Anna Korotysheva

Perturbation bounds and truncations for a class of Markovian queues

We consider time-inhomogeneous Markovian queueing models with batch arrivals and group services. We study the mathematical expectation of the respective queue-length process and obtain the bounds on the rate of convergence and error of truncation of the process. Specific queueing models are shown as examples.

Perturbation Bounds for M t /M t /N Queue with Catastrophes

This articles focuses on M t /M t /N queue with catastrophes and obtains stability bounds for the main characteristics of the respective queue-length process.

Truncation Bounds for Approximations of Inhomogeneous Continuous-Time Markov Chains

Weakly ergodic continuous-time countable Markov chains are studied. We obtain uniform in time bounds for approximations via truncations by analogous smaller chains under some natural assumptions.

On a Queueing Model with Group Services

An analogue of M t /M t /S/S Erlang loss system for a queue with group services is introduced and considered. Weak ergodicity of the model is studied. We obtain the bounds on the rate of convergence to the limiting characteristics and consider two concrete queueing models with finding of their main limiting characteristics.

Modeling and analyzing licensed shared access operation for 5G network as an inhomogeneous queue with catastrophes

The framework of licensed shared access (LSA) to spectrum seems to become one of the trends of 5G wireless networks. The framework assumes the simultaneous access to spectrum by at least two parties - the primarily owner (incumbent), which has the highest priority, and several secondary users (licensees), which have lower priorities. The critical up-to-date problem is the development of the corresponding radio admission control and load balancing algorithms that form an essential part of the LSA agreement between the parties. The algorithm of binary use of spectrum gives an absolute priority to the incumbent, e.g. the airport using spectrum for aeronautical telemetry purposes. In the paper, capturing the inhomogeneous in time nature of rates of requests for access to spectrum and average times of spectrum use, we propose a queuing model of binary access to spectrum as seen from the licensees point of view. The queue is described by an inhomogeneous birth and death process with catastrophes and repairs. The main aim of the paper is to find the bounds on the rates of convergence to the limiting characteristics of the queue - average number of users, blocking probability, and probability of service interruption due to the incumbents need for spectrum. Not only the acceptable upper thresholds on the limiting characteristics are important for consideration but also the corresponding bounds showing the moment in time when the system becomes stable and the LSA licensee could really access to spectrum.

Ergodicity and truncation bounds for inhomogeneous birth and death processes with additional transitions from and to origin

ABSTRACT In this paper, we present the extension of the analysis of time-dependent limiting characteristics the class of continuous-time birth and death processes defined on non-negative integers with special transitions from and to the origin. From the origin transitions can occur to any state. But being in any other state, besides ordinary transitions to neighboring states, a transition to the origin can occur. All possible transition intensities are assumed to be non-random functions of time and may depend on the state of the process. We improve previously known ergodicity and truncation bounds for this class of processes that were known only for the case when transitions from the origin decay exponentially (other intensities must have unique uniform upper bound). We show how the bounds can be obtained if the decay rate is slower than exponential. Numerical results are given in the queueing theory context.

