Ksenia Kiseleva

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

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.

Two-Sided Truncations for a Class of Continuous-Time Markov Chains

We consider nonstationary 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.

Bounds For Markovian Queues With Possible Catastrophes.

We consider a general Markovian queueing model with possible catastrophes and obtain new and sharp bounds on the rate of convergence. Some special classes of such models are studied in details, namely, (a) the queueing system with S servers, batch arrivals and possible catastrophes and (b) the queueing model with “attracted” customers and possible catastrophes. A numerical example illustrates the calculations. Our approach can be used in modeling information flows related to high-performance computing. INTRODUCTION There is a large number of papers devoted to the research of Markovian queueing models with possible catastrophes, see for instance, [1], [3], [2], [10], [11], [17], [18], [19], [21], [24], [25] and the references therein. Such models are widely used in simulations for hight-performance computing. In particular, in some recent papers the authors deal with more or less special birth-death processes with additional transitions from and to origin [1], [2], [3], [10], [11], [21], [24], [25]. In the present paper we consider a more general class of Markovian queueing models with possible catastrophes and obtain key bounds on the rate of convergence, which allow us to compute the limiting characteristics of the corresponding processes. Namely, we suppose that the queue-length process is an inhomogeneous continuous-time Markov chain {X(t), t ≥ 0} on the state space E = {0, 1, 2 . . . }. All possible transition intensities are assumed to be non-random functions of time and may depend on the state of the process. From any state i the chain can jump to any another state j > 0 with transition intensity qij(t). Moreover, the transition functions from state i > 0 to state 0 (catastrophe intensities) are βi(t). Denote by pij (s, t) = P {X (t) = j |X (s) = i}, i, j ≥ 0, 0 ≤ s ≤ t the probability of transition X (t), and by pi (t) = P {X (t) = i} the corresponding state probability that X (t) is in state i at the moment t. Let p (t) = (p0 (t) , p1 (t) , . . . ) T be the vector of state probabilities at the moment t. Throughout the paper we suppose that for any i, j P (X (t+ h) = j|X (t) = i) = =  qij (t)h+ αij (t, h) , if j 6= i, βi (t)h+ αi0 (t, h) = qi0(t) + αi0(t, h), if j = 0, i > 1, 1− ∑ j 6=i qij(t)h+ αi (t, h) , if j = i, (1) where sup i |αi(t, h)| = o(h). (2) Let Q(t) be the corresponding intensity matrix. We suppose that all intensity functions are non-negative and locally integrable on [0,∞). Put aij (t) = qji (t) for j 6= i and aii (t) = − ∑

Two-Sided Truncations For The Mt/Mt/S Queueing Model.

