Matrix Exponential Form Research Articles

Finding the Exact Distribution of (Peak) Age of Information for Queues of PH/PH/1/1 and M/PH/1/2 Type

Bufferless and single-buffer queueing systems have recently been shown to be effective in coping with escalated Age of Information (AoI) figures arising in single-source status update systems with large buffers and FCFS scheduling. In this paper, for the single-source scenario, we propose a numerical algorithm for obtaining the exact distributions of both the AoI and the peak AoI (PAoI) in (i) the bufferless $PH/PH/1/1/P(p)$ queue with probabilistic preemption with preemption probability $p$, $0 \leq p \leq 1,$ and (ii) the single buffer $M/PH/1/2/R(r)$ queue with probabilistic replacement of the packet in the queue by the new arrival with replacement probability $r$, $0 \leq r \leq 1$. The proposed exact models are based on the well-established theory of Markov Fluid Queues (MFQ) and the numerical algorithms are matrix-analytical and they rely on numerically stable and efficient vector-matrix operations. Moreover, the obtained exact distributions are in matrix exponential form, making it amenable to calculate the tail probabilities and the associated moments straightforwardly. Firstly, we validate the accuracy of the proposed method with simulations, and for sume sub-cases, with existing closed-form results. We then comparatively study the AoI performance of the queueing systems of interest under varying traffic parameters.

A matrix-exponential decomposition based time-domain method for calculating the defect states of scalar waves in two-dimensional periodic structures

A time-domain method for calculating the defect states of scalar waves in two-dimensional (2D) periodic structures is proposed. In the time-stepping process of the proposed method, the column vector containing the spatially sampled field values is updated by multiplying it with an iteration matrix, which is written in a matrix-exponential form. The matrix-exponential is first computed by using the Suzuki's decomposition based technique of the fourth order, in which the Floquet–Bloch boundary conditions are incorporated. The obtained iteration matrix is then squared to enlarge the time-step that can be used in the time-stepping process (namely, the squaring technique), and the small nonzero elements in the iteration matrix is finally pruned to improve the sparse structure of the matrix (namely, the pruning technique). The numerical examples of the super-cell calculations for 2D defect-containing phononic crystal structures show that, the fourth order decomposition based technique for the matrix-exponential computation is much more efficient than the frequently used precise integration technique (PIT) if the PIT is of an order greater than 2. Although it is not unconditionally stable, the proposed time-domain method is particularly efficient for the super-cell calculations of the defect states in a 2D periodic structure containing a defect with a wave speed much higher than those of the background materials. For this kind of defect-containing structures, the time-stepping process can run stably for a sufficiently large number of the time-steps with a time-step much larger than the Courant–Friedrichs–Lewy (CFL) upper limit, and consequently the overall efficiency of the proposed time-domain method can be significantly higher than that of the conventional finite-difference time-domain (FDTD) method. Some physical interpretations on the properties of the band structures and the defect states of the calculated periodic structures are also presented.

Throughput Analysis of Hybrid-ARQ—A Matrix Exponential Distribution Approach

We propose a novel performance analysis framework for lossless- and truncated-hybrid automatic repeat request (HARQ) that enables neat, general, closed-form throughput expressions in a matrix exponential (ME) distribution form. This approach is applicable to all HARQ schemes for which the probability density function of the effective channel can be characterized by a rational Laplace transform, or equivalently, an ME-distribution. This includes, for example, repetition redundancy HARQ in ME distributed channels. Throughput expressions are also given for the $K$ -truncated-HARQ $N$ -fold diversity, ARQ $N$ -fold diversity, and lossless-HARQ 2-fold diversity cases in the ME distributed channel. Schemes with effective channels of non-rational Laplace transforms, such as IR-HARQ, are explored using truncated continued fractions. A novel integration trick is developed for the integration of ME distributions with singular matrices and yields the simple throughput expression of lossless-HARQ. We also give general analytical expressions for the optimal throughput and optimal rate point that benefit from the compact ME-distribution form proposed.