Two-Sided Truncations Of Inhomogeneous Birth-Death Processes

We consider a class of inhomogeneous birth-death queueing models and obtain uniform approximation bounds of two-sided truncations. Some examples are considered. Our approach to truncations of the state space can be used in modeling information flows related to high-performance computing. INTRODUCTION It is well known that explicit expressions for the probability characteristics of stochastic birth-death queueing models can be found only in a few special cases. Therefore, the study of the rate of convergence as time t → ∞ to the steady state of a process is one of two main problems for obtaining the limiting behavior of the process. If the model is Markovian and stationary in time, then, as a rule, the stationary limiting characteristics provide sufficient or almost sufficient information about the model. On the other hand, if one deals with inhomogeneous Markovian model then, in addition, the limiting probability characteristics of the process must be approximately calculated. The problem of existence and construction of limiting characteristics for time-inhomogeneous birth and death processes is important for queueing and some other applications, see for instance, [1], [3], [5], [8], [15], [16]. General approach and related bounds for the rate of convergence was considered in [13]. Calculation of the limiting characteristics for the process via truncations was firstly mentioned in [14] and was considered in details in [15], uniform in time bounds have been obtained in [17]. As a rule, the authors dealt with the so-called northwest truncations (see also [9]), namely they studied the truncated processes with the same first states 0, 1, . . . , N In the present paper we consider a more general approach and deal with truncated processes on state space N1, N1 + 1, . . . , N2 for some natural N1, N2 > N1. Let X = X(t), t ≥ 0 be a birth and death process (BDP) with birth and death rates λn(t), μn(t) respectively. Let pij(s, t) = Pr {X(t) = j |X(s) = i} for i, j ≥ 0, 0 ≤ s ≤ t be the transition probability functions of the process X = X(t) and pi(t) = Pr {X(t) = i} be the state probabilities. Throughout the paper we assume that P (X (t+ h) = j|X (t) = i) = = qij (t)h+ αij (t, h) if j ̸= i, 1− ∑ k ̸=i qik (t)h+ αi (t, h) if j = i, (1) where all αi(t, h) are o(h) uniformly in i, i. e. supi |αi(t, h)| = o(h). Here all qi,i+1 (t) = λi(t), i ≥ 0, qi,i−1 (t) = μi(t) i ≥ 1, and all other qij(t) ≡ 0. The probabilistic dynamics of the process is represented by the forward Kolmogorov system of differential equations:  dp0 dt = −λ0(t)p0 + μ1(t)p1, dpk dt = λk−1(t)pk−1 − (λk(t) + μk(t)) pk+ +μk+1(t)pk+1, k ≥ 1. (2) By p(t) = (p0(t), p1(t), . . . ) , t ≥ 0, we denote the column vector of state probabilities and by A(t) = (aij(t)) , t ≥ 0 the matrix related to (2). One can see that A (t) = Q⊤ (t), where Q(t) is the intensity (or infinitesimal) matrix for X(t). We assume that all birth and death intensity functions λi(t) and μi(t) are linear combinations of a finite number of functions which are locally integrable on [0,∞). Moreover, we suppose that λn(t) ≤ Λn ≤ L < ∞, μn(t) ≤ ∆n ≤ L < ∞, (3) Proceedings 30th European Conference on Modelling and Simulation ©ECMS Thorsten Claus, Frank Herrmann, Michael Manitz, Oliver Rose (Editors) ISBN: 978-0-9932440-2-5 / ISBN: 978-0-9932440-3-2 (CD) for almost all t ≥ 0. Throughout the paper by ∥ · ∥ we denote the l1-norm, i. e. ∥x∥ = ∑ |xi|, and ∥B∥ = supj ∑ i |bij | for B = (bij)i,j=0. Let Ω be a set all stochastic vectors, i. e. l1 vectors with nonnegative coordinates and unit norm. Then we have ∥A(t)∥ ≤ 2 sup(λk(t) + μk(t)) ≤ 4L, for almost all t ≥ 0. Hence the operator function A(t) from l1 into itself is bounded for almost all t ≥ 0 and locally integrable on [0;∞). Therefore we can consider the system (2) as a differential equation dp dt = A (t)p, p = p(t), t ≥ 0, (4) in the space l1 with bounded operator function A(t). It is well known (see, for instance, [2]) that the Cauchy problem for differential equation (1) has unique solutions for arbitrary initial condition, and moreover p(s) ∈ Ω implies p(t) ∈ Ω for t ≥ s ≥ 0. Therefore, we can apply the general approach to employ the logarithmic norm of a matrix for the study of the problem of stability of Kolmogorov system of differential equations associated with nonhomogeneous Markov chains. The method is based on the following two components: the logarithmic norm of a linear operator and a special similarity transformation of the matrix of intensities of the Markov chain considered, see the corresponding definitions, bounds, references and other details in [4], [5], [13], [15], [17]. Definition. A Markov chain X(t) is called weakly ergodic, if ∥p∗(t) − p∗∗(t)∥ → 0 as t → ∞ for any initial conditions p∗(0),p∗∗(0). Here p∗(t) and p∗∗(t) are the corresponding solutions of (4). Put Ek(t) = E {X(t) |X(0) = k } ( then the corresponding initial condition of system (4) is the k − th unit vector ek). Definition. Let X(t) be a Markov chain. Then φ(t) is called the limiting mean of X(t) if lim t→∞ (φ(t)− Ek(t)) = 0

Some results for inhomogeneous birth-and-death process with application to staffing problem in telecommunication service systems

This paper proposes some analytical results that may facilitate long-term staffing problem in high-level telecommunication service systems (such as information call centers) in which rates of processes, that govern their behaviour, depend on time. We assume that except for arrivals of requests and their service there happen periodic system breakdowns (possibly with very long inter-breakdown periods). The staffing objective is “immediate service of a given percentage of incoming requests”. A natural model for such a time-varying processes is an innhomogeneous birth-death process for which we propose some general theoretical results concerning its ergodicity conditions and limiting behaviour. As an example we show that if the service system is modelled by multiserver queue Mt/Mt/S with state-dependent periodic arrivals, services and breakdown rates, then using obtained results one can calculate the quantities needed for the solution of optimization problem. Accuracy of approximation is briefly discussed.

Bounds on the rate of convergence for one class of inhomogeneous Markovian queueing models with possible batch arrivals and services

Abstract In this paper we present a method for the computation of convergence bounds for four classes of multiserver queueing systems, described by inhomogeneous Markov chains. Specifically, we consider an inhomogeneous M/M/S queueing system with possible state-dependent arrival and service intensities, and additionally possible batch arrivals and batch service. A unified approach based on a logarithmic norm of linear operators for obtaining sharp upper and lower bounds on the rate of convergence and corresponding sharp perturbation bounds is described. As a side effect, we show, by virtue of numerical examples, that the approach based on a logarithmic norm can also be used to approximate limiting characteristics (the idle probability and the mean number of customers in the system) of the systems considered with a given approximation error.

On Sharp Bounds on the Rate of Convergence for Finite Continuous-Time Markovian Queueing Models

Finite inhomogeneous continuous-time Markov chains are studied. For a wide class of such processes an approach is proposed for obtaining sharp bounds on the rate of convergence to the limiting characteristics. Queueing examples are considered.