The paper deals with the problem of existence and construction of limiting characteristics for timeinhomogeneous birth and death processes which is important for queueing applications. For this purpose we calculate the limiting characteristics for the process via the construction of two-sided uniform in time truncation bounds. We consider the Mt|Mt|S queueing model and obtain uniform approximation bounds of two-sided truncations. A numerical example is considered. Our approach to truncations of the state space can be used in modeling information flows related to high-performance computing. INTRODUCTION Explicit expressions for the probability characteristics of stochastic birth-death queueing models can be found in a few special cases. One of main problems for obtaining the limiting behavior of the process is studying of the rate of convergence as time t → ∞ to the steady state of a process. The problem of existence and construction of limiting characteristics for time-inhomogeneous birth and death processes is important for queueing applications, see [2], [4], [5], [9]. Calculation of the limiting characteristics for the process via “north-west” truncations was firstly mentioned in [7] and was considered in details in [9]. Uniform in time north-west truncation bounds have been obtained in [10], [11] for birth-death processes and general Markov chains respectively. Two-sided uniform in time truncation bounds were firstly studied in our previous paper [6]. This paper is the continuation of [6]. Namely, we apply this approach for a specific class of queueing models. 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 X = X(t) and pi(t) = Pr {X(t) = i} the state probabilities. Also we assume that P (X (t+ h) = j|X (t) = i) = = qij (t)h+ αij (t, h) if j 6= i, 1− ∑ k 6=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) Proceedings 31st European Conference on Modelling and Simulation ©ECMS Zita Zoltay Paprika, Péter Horák, Kata Váradi, Péter Tamás Zwierczyk, Ágnes Vidovics-Dancs, János Péter Rádics (Editors) ISBN: 978-0-9932440-4-9/ ISBN: 978-0-9932440-5-6 (CD) 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). Moreover, A (t) = Q> (t), where Q(t) the intensity (or infinitesimal) matrix for X(t). We assume that all birth and death intensity functions λi(t) and μi(t) are locally integrable on [0,∞). We suppose that λn(t) ≤ Λn ≤ L <∞, μn(t) ≤ ∆n ≤ L <∞, (3) for almost all t ≥ 0. 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. 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;∞). We 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). The Cauchy problem for differential equation (1) has unique solutions for arbitrary initial condition (see, for instance, [1]), and moreover p(s) ∈ Ω implies p(t) ∈ Ω for t ≥ s ≥ 0. We 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, see the corresponding definitions, bounds, references and other details in [3], [4], [8], [9], [10]. 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), where 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 for any k. TWO-SIDED TRUNCATIONS OF INHOMOGENEOUS BIRTH-DEATH PROCESSES By introducing pi(t) = 1 − ∑ j 6=i pj(t), (for arbitrary fixed i and p(t) ∈ Ω, t ≥ 0) we have the following system from (4) dz(t) dt = B(t)z(t) + f(t), (5) where z (t) is p (t) without coordinate pi, namely, z (t) = (p0, p1, . . . , pi−1, pi+1, . . . ). Hence we obtain f (t) = (0, 0, . . . , μi, λi, 0, . . . ), and the corresponding B (t). Let D∗ be a matrix D ∗ =  0 i − 2 i − 1 i + 1 i + 2 i + 3 0 −1 · · · 0 0 0 0 0 · · · · · · · · · · · · · · · · · · · · · · · · · · · i − 2 −1 · · · −1 0 0 0 0 · · · i − 1 −1 · · · −1 −1 0 0 0 · · · i + 1 0 · · · 0 0 1 1 1 · · · i + 2 0 · · · 0 0 0 1 1 · · · i + 3 0 · · · 0 0 0 0 1 · · · · · · · · · · · · · · · · · · · · · · · · · · ·  , then D∗BD∗−1 =  −μ1 − λ0 μ1 0 0 · · · λ1 −μ2 − λ1 μ2 0 · · · 0 λ2 −μ3 − λ2 μ3 · · · 0 0 λ3 −μ4 − λ3 · · · 0 0 0 λ4 · · · · · ·  Let now {dk} be a sequence of positive numbers, and D∗∗ = diag (d0, d1, . . . , di−1, di+1, di+2, . . . ). Put D = D∗∗D∗, D =  −d0 0 0 0 0 · · · −d1 −d1 0 0 0 · · · . . . −di−1 −di−1 · · · −di−1 0 · · · 0 0 · · · 0 di+1 di+1 · · · 0 0 · · · 0 0 di+2 · · · · · ·  . Let l1D be the space of sequences: l1D = {z = (p0, p1, ..., pi−1, pi+1, ...) > : ‖z‖1D ≡ ‖Dz‖ < ∞}. We introduce the auxiliary space of sequences l1E as l1E = {z = (p0, p1, ..., pi−1, pi+1, ...)> : ‖z‖1E ≡ ∑ k 6=i k|pk| <∞}. Consider the expressions: αk (t) =  λk (t) + μk+1 (t) − dk+1 dk λk+1 (t) − dk−1 dk μk (t) , k < i − 1 λi−1 (t) + μi (t) − di+1 di−1 λi (t) − di−2 di−1 μi−1 (t) , k = i − 1 λi (t) + μi+1 (t) − di+2 di+1 λi+1 (t) − di−1 di+1 μi (t) , k = i λk (t) + μk+1 (t) − dk+2 dk+1 λk+1 (t) − dk dk+1 μk (t) , k > i (6) and α (t) = inf k≥0 αk (t) . (7) Considering (5) as a differential equation in the space l1D, we have its solution: z(t) = V (t, 0)z(0) + ∫ t 0 V (t, τ)f(τ) dτ, (8) where V (t, z) is the Cauchy operator of (5), see [8]. We obtain ‖f(t)‖1D = di−1μi(t) + di+1λi(t) ≤ di−1∆i + di+1Λi for almost all t ≥ 0. On the other hand, putting βk (t) =  λk (t) + μk+1 (t) + dk+1 dk λk+1 (t) + dk−1 dk μk (t) , k < i − 1 λi−1 (t) + μi (t) + di+1 di−1 λi (t) + di−2 di−1 μi−1 (t) , k = i − 1 λi (t) + μi+1 (t) + di+2 di+1 λi+1 (t) + di−1 di+1 μi (t) , k = i λk (t) + μk+1 (t) + dk+2 dk+1 λk+1 (t) + dk dk+1 μk (t) , k > i. (9)

