Galina Shilova

Some universal limits for nonhomogeneous birth and death processes

In this paper we consider nonhomogeneous birth and death processes (BDP) with periodic rates. Two important parameters are studied, which are helpful to describe a nonhomogeneous BDP X = X(t), t≥ 0: the limiting mean value (namely, the mean length of the queue at a given time t) and the double mean (i.e. the mean length of the queue for the whole duration of the BDP). We find conditions of existence of the means and determine bounds for their values, involving also the truncated BDP XN. Finally we present some examples where these bounds are used in order to approximate the double mean.

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.

On Truncations For SZK Model.

We consider a class of inhomogeneous Markovian queueing models with batch arrivals and group services. Bounds on the truncation errors in weak ergodic case are obtained. Two concrete queueing models are studied as examples.

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

On Sharp Bounds Of The Rate Of Convergence For Some Queueing Models.

We consider a class of homogeneous SZK models with finite state space and consider sharp bounds on the rate of convergence to stationary distribution.

On Mt /Mt /S Type Queue With Group Services.

We consider Mt/Mt/S-type queueing model with group services. Bounds on the rate of convergence for the queue-length process are obtained. Ordinary Mt/Mt/S queue and Mt/Mt/S type queueing model with group services are studied as examples.

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 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.