Maximum queue lengths during a fixed time interval in the [formula omitted] retrial queue

We are concerned with the problem of characterizing the distribution of the maximum number Z(t0) of customers during a fixed time interval [0,t0] in the M/M/c retrial queue, which is shown to have a matrix exponential form. We present a simple condition on the service and retrial rates for the matrix exponential solution to be explicit or algorithmically tractable. Our methodology is based on splitting methods and the use of eigenvalues and eigenvectors. A particularly appealing feature of our solution is that it allows us to obtain global error control. Specifically, we derive an approximating solution p(x;t0)≡p(x;t0;ε) verifying |P(Z(t0)⩽x|X(0)=(i,j))-p(x;t0)|<ε uniformly in x⩾i+j, for any ε>0 and initial numbers i of busy servers and j of customers in orbit.

MAXIMUM POPULATION SIZES IN HOST–PARASITOID MODELS

We are concerned with the problem of characterizing the distribution of the maximum number of individuals alive during a fixed time interval in host–parasitoid models, which is shown to have a matrix exponential form. We present simple conditions on the rates of change of population sizes for the matrix exponential solution to be explicit or algorithmically tractable. A particularly appealing feature of our solution based on splitting methods is that it allows us to obtain global error control.

Complex and hypercomplex discrete Fourier transforms based on matrix exponential form of Euler’s formula

We show that the discrete complex, and numerous hypercomplex, Fourier transforms defined and used so far by a number of researchers can be unified into a single framework based on a matrix exponential version of Euler’s formula ejθ=cosθ+jsinθ, and a matrix root of -1 isomorphic to the imaginary root j. The transforms thus defined can be computed using standard matrix multiplications and additions with no hypercomplex code, the complex or hypercomplex algebra being represented by the form of the matrix root of -1, so that the matrix multiplications are equivalent to multiplications in the appropriate algebra. We present examples from the complex, quaternion and biquaternion algebras, and from Clifford algebras Cℓ1,1 and Cℓ2,0. The significance of this result is both in the theoretical unification, which permits comparisons between transforms in different hypercomplex algebras, and also in the scope it affords for insight into the structure of the various transforms, since the formulation is such a simple generalization of the classic complex case. It also shows that hypercomplex discrete Fourier transforms may be computed using standard matrix arithmetic packages without the need for a hypercomplex library, which is of importance in providing a reference implementation for verifying faster implementations based on hypercomplex code.

Sojourn time distributions in the queue defined by a general QBD process

We consider a general QBD process as defining a FIFO queue and obtain the stationary distribution of the sojourn time of a customer in that queue as a matrix exponential distribution, which is identical to a phase-type distribution under a certain condition. Since QBD processes include many queueing models where the arrival and service process are dependent, these results form a substantial generalization of analogous results reported in the literature for queues such as the PH/PH/c queue. We also discuss asymptotic properties of the sojourn time distribution through its matrix exponential form.

Hitting probabilities in a Markov additive process with linear movements and upward jumps: Applications to risk and queueing processes

Motivated by a risk process with positive and negative premium rates, we consider a real-valued Markov additive process with finitely many background states. This additive process linearly increases or decreases while the background state is unchanged, and may have upward jumps at the transition instants of the background state. It is known that the hitting probabilities of this additive process at lower levels have a matrix exponential form. We here study the hitting probabilities at upper levels, which do not have a matrix exponential form in general. These probabilities give the ruin probabilities in the terminology of the risk process. Our major interests are in their analytic expressions and their asymptotic behavior when the hitting level goes to infinity under light tail conditions on the jump sizes. To derive those results, we use a certain duality on the hitting probabilities, which may have an independent interest because it does not need any Markovian assumption.

Solving seismic wave propagation in elastic media using the matrix exponential approach

Three numerical algorithms are proposed to solve the time-dependent elastodynamic equations in elastic solids. All algorithms are based on approximating the solution of the equations, which can be written as a matrix exponential. By approximating the matrix exponential with a product formula, an unconditionally stable algorithm is derived that conserves the total elastic energy density. By expanding the matrix exponential in Chebyshev polynomials for a specific time instance, a so-called “one-step” algorithm is constructed that is very accurate with respect to the time integration. By formulating the conventional velocity-stress finite-difference time-domain (VS-FDTD) algorithm in matrix exponential form, the staggered-in-time nature can be removed by a small modification, and higher-order in time algorithms can be easily derived. For two different seismic events the accuracy of the algorithms is studied and compared with the result obtained by using the conventional VS-FDTD algorithm.

