the rate of retrials is constant regardless of the number of customers in the orbit or this rate is constant when the number of orbital customers exceeds a certain threshold. Such assumptions essentially simplify the mathematical analysis of the system, but do not reflect the nature of the majority of real-life retrial processes. The main feature of the model under study is that we considered the classical retrial strategy under which the retrial rate is proportional to the number of orbital customers. However, in this case, the assumption of the non-exponential distribution of inter-retrial times leads to insurmountable computational difficulties. To overcome these difficulties, we supposed that inter-retrial times have a phase-type distribution if the number of customers in the orbit is less than or equal to some non-negative integer (threshold) and have an exponential distribution in the contrary case. By appropriately choosing the threshold, one can obtain a sufficiently accurate approximation of the system with a PH distribution of the inter-retrial times. Thus, the model under study takes into account the realistic nature of the retrial process and, at the same time, does not resort to restrictions such as a constant retrial rate or to rough truncation methods often applied to the analysis of retrial queueing systems with an infinite orbit. We describe the behavior of the system by a multi-dimensional Markov chain, derive the stability condition, and calculate the steady-state distribution and the main performance indicators of the system. We made sure numerically that there was a reasonable value of the threshold under which our model can be served as a good approximation of the BMAP/PH/N queueing system with the PH distribution of inter-retrial times. We also numerically compared the system under consideration with the corresponding queueing system having exponentially distributed inter-retrial times and saw that the latter is a poor approximation of the system with the PH distribution of inter-retrial times. We present a number of illustrative numerical examples to analyze the behavior of the system performance indicators depending on the system parameters, the variance of inter-retrial times, and the correlation in the input flow.