Ergodicity bounds for birth-death processes with particularities

We introduce an inhomogeneous birth-death process with birth rates λk(t), death rates µk(t), and possible transitions to/from zero with rates βk(t), rk(t) respectively, and obtain ergodicity bounds for this process.

On Truncations For A Class Of Finite Markovian Queuing Models.

We consider a class of finite Markovian queueing models and obtain uniform approximation bounds of truncations. INTRODUCTION It is well known that explicit expressions for the probability characteristics of stochastic models can be found only in a few special cases, moreover, if we deal with an inhomogeneous Markovian model, then we must approximately calculate the limiting probability characteristics of the process. The problem of calculation of the limiting characteristics for inhomogeneous birth-death process via truncations was firstly mentioned in (Zeifman 1991) and was considered in details in (Zeifman et al. 2006). In (Zeifman et al. 2014b) we have proved uniform (in time) error bounds of truncation this class of Markov chains. First uniform bounds of truncations for the class of Markovian time-inhomogeneous queueing models with batch arrivals and group services (SZK models) introduced and studied in our recent papers (Satin et al. 2013, Zeifman et al. 2014a), were obtained in (Zeifman et al. 2014c). In this note we deal with approximations of finite SZK model via the same models with smaller state space and obtain the correspondent bounds of error of truncation bounds. Consider a time-inhomogeneous continuous-time Markovian queueing model on the state space E = {0, 1, . . . , r} with possible batch arrivals and group services. Let X(t), t ≥ 0 be the queue-length process for the queue, pij(s, t) = P {X(t) = j |X(s) = i}, i, j ≥ 0, 0 ≤ s ≤ t, be transition probabilities for X = X(t), and pi(t) = P {X(t) = i} be its state probabilities. Throughout the paper we assume that P (X (t+ h) = j|X (t) = i) = = { qij (t)h+ αij (t, h) , if j 6= i, 1− ∑ k 6=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). We also assume qi,i+k (t) = λk(t), qi,i−k (t) = μk(t) for any k > 0. In other words, we suppose that the arrival rates λk(t) and the service rates μk(t) do not depend on the queue length. In addition, we assume that λk+1(t) ≤ λk(t) and μk+1(t) ≤ μk(t) for any k and almost all t ≥ 0. Hence, X(t) is a so-called SZK model, which was studied in (Satin et al. 2013, Zeifman et al. 2014a, 2014c). We suppose that all intensity functions are locally integrable on [0,∞), and λk(t) ≤ λk, μk(t) ≤ μk, (2) for any k and almost all t ≥ 0, and put