Erlangian Approximations for Finite-Horizon Ruin Probabilities

AbstractFor the Cramér-Lundberg risk model with phase-type claims, it is shown that the probability of ruin before an independent phase-type time H coincides with the ruin probability in a certain Markovian fluid model and therefore has an matrix-exponential form. When H is exponential, this yields in particular a probabilistic interpretation of a recent result of Avram &amp; Usabel. When H is Erlang, the matrix algebra takes a simple recursive form, and fixing the mean of H at T and letting the number of stages go to infinity yields a quick approximation procedure for the probability of ruin before time T. Numerical examples are given, including a combination with extrapolation.

A matrix exponential form for hitting probabilities and its application to a Markov-modulated fluid queue with downward jumps

We consider a fluid queue with downward jumps, where the fluid flow rate and the downward jumps are controlled by a background Markov chain with a finite state space. We show that the stationary distribution of a buffer content has a matrix exponential form, and identify the exponent matrix. We derive these results using time-reversed arguments and the background state distribution at the hitting time concerning the corresponding fluid flow with upward jumps. This distribution was recently studied for a fluid queue with upward jumps under a stability condition. We give an alternative proof for this result using the rate conservation law. This proof not only simplifies the proof, but also explains an underlying Markov structure and enables us to study more complex cases such that the fluid flow has jumps subject to a nondecreasing Lévy process, a Brownian component, and countably many background states.

A matrix exponential form for hitting probabilities and its application to a Markov-modulated fluid queue with downward jumps

We consider a fluid queue with downward jumps, where the fluid flow rate and the downward jumps are controlled by a background Markov chain with a finite state space. We show that the stationary distribution of a buffer content has a matrix exponential form, and identify the exponent matrix. We derive these results using time-reversed arguments and the background state distribution at the hitting time concerning the corresponding fluid flow with upward jumps. This distribution was recently studied for a fluid queue with upward jumps under a stability condition. We give an alternative proof for this result using the rate conservation law. This proof not only simplifies the proof, but also explains an underlying Markov structure and enables us to study more complex cases such that the fluid flow has jumps subject to a nondecreasing Lévy process, a Brownian component, and countably many background states.

Markov processes whose steady state distribution is matrix-exponential with an application to the GI/PH/1 queue

This paper is concerned with a bivariate Markov process {Xt, Nt ; t ≧ 0} with a special structure. The process Xt may either increase linearly or have jump (downward) discontinuities. The process Xt takes values in [0,∞) and Nt takes a finite number of values. With these and additional assumptions, we show that the steady state joint probability distribution of {Xt, Nt ; t ≧ 0} has a matrix-exponential form. A rate matrix T (which is crucial in determining the joint distribution) is the solution of a non-linear matrix integral equation. The work in this paper is a continuous analog of matrix-geometric methods, which have gained widespread use of late. Using this theory, we present a new and considerably simplified characterization of the waiting time and queue length distributions in a GI/PH/1 queue. Finally, we show that the Markov process can be used to study an inventory system subject to seasonal fluctuations in supply and demand.

Markov processes whose steady state distribution is matrix-exponential with an application to the GI/PH/1 queue

This paper is concerned with a bivariate Markov process {Xt, Nt; t ≧ 0} with a special structure. The process Xt may either increase linearly or have jump (downward) discontinuities. The process Xt takes values in [0,∞) and Nt takes a finite number of values. With these and additional assumptions, we show that the steady state joint probability distribution of {Xt, Nt; t ≧ 0} has a matrix-exponential form. A rate matrix T (which is crucial in determining the joint distribution) is the solution of a non-linear matrix integral equation. The work in this paper is a continuous analog of matrix-geometric methods, which have gained widespread use of late. Using this theory, we present a new and considerably simplified characterization of the waiting time and queue length distributions in a GI/PH/1 queue. Finally, we show that the Markov process can be used to study an inventory system subject to seasonal fluctuations in supply